YALE UNIVERSITY LIBRARY J935 THE Mathematical Principles o F geography. CONTAINING, I. An Account of the various Properties and Affedlions of the Earth and, Sea j with a Defcription of the feveral Parts thereof. And a Table of the Latitude and Longitude of Places. II. The Ufe of the Artificial or " Terreftrial Globe, in folving Problems. III. The Principles of Spherical and Sphe roidical Sailing j with the Solution of the feveral Cafes in Numbers, by the common Tables, according to the Spheroidical Figure of the Earth. - ¦ I Tit *,-.._ ^ _ ¦ Mount where Science guides-, Go meafure earth, weigh air, andfiate the tides. Pope's Efiay. LONDON: Printed for J. Noorse, in the Strand ; Bopkfeller in Ordinary to his MAJESTY. M DCC LXX. C i 1 THE PREFACE. T-1AVING treated of the world, in its full extent under the title of Aftronomy, we come now to that Jingle part of it called the Earth, which is our deftin'd habitation. \This terreftrial globe on which we are feated, tho' it feems large to us that live upon- it, yet when compared to the magnitude of the hea vens, is only like a point in it, being in a manner loft in that v aft fabric. But fmall as it is, it is of the greatejl confequence to us, for here we live and move and have our being, here pur whole bufinefs and'tranf- atlions are performed, here we muft abide, and can not remove into another planet, nor have any commu nication therewith, nor with the inhabitants thereof, if any fuch there be ; nor can we tell what they are doing, no more than they know what we are about. All we can do is to remove from one place to another upon this little ball, and pafs from one region to ano ther, and by this means we get acquaintance with our fellow-creatures on different parts of the globe, with whom we can converfe, carry on a trade, or" tranfacl any bufinefs, by which we make ajhift to get a living. Andjince we are confined to this globe, and are obliged to have various tranfatlions with' others, both at home and abroad', therefore Geography (which is the knowledge and defcription of the earth) becomes a ne- Cgjfary branch of knowledge to us. For without it, we Jhould be ignorant of the Jituation of the Jeveral countries, with which we traffick, how they lye from one another, and the way thither. A 2 Geogra* w PREFACE. Geography teaches us the Jituation of all countries and- kingdoms, and the limits thereof, how they are bounded by fea and land, and by one another ; into what provinces and diftricls they are divided; the Ji tuation, latitude and longitude, of all cities and towns therein ; what woods, for efts, mountains, lakes, rivers', mines, &c. each contains ; what commodities it af fords ; what matters of wonder,^ curiofity, or anti quity -, what buildings, caftles, towers are therein ; what ports and havens ; what rocks, fands, fhoals, and fuch places of danger there are ; and the manners and employment of the people. This art teaches us ta draw maps, or reprefentations of dike feveral countries . of the earth, which gives a •*<¦•• »¦*" a of their Jitua tion, and magnitudes, by expq/ing "them all to our fight,' And thefe geographical maps are of very great ufe to travellers, as well' as to failors ; for by help of thefe they find the neareft way from one place to another. Without^ the knowledge of Geography, Hiftory is very imperfect ; for we can have but a very fender no tion of any tranfatlion, , when we are ignorant of the place it was tranfacled in. Without it we cannot tell how the moft memorable enterprizes of the world have- been carried on and executed. Without it we are igno rant of the rife, growth, flour ifhing, and fall of the fe veral great monarchies in the world, which fix a begin ning to all Hiftory. Without it we cannot judge of the government, commodities, riches, and number of people in other nations ; nor judge of thefirength of jc'uf^eqemies ; nor underftand-the limits between one king dom and another -, nor diftinguifa between the names of 'places and the names of people. But by the help of Geography we can eafily know all thefe things, and much more ; without expofmg our bodies to long tedious travels and voyages. How prejudicial the ignorance of Geography has been to Princes in foreign expeditions againft -their enemies ; hiftory gives many inftances of their ill fuccefs. And theft PREFACE. in ihefe mifcarriages have been principally owing to their ignorance of the country they were going to invade» When any perfen goes upon fuch an expedition, he ought. to have a map of the country, with all the feveral p&Jfages in itfrorn one place to another, the mountains, woods, rivers, marjhes; and what rocks or fands lie near it. From which may be found where the fafefl place for landing is, where one may avoid this rock or that f and, where fuch a river may be paft, and which is the moft commodious place for giving his enemy bat tle, and what advantages and difadvantages there is in hisfituation.And as thisfcience is fo beneficial and ufeful, it is no lefs pleafant and delightful. It at once, pleafes the eye and inftrucls the mind. It gives fuch a vail variety tf objetls to contemplate, that we are ftruck with ad miration, with thefe beautiful fanes of nature, which this earth of ours affords. \ Princes have not thought it below them to make it their ftndy, but to the great danger of their perfons, have travelled into foreign countries to make difcoveries. And at great expence harae fent philofophers and mathematicians to remote places to make proper obfervations ; as upon the Jit ua^- ' tion of places, the phenomena of the celejlial bodies, and fuch like; by which this art has" been very much improved. Tet this art atpfefenl is far from being at perfetlion* For there ar every few places whofe latitude and longitude are truly determined. And there areftill many countries* which remain undijcovered, and mufl wait for the induf try of future ages to find out. There are likewife many Continents of great extent, of which we know little more than the fea Coafts.s And even in places .which we know, and daily travel over, there are many towns whofe Jituation is very uncertain. This appears from the dif agreement of the feveral geographical maps made, cf the fame country. Tet this art is daily improving ; A 2 ft? IV PREFACE. for long voyages have made many new difcoveries; and fucceeding times willftill add more, and will increafe our knowledge, andfhew us the errors of former ages. The ancients thought the torrid zone uninhabitable, as well as the polar regions ; and that there was no fuch thing as antipodes, or people ftanding contrary ways on oppqfite fides of the earth ; for indeed they thought the earth flat. But later experience has taught us better. And fo will the experience of future ages de~ tecl our ignorance in many things. I took notice that the earth compared to the heavens is no more than a point ; the leaft ftar we can fee, far exceeds it in magnitude. And yet this earth, J mall as it is, is that which with fire and fword is divided a- mong fo many nations ; who are always contending, and never can agree, about their Jh ares in it. -And yet this is not fo much owing to the fmallnefs of it, as to the infatiable dejires of men, after too large a fhare of it ; and their- unquenchable thirft and luft after riches. For the earth produces enough for all its inhabitants, if people had but as much humanity as to fuffer their fellow creatures to enjoy a reafonable Jhare along with them. ' Inftead of that, men are in perpetual war and ftrife who floall get the moft, HU old Time overtakes them, when all on afudden they drop into the bofom of our common mother the earth ; and then the greateft monarch is no richer than the pooreft beggar. Navigation is an art as ufeful and beneficial as Geography, and like twofifters, they go hand in hand, and one is ever- affifting to the other. By this art men can pafs from one country to another, or from one port to another with great eafe and difpatch. By this all trade is carried on, and goods exported and imported at tleafure, to or from diftant places as occajion requires. It is a moft ufeful art upon many accounts, for by this we can not only increafe our knowledge by travelling to foreign parts, but our riches alfo by merchandizing. " And therefore it is neceffary to be known for the fake of PREFACE. of trade. But neceffity may be underftood two ways, either for abfolute need, without which a thing can not be; or merely for a conveniency, without which a thing cannot well be. Now it is certain that many places arefo poor, as not to be able to maintain a po pulous nation, without the help of foreign trade, at this time when the world is grown fo full of people. In this cafe there is an abfolute neceffity for Navigation, to carry on the bufinefs of merchandizing, without which the inhabitants could not live. Trading, which is neceffary tofome, is certainly very beneficial to all Nations ; as is evident for many rea- fons. For, i . By this means one nation may traff.k with another, by exporting fuch goods as they have in too great abundance, and importing others that they want. For fome commodities are fo plentiful in fame countries, that they are a mere drug ; whilft others have little or nothing of the fort. Therefore tranfporting them to thefe deficient places, is a great advantage. 2. Goods cannot be conveyed from one place to another, fo eajily, nor fo cheap, nor fo foon, nor in fuch quanti ties, by any methods, as by fhipping. 3. By means of Navigation, arts and fciences are promoted and im proved, and conveyed to diftant places. And for this reafon, many famous Philofophers have pajfed the feas and travelled far, to converfe with men of learning, for the improvement of arts. And hence the principal difcoveries in Geography are much owing to Naviga tion. 4. Likewife when a nation is overburthened with inhabitants, colonies may by this means be tranfported to diftant countries not fo populous. And thus many idlers are frequently carried to Virginia and other places. 5» Navigation is of great ufe for the defence of a na tion againft foreign enemies, where a fight at fea does not a quarter of the damage as at land. To which we may add 'the plea fur e of converfing with foreigners, end holding fociety with men of learning, as well as A 4 of vi PREFACE, of mutual commerce with them; for ficiety is natural to mankind. Navigation has alfo been highly encouraged by many princes and States ; and great honours have alfo been paid to it. And for the encouragement of foreign trade, focieties have been formed in feveral nations, and en dowed with ample privileges, by which many perfons have become immenfely rich. It is time now to give fome account of what is con* tdined in this Trail. The firft feclion treats of the figure of the earth, bothasafphereandfpheroid; of fever dl properties of the fea •, of the origin of fprings and ri^ vers; of finding the diftances of places; of making maps,; ajhort account of the feveral kingdoms. of the world; and a table of the latitude and longitude of places. Here the longitude is reckoned from the iftand of Ferro, ac cording to the antient geographers, which is about the fame longitude as the weftermofi part of the continent of Africa, and therefore is a very proper and fit place for the beginning of longitude. And was foolijhly altered by later Geographers, by which they have coh- fufed all reckonings. The fecond feclion contains the ufe of the terreftrial globe in folving problems of the fphere. The praclice of this is very eafy and delightful. The third feclion is Navigation, andjhews the prin ciples both of fpherical and fpheroidical failing. And as fpheroidal failing has madefo much noife in the world, I have given the folution of all the common cafes in numbers by my method. This Treatife is moftly mathematical, the hidorical account of kingdoms, &c. not being my dejign. Of all the books( of Navigation that have come into my hands, I have met with few or none, that give a true notion of departure; the moft of them confounding it with meridional diftance. And fome of them tell us that it is quite ufelefs in navigation ; and yet at the fame time, they cannot fmd the longitude without it, or PREFACE. vii or fome thing equivalent to it, but not fojimple as it ; and therefore ufe it for that purpofe. For this reafon I have made fome remarks in the Schol. of Prop. I. Seel. III. t As to fpheroidal failing, I have given my thoughts of it in my book of Navigation, and have Jhewn that in a, day's run, the difference between that and the fphere is quite infenjible. And ajhip muft reckon her way every day, and Jo day after day, thro* the whole voyage. And when an obfervation is had, this f weeps away all irregularities from every caufe, and fets all right, as far as there is a poffibility to do it ; and furely an ob fervation is the only thing to be depended on in a reckon ing, and ought never to be neglecled. And therefore as no apparent advantage is got by this way of failing, I fet it qfide, and kept to thejimple and eafy way by the fphere. For who will think it worth their while to fpend a deal of fuperfluous time and labour, to ob tain a degree of accuracy, which can never be wanted ? No bgdy will, but fuch as are fond of novelties, and therefo) e they prefer fuch things, becaufe they are new, tho' they have no advantage above other methods, but a manifeft difadvantage, of embarraffing the cal culation, and making more work for the Jailor ; for which I believe, he will never thank them. But altho* this method is really of little ufe in Na vigation, yet becaufe fome people may think otherwife, and may fuppofe there will refult a greater difference between the, two methods, than there really is in prac tice ; I have therefore laid down the principles of that method, and folyed all the cafes thereby ; and if it ferve for nothing elfe, it will ferve for an amufement to fuch people as are delighted with mathematical en quiries. Thefe cafes are all refolved according to Mauper- tuis! 's figure of the earth, which is by far theflatteft earth that has ever been fuppofed. That of d' Juan and Ulloa are not half fo flat, and differ very little from viii P R E F A C E. from Sir J. Newton's earth. But after all the expe riments, that have been made to determine the figure of it,^ the refult is, that its figure is inconjS/lent with that of any fpherOid. And if the earth is not afpheroid, thefe Virtuofi that want to be exacl, will have neip, rules tofeek out, or elfe be forced to take up with im- perfecl ones. Laftly, if itjhould happen that the earth is nearly in the form that Juan has determined it ; then thoje that follow Maupertuis will be farther from the truth, than tbofe that fuppofe it a fphere; and we know nothing to the contrary-. For which reafon' we may well content ourfelves with the fpherical figure of the earth as heretofore, in the praclice of Navigation. W. Emerfon. con- • ¦I » THE CONTENTS. SECT. I. OF the figure of the earth and fea. Of finding the la titude and longitude of places. The phoenomena of the celeftial motions, to the inhabitants of different fpheres and zones. Of meafuring a degree of the earth. The meafure of the earth. To find the polar and equi noctial diameters. A table of the length of a pendulum in all latitudes. Of the ebbing and flowing of the fea ; finding its depth. Of finding the diftances of places. The origin of fprings and rivers. Of drawing maps. Of gaining or lofing a day in going round the earth. Pro perties of the earth and fea. Of the changes that have happened in the earth. Its divifion into kingdoms. A table of the latitude and longitude of places. SECT. II. The ufe of the terreftrial globe, where 20 problems of the fphere are folved by it. SECT. III. The theory of Navigation, relating to plane failing, parallel failing, middle latitude failing, Mercator's failing, and that by the log. tangents. Three uncommon cafes in 11 The C T B NTS. in Navigation folved. The elements of fpheroidal failing. Three propositions containing rules,; for the folufion of the cafes of* failing by the fpheroid. The aclual-folution ,of the 5 cafes of failing by the fpheroid, and 3 of pa-v rallel failing. Correcting a reckoning at fea. Of finding the meridian by two equal altitudes. To take an altitude without a horizon. ERRATA, Geography. page line read 4 17 length of days 5 9 Afcii. At N 8b frigid zone, thefe 23 1 whofe diftance 40 9b. pole, A and B 57 8 oblate fpheroid ' 79 9 eaft by Spain, 82 6 as the land of . 123 zb long night be ,- 128 '4 dif. lat. AR. v,;". 4b velocity Ak. 130 »3 ts + sr, &c. »3» 16 Jer than Az j ' 137 10 whence js r: 2.30 144 10 1 HI r= BE X 1 + iqss, 162 L *z I jig. 18. - GEOGRAPHY. [ I ] GEOGRAPHY. G DEFINITIONS. D E F. I. EOGRAPHT, is a fcience which teaches the defcription of the earth and its feveral parts j as it is a globe confifting of land and water. D E F. II. The axis of the earth is a line palling thro' the center of it, upon which it is fuppofed to turn round. D E F. III. The poles, are the two extremities of the axis,, whereat cuts the furface of the earths the one the north,- the other the fouth pole. D E F. IV. The equinoilial is a great circle of the earth, go degrees diftant from the two poles. This divides the globe into two hemifpheres, the northern and fouthern. D E F. V. Meridians or hour circles, are great circles of the earth palling thro' the two poles, and cutting the equinoctial at right angles. The firft meridian is that from which the Geographers have agreed to reckon from, which anciently went thro' the ifle of Ferro. B DEF. * D E F I N I'T'I O N S/ D E F. VI. The horizonis^hns. tpucl>ing_theu earth, at^ the place where we ftand ; which plane extended to the heavens, divides the upper from the lower hernif- phere •, this is called th$\ fenfible horizon. And a plane drawn thro'- the center of the earth parralkl to the former, and extended to the heavens, the great circle made thereby, is called the rational ho rizon. D E F; VII. Parallels of latitude, are- lqffer circles parallel to the equator,-, of which the principal, are the two tropics, and the two polar circles. The tropics are 234. deg. from the equinoctial, the northern tropic is called the Tropic of Cancer ; the fouthern, the Tropic of Capricorn, The polar, circles are 23^- deg. from the poles. The. nprthern is called the Artie circle, and the fouthern the Afitarilic D E F. VIII. Latitude of a place is the diftance from the equa tor to that place, reckoned upon the meridian of the place. If the place be in the nocth heraifphere, it is north latitude ; if in the fouth, it is fouth. latitude. DEF, \\. Longitude of a place, is the diftaace. from thefirft meridian to the meridian of the place, and is count ed on the equinoctial. Formerly longitude was counted from the firft meridian eaftward, quite round the globe. DEF. X. Rhombs, are the divifions of the horizon, into fe veral parts, which are 32 in number; thefe. are the feveral points of the compafs ; all of thefe have particular names, expreffed on a compafs card. The 4 cardinal points areeaft, weft, north and fouth. If DEFINITIONS. 3 If you fet your face to the north, as all geographers dp, then the eaft is on your right hand, the welt on your left ; the north before your face, and the fouth behind your back. Thefe points of the com pafs are equivalent to fo many azimuths. DEF. XI. A Rumb line, is a fpiral, drawn or fuppofed to be drawn upon the earth, which cuts all the me ridians at the fame angle, which is the proper an gle of that rumb. This being continued never re turns into itfelfj except it happen to be eaft or weft, and then it coincides with fome parallel circle. The circles here defined being extended, coincide with their refpective circles in the heavens, or the celeftial circles of the fame name, which have been defined in Sect. II. of the Aftronomy. .And thi ther I refer the reader for the definition of the reft of the circles, which more properly belong to Af tronomy. We muft next add fome things in regard to the inhabitants of the earth. DEF XII. A parallel fphere, is that pofition the earth has,. when the horizon coincides wiith the equator. Here thepoles'are in the zeoith and nadir; and the in habitants of this fphere -live juft at the poles. DEF. XIII. A right fphere, is that pofition of the earth, where the equinoctial paries thro' the zenith, and the poles are in the horizon ; and the inhabitants here live juft at the equator. D E F. XIV. An oblique fphere, is that pofition of the earth, where the equator and horizon make an oblique angle. Here one of the poles is elevated above the horizon, and the other deprefied below it. The ' v ' B 2. inha- +' DEFINITIONS. inhabitants here live any way between the poles and the equator. There is alfo a divifion of the earth according to the fituations of different inhabitants ; as, DEF. XV. Antaci, are thofe that live under the fame femi- meridian, and on different fides of the equinoctial, and equally diftant from it. Therefore their longi tudes are the fame, and the latitudes equal but con trary. Noon and midnight, and all the hours of day and night, are the fame to both. The length of the days to one is always equal to the length of the nights to the other. The feafons of the year are contrary, being furomer in one, when 'tis winter to the other. They have the fame feafons, the fame heat in fummer, the fame cold in winter, the fame length jofdays and nights ; but all at different times of the year. DEF. XVI. PerUci, are thofe people that live in the oppofite points of the fame parallel. And therefore when it is noon to one, it is midnight to the other. They have the fame feafons, the fame temper of the air, the fame fummer, the fame winter, the fame length of days and nights, all at the fame time. DEF. XVII. The Antipodes, are thofe inhabitants that live in places of the earth diametrically oppofite. Here both their latitudes and longitudes are contrary ; confequently when 'tis noon to one, 'tis midnight to the other ; when one has the longeft day the other has the longeft night, and when one has the fhorteft day the other has the fhorteft night. The feafons of the year are all contrary in both. They have the fame feafons, the fame length of days and nights, but at different times. When they ftand, their DEFINITIONS. 5 their feet are towards one another, and their heads oppofite. The inhabitants are alio confidered in regard to their fhadows ; as, DEF. XVIII. Amphifcii or Amphifcians, are thofe people that eaft their fhadows both north and fouth, at different times of the year. Thefe people live any way be tween the tropics. When the fun is in their zenith, they are called Afcii. At that time the Afcians eaft no fhadow at all. DEF. XIX. Heterofcii, are thofe inhabitants that eaft their fhadows all one way, either north or fouth. The Heterofcians live between the tropics and the polar circles. DEF. XX. Perifcii, are thofe people whofe fhadows turn round about them. The Perifcians live within the arctic or antarctic circle, where their fhadows in one day are directed to all points of the compafs. DEF. XXI. A Zone is a portion of the earth's furface contained between two parallel circles, or at leaft within one parallel. Thus the whole fpace contained between the two tropics, is called the Torrid zone. The fpace between the tropic and polar circle, either north or fouth, is the Temperate zone, thefe are two. The fpace contained within the polar circle is the frozen or frigid^*'thefe are alfo two. The ancients thought the torrid zone not ha bitable, by reafon of the great heat of the fun, but experience has fhewn the contrary. For the heat there is much diminifhed on account of fea breezes, long nights, frequent rains, -&c. In the middle of this zone, is the equinoctial circle ; and thofe that live there have perpetually their days and nights B 3 equal. 6 D E F I N ITIONS. equal. But here the twilight is very tfhort, being little more than an hour. As the fun , goes twice in the year over their heads in this zone, ,it caufes two fummers and two winters every year, which does not happen in the other zon$s. . The frozen zones are fo cold, that the ancients did not think them habitable ; and indeed thefe pla ces muft be exceffive cold in the winter, where they want the fun for almoft half a yean But then to make amends, he ftays with them almoft half a year in fummer. And this fo warms and nourifhes .the earth, 'that it is able to refift the cold. the moft part of winter., So that we find that even the cold- eft of thefe places are. inhabited, DEF. XXII. The Climates, are certain parts of the earth's fur- face, contained between parallel? of latitude, in fuch manner, that at every fucceeding parallel, reckon ing from the equinoctial, the longeft day fhall in creafe half an hour. Whence from the equator to the polar circle there will be 24 of thefe climates, t And later geographers have added 6 more, where the longtft day is fuppofed to encreafe by a month at each climate. So that now there are 30,. or in all 60, confide ring the whole globe of fhe earth. A Table GEOGRAPHY. A Table of Climates, -fhewing in what Latitude ' each ends. clim.j tat. ton. day. p ,9!w,. ,iat. Ion. clay t; ¦ 8 25 12th. '13 59 58 i8fh 2 16 25 13 H 61 18 . 19 , 3 23 50 i3v *5 62 25 i9t 14: 3Q ,„20 . JL4_.. 1 :.i6, M 22 20 5 36 28 t4t '7 64 6 20i ¦br 41 22 '15 : 18 64 49 21 •T 45 r«9 •»5f 19 65 21 :2lf 8 49 1 16 20 65 47 22 . 9 5» ;£« .'f6t '"' 2 I : 66 6 224 io 54 27 *7 421' 66 20 23 n 56 -37 '»H • 23 66 "28 Xtf 12 58 29 18 • -24 66 31 24 > Climates cori dntie d to the Pole. , ,clitn. -,lat. Ion. day clira, 28 Jat. ion. day *5 67 21 1 mon. 7» 3° 4 mon z6i "69 -48 2 > j > 29 S4 5 5 27. 73.^.37\ 3- .-' - .39 J 39 c 6 It is evident, this divifion into climates' is, very irreguiarj for near the polar circle, they are fo fmall, that they are hardly diftinguilhable. It had been' better to have divided them at the dif tance of every 5 degrees quite to the pole. How ever the climates as here fet out, ferve to fhew r what length the longeft day is of, by' having the Climate given ; and that is by taking half the num ber of climates, and adding that to 12. Or know ing ! the length of the longeft day, the climate' may be found, by fubtracting 12 from it, and doub ling the remainder. B4 An 8 GEOGRAPHY. An Explanation of Terms. Bay, a fmall part of the fea, encompaffed with the land, having only a narrow paffage into it. Bed of a river, is the hollow or channel in which the water runs. Borough, a fmall corporate town. Brook, Bourn, or Beck, a narrow current of wa ter, that runs continually. Canal, a deep ftream or current of water, inclo- fed on both fides by banks either natural or artifi cial. Cape, or head land, a high part of the land Handing by the fea fide. Cafcade, a fall of water in a fiver, which may be either natural or artificial. Cataracl, a high fall of water in a large river, that makes a great noife, fo as to be heard feveral miles. Champion country, a flat open country of great extent. Channel, a ftreight or narrow place of the fea, leading from one part of the fea to another. Chorography, the defcription or reprefentation of any country or kingdom. City, a large corporate town, with feveral privi leges. Cliff, a high fteep rock on the fea fide. Continent, a large continued tract of land, con taining feveral countries or kingdoms. Coaft, that part of the land which is next the fea. Country, a tract of land under a king or prince. This is oppofed to town or city. Creek, an arm of the fea running into the land. Current, a rapid motion of the fea. Depart, a quantity of ground uninhabited. Downs, hills of fand by the edge of the fea. Mbb, the fettling of the fea after the tide. Fens, GEOGRAPHY. 9 Fens, places full of bogs or ftanding waters. Floods the rifing of the fea in the tide. Ford, a fhallow place in a river to go or ride through. Foreft, a large extent of ground with trees in it, and fometimes wild beafts. Grove, or Thicket, a fmall place fet with trees, made for pleafure. Gulph, a large part of the fea running a great way into the land. Hamlet, a fmall village, or fome divifion of a large town. Harbour, , a place where fhips may lye at anchor fafely. Haven, an entrance of the fea within the land, at the mouth of fome river or creek, where fhips may lye. Hill, a high piece of ground. Hydrography, the defcription of water, as the fea, rivers, lakes, &c. Ifle, or Ifland, a part of the earth encompaffed round by the water. Ifthmus, a neck or narrow piece of land running into the fea. Lake, a great collection of ftanding water, in the land. Map, a geometrical defcription of any country upon paper, &c, Marfhes, fee Fens. Monfoons, periodical winds, which blow for a certain time one way, and as long the contrary way,. Mountain, a very high part of the land in any country. Ocean, that great collection of water that fur- rounds the whole earth. Parifh, fuch an extent of land as belongs to a xhurcht P'ark, 10 G E 0 G R A P H rY. Park, a parcel of -ground ertcTbfe'd' for- theiieep- ing of deer. » "' Peninfula, a. piece of land encompaffed by the fea, except a narrow place or entrance into it. .¦': Pond, Vfmalr collecticJh of ftahdihg water in the land. Port, a place where fhips lye, or where they load and unload. - Precipice,' a' yety'^iglC arid fteep [ place, v'as the brow of a hill. , , « . . ^-Promontory, a high land ftretchiftg into the fea. Reflux, fee ebb. . Y - •' Region, a great 'fpace of land containing many inhabitants, under fome king, &c. -Rill, a fmall "rivulet of riiHiuhg'' water. -v River, a "iarge current of 'water, able to carry fhips, &c. efpecially where it runs into the fea. Rivulet, a fmall river, '•>-¦ Road, - a 'place" hear the cbaft, where fhijps may lye at anchor. Sea, a part of "the ocean 'that lyes between one country and another. ¦Shelves, rocks arid' fahds lying under Water. Shoar, dry land next the fea. ' Sphere artificial, ,' an artificial1 globe, with the countries of the earth drawn upon it. Springs a fmall current of frefh water riling out of the earth. Staple town, is' a port town where merchants traffick. Strand, that part of the coaft which the : fea covers and uncovers, as it flows and ebbs. Streight, a narrow, paffage between two lands, in going into a larger fea. Thicket, fee grove. Topography, a reprefentation of a fmall particu lar place of the earth. * Town^ GEOGRAPHY. u Town, a collection of houfes near together. Valley, vale or dale, a low or hpllow place of the earth between feveraj hills: „ ^ Village, a fmall town, whofe inhabitants have no "particular privileges. ,.ni .,-;.. \ ¦ x Whirlpool, ^ very deep place, where the water turns round and draws every thidg into it, and finks it. , o . .. . ,* u n- Wildernefs, ,a large uncultivated piece of ground, growing nothing but ufelefs fhrubs arid bufhes. Wood, a piece of ground planted full of trees. SECT. 12 GEOGRAPHY. SECT. I. Of the figure and magnitude of the Earth. The ebbing and flowing and depth of the Sea. To find the dif~ tances of Places. Of Springs and Rivers. Of drawing Maps. AdjunEis and properties of the Earth, and the changes that have happened in it. The divifion of it into Kingdoms and Coun tries. A Table of the latitude and longitude of Places. P R O P. I. The figure of the earth as compofed of land and wa ter is nearly fpherical. THAT the earth is fpherical, or nearly fo, will appear from the following particulars. i. Perfons that are a fhip-board failing north wards obferve the pole ftar to be higher and high er the further they go. Likewife they obferve new ftars continually afcending above the horizon which were hid before, and thefe ftars continue to rife higher and higher the farther north they go. Like- wife, fuch ftars as are fouthwards are obferved to grow continually lower and lower, till by degrees they difappear below the horizon. Likewife fuch ftars as were at firft vertical, move gradually to wards the fouth, appearing ftill lower and lower. On Sea. I. GEOGRAPHY. On the contrary, when a fhip is failing towards the fouth, the pole ftar is obferved to grow lower and lower; and all ftars that are northward are obferved to defcend gradually, and that in propor tion to the diftance failed. And at the fame time, fuch ftars as are fouthward, continually afcend ; and new ftars, that were hid before, are obferved continually to emerge above the horizon, and to rife higher and higher, fo long as the fhip's courfe is continued towards the fouth. Now thefe phe nomena, can be owing to nothing but the fpheri cal figure of the earth ; for if the earth was a plane, all thefe ftars would appear always of the fame height. But as the earth is round, new ftars muft continually come into the zenith, as the traveller goes north or fouth. Therefore the earth is round , from north to fouth. 2. The fame is evident from the different longi tude of places ; for when any eclipfe of the moon happens, it is obferved fooner, by thofe that live eaftward, than by thofe that live weftward. Con- ftant experience fhews, that for every 15 degrees difference of longitude, an eclipfe begins fo many hours fooner in the eaft, or later in the weft ; which arifes from the fpherical figure of the earth. But if the earth was flat, the eclipfes would happen at the fame time in all places ; alfo day and night would alfo be at the fame time. But as thefe things are contrary to experience ; it is plain the earth is round from eaft to weft. 3. Again, a fhip failing in any part of the world, and upon any courfe ; at leaving any coaft,. all high towers, or high mountains, gradually difappear; firft the bottom part becomes invifible, then the middle part, and laftly the top ; appearing to fink gradually under the horizon, all they be quite out of fight. The fame is true of .(hips at a great dif tance off at fea ; firft their top mafts appear, and as *3 14 G E OrG R.A.PH Y. Fig. as they draw nearer, their lower mafts and rigging, an/TIaftlythe whole fhip.5 'Arid this cafi be owing id nothing but the fpherical figure of th6 earth. ' 4. In all lunar eclipfes/ that ever 'were obferved, and- in' whatever pofition the earth is:' at that time, ftill the fhado'w of the earth upon the: moon's difk, always as to fenfe, appears circular. And therefore it follows that the earth, which is the body that calls the fhadow, muft1 be round on all fides ; that is, it muft "be fpherical." And further, fince all cal culations of eclipfes, and of the planets' places, are madeupbri this fuppofttion ; and all anfwef to the true1 times ; which they do not upon any other fup pofition; therefore this confirms the propofition, thatfhe earth is round. 5. Several people have failed round the earth, fetting off weftward, arid continuing their courfe weftward continually, till they arrived again at the place they departed from."'1 And nothing but a jto'uhd figure will admit of this. Cor. 1. Hence the fea has the fame convexity :as the earth, and both* together make one globe; and therefore it is called- the terraqueous globe. Cor. 2. If the difference of longitude of two places, be 15 degrees; the people in the eaft' will reckon the time' of the day fooner by an hour,' than thofe in the weft. And for every 15 degrk's dif. longitude*, th-ofe in the eaft will reckon their time fo many hours fooner, and thefe in the weft fo many hours later, than the others. This is an immediate confequence from the fpherical figure of the earth. ^For it is 12 a clock in any place, when the fun is in the meridian of that place. And as the fun apparently moves from eaft to weft, thro' 360 degrees in" 24 hours ; it will be art hour of moving from the meridian of anv place, to the meridian of any other place which is ..¦'-¦*¦¦ l5 Sect. I. GEO G R A, P H Y. JS 15 degrees more wefterly. That is, it is 12 a clock pig. at the eaftermoft place, an hour before it is 12 a clock at the weftermoft ; , and the like for the other hours. "Arid fo the difference of apparent times, will be proportional v to the difference Of longitude. Scholium. It is not here meant that the earth is a perfect geometrical fphere. For experience fhews, that its" fu'rface"is fuirbf mountains arid valleys.' And befides it is higher at the equinoctial than at the poles by a 2,30th part, which amounts to iy miles, which is far rhore than' the* higheft mountains. And yet none of thefe irregularities are difcoverable to fenfe. Arid they no more hinder the earth from being reckoned fpherical, than the roughnefs of an orange or a lemon,' hinders it from being efteemed round. PROP. II. The latitude of any place on the earth, is equal to the height -of the pole above the horizon. rs r - ' '' Let HO be the horizon of the place,. Z the I. zenith, EQ^ the equinoctial, P die, pole, Pp the earth's axis, then (Def. 8) EZ is the latitude of the place." But EC P = a right angle = ZCO, from which taking the common angle ZCP, there remains the latitude ECZ = height of the pole PCPf Gor. Hence the height of the eauinoclial ECH 1= complement of the latitude ZCP. ' For ECH = comp. of ECZ = comp. OCP = ZCP, PROP, 16 GEO G R A P H Y. Fig. PROP. III. The direclion of motion of heavy bodies at the earth's furface, is perpendicular to the furface, or plane of the horizon, in that place. Here the earth's furface is fuppofed to be per fectly even and regular in that place, like the Imooth furface of ftanding water. Suppofe then a body defcending to the earth by the force of gra vity, and that it is covered with water at that very place; it is plain, if the line of direction of the falling body is not perpendicular to its furface, the water would not reft in that pofition, but will run down continually towards that fide which is loweft, that is, where the angle is obtufe ; or from the acute fide to the obtufe fide ; till fuch time as the fluid makes right angles on all fides with that line of .direction. And by the laws of hydroftatics, it can never reft till it get into that pofition. And therefore this, line of direction muft be perpendicu lar to that furface, or to the plane of the horizon in that place. Cor. If' a body be fufpended by a firing ; the firing continued to the earth, is perpendicular to the horizon. PROP. IV. In the globe of the earth, it is probable there is more folid- earth than water; and more fuperficial water than earth. The former feems to be evident from the mo tion of the tides, requiring generally 3 hours for the time of high water, after the moon's fouthing ; which could not be, if the fea was of a very great deptfy ; for then the fea would have nothing to do but Sect. I. GEOGRAPHY. 17 but to rife perpendicularly under the moon. For Fig. in any column of water, reaching near the center of the earth, the gravity of every particle of it is leffened by the approach of the moon ; therefore every particle of it will endeavour to rife at once, which therefore it will do inftantly, requiring no fenfible time; the water coming in laterally from the other columns at that depth, to affifl the motion -, which is done in a very little time, as the water has but a few feet to rife in that direction. But- in fhallow feas the cafe is quite contrary, for this tide of water cannot now be fupplied from an abyfs below ; but muft come in laterally, from ail parts around, tending to the place where the moon is vertical ; and having many miles to go, muft create a great current, and require a deal of time to perform the motion in, juft as we fee it does in fact. And this proves the fhallownefs of thefe feas where fuch currents happen to be. That the fea is of no great depth, appears alfo by the greatf number of Iflands difperfed all over the broadeft feas. And fome have fuppofed the depth of the fea to anfwer to the height "of the mountains. Thatthtre is more water than land in the furface, is evident by infpecting the terreftrial globe. And fome who pretend to have meafured both, or rather to have weighed them, tell us, that there is near three' times as much water as land. Cor. Hence the depth of the fea may be judged of, by the motion or current of the tides. For the deeper the fea is, the lefs the current will be-, as more of the water is fupplied from the abyfs below. PROP. V. Prob. To. find the latitude of any place on the earth. Altho' this has been fhewn in the Aftronomy {Prob. I. Seil. IV.) yet I fhall fhew other methods C of 18 GEOGRAPHY. Fig. of doing it here, as it is a very neceffary requifite in Geography, i Way. With a very good inftrument take the height of the pole ftar, which note down. Then at twelve hours diftance, take its altitude again, which al fo note down. Then add thefe two altitudes to gether, and take half the fum, and this will be the latitude of the place. Inftead of the pole ftar, any other ftar will do, that is not too far from the pole; for if it be low at one obfervation it will be affected with re fraction, which muft be allowed for, or elfe the lat. will not be exact. 2 Way. If you know the declination of. a ftar ; take the meridian altitude of that ftar; from this fubftract the declination if it be north ; or add it, if fouth-, and you have the height of the equinoctial. Sub tract the height of the equinoctial from 90 degrees and you have the latitude. Inftead of a ftar, you may make ufe of the fun, if his declination be known. But refraction muft be allowed for in both. 1 PROP. VI. Prob. To find the longitude of any place upon the earth. This Prob. is very neceffary in Geography, tho5 it ha's been folved before in the Aftrbnomy. 1 Way. Having the time of the moon's fouthing at any place, whofe longitude is known ; the longitude at "any other place may be found thus. Having a meridian line, and a clock or watch exactly fet to the Sect. I. GEOGRAPHY. 19 the time ; obferve the time of the moon's fouthingFig. at your place ; then by fubtra&ion, find the differ ence of times, at this place and that of known lon gitude. Then as 48 minutes is the difference of the times of fouthing of the fun and moon, in 24 hours. Say, as 48 minutes to 3600 : : fo is the faid difference of times, to the difference of longi tude. And this difference added to the longitude of the known place, when your time of fouthing was fooner, or fubtracted, when later; gives the longitude of your place. 2 Way. Having the time of the moon's fouthing at any place whofe longitude is known, as alfo the time of the fouthing of fome ftar near the moon •, the longitude of any other place may be found. Ob ferve by a clock or watch, by your meridian, how long after or before the ftar, the moon -fouths. By fubtraction find the difference of times at the firft place ; and having the difference of times at your place, fubtract one from the other for the fe- cond difference. Then fay, as 48 -minutes, to that fecond difference ; £0' 360 degrees, to the dif ference of longitude. If the ftar fouths firft, that place is eaft of the other, whofe difference of times is the leaft. If the moon fouths firft, that place is eaft, whofe difference of times is greateft. But the moon may alfo fouth the firft at one place, and the laft at the other. In that cafe you muft take the fum of the times, inftead of the fe cond difference before mentioned ; and that place is eaftward where the moon fouths firft. Therefore in any cafe your latitude will be known ; by ad ding or fubtracting your lat. to or from the known lat. as the cafe requires. C 2 Scho- so GEOGRAPHY. Fig. Scholium. In this laft method, the times of fouthing at the place whofe longitude is known, may be found either by obfervation, or by the aftronomical tables. If by obfervation, then you need not know the preeife times of their fouthing, but only the differ ence of times, by a correct clock. And therefore if proper obfervations be made at the two places, on any night agreed on; that fecond difference will be found, and from thence the diff. longitude. And both ways, to be more exact, inftead of 48 min. the motion of the moon from the fun in 24 hours, ought to be taken from the Aftronomical tables. PROP. VII. Prob. To defcribe the phenomena happening to the inha bitants of the feveral fpheres and zones. I. In a right fphere. 1 . Here the people live under the equinoctial, and therefore have no latitude. 2. The inhabit ants here enjoy a, perpetual equinox, their days and nights being equal thro' the whole year. 3. All the conftellations are vifible to them, and every ftar is 1 2 hours above and 1 2 hours below the ho- 'rizon. 4. The fun is twice in the year directly over their heads ; and twice in the folftices ¦, in one, declining 2 34- degrees towards the north ; in the other, as far towards the fouth. 5. They have two winters and two fummers in a year, as al fo two fprings and two autumns ; the fummers be ing when the fun is vertical. II. In a parallel fphere. 1. Here the .people live under the poles, and confequencly their latitude is 90. degrees, which is the Sect. I. GEOGRAPHY. 21 the greateft that can be. 2. They have no eaft orFig. weft, or any other point but north and fouth. Thofe that five under the north pole have nothino- but fouth ; and thofe under the fouth pole, no thing but north. 3. They have but one day and night in the year, and each of them is 6 months long. 4. Thofe at the north pole have day and fummer, when thefe at the fouth pole have night and winter ; and vice verfa. 5. The ftars never rife and fet, but move round them in circles paral lel to the horizon ; and therefore are always at ' the fame height. 6. The fun moves continually round about them, riling higher and higher every day, till he be 234. deg. high, or at the tropic. And afterwards defcends again, by the fame de grees, till he fets in the horizon. * III. In the oblique fphere. 1. Here, there is equal day and night only twice in the year, and that is when the fun is in the equa tor. 2. In fummer the days continually increafe, and the nights decreafe, till the fun is at the tro pic towards their pole, after which the days con tinually decreafe and the nights increafe, till the fun be at the other tropic in winter. 3. The'great- er latitude any place is in, the longer are the long eft days, and the fhorter the fhorteft. 4. Some ftars about their pole are always above the horizon, and therefore always vifible ; and fome ftars about the contrary pole, are always below the horizon, and therefore never feen. 5. To the inhabitants of the oppofite hemifpheres, the feafons are all contrary. 6. To all people in the fame parallel, the days and nights are of the fame length at all times of the year. 7. The, greater the latitude of the place is, the longer twilight continues. C 3 IV. 22 GEOGRAPHY. Fig. IV. In the torrid Zone. r. Thofe that live in the torrid zone, have the fun twice over their heads in a year. 2. They have two fummers and two winters in the year, but of unequal lengths. 3. Their days and nights are. unequal ; and the more fo, the further they are from the equator. 4. Thofe that live under the tropic, have the fun but once a year over their heads, and have only one winter and one fummer ; ^and to them all the ftars within their polar circle al ways appear, but none of thefe in the oppofite polar circle. V. In the frigid Zone. 1. Here the fun continues above the horizon for feveral days in fummer, and below the hori zon for as many days in winter ; and the nearer the pole, the more days it continues. 2. Here is but one' fummer, and one winter in the year. 3. In contrary zones, the one has fummer, when the other has winter. 4. Thofe that live under the polar circle, fee the fun going quite round them, when he is in their tropic ; and when he is in the contrary tropic, he does not rife for a day. To them the ftars between the tropics rife and fet : the ftars between their own tropic and the pole never fet, and thofe between the other pole and tropic never rife. VI. In the. temperate Zone. 1. The meridian fun is always one way, viz. fouth in north latitude, and north in fouth latitude: and therefore never coaies over their heads. 4. They have but one winter and one fummer in tl\t year. 3. The days and nights are always of. unequal lengths except at the equinoxes, 4. Thofe' ftars about their own pole never fet, which are „ within Sect. I. GEOGRAPHY. 23 within that parallel whofe diftani from the pole isFig. equal to the latitude. And thofe about the oppo fite pole never rife. 5. In the oppofite zones, the feafons are contrary, being winter in one, when it is fummer in the other. PROP. yill. Prob. To find the length of a fmall arch of the earth, as a degree, &c. 1 Way. Chufe two ftations A and B on the tops of two 2. mountains, and as far diftant as pofiible, fo as to be feen from one another. And having fet two marks at A and B, which may be feen by help of an inftrument with telefcopie fights. Place your- felf at one ftation A, and with the inftrument take the angle BAC contained between the perpendicu lar AC, and the vifual ray AB. Then remove to B the other ftation, and in like manner take the angle ABC contained between the perpendicular. BC and the vifual ray BA. Then by adding thefe angles CAB and CBA together, you have the fum, which taken from 1 80 degrees, gives the an gle ACB. Laftly, meafure the line AB, leading directly from one ftation to the other, and you will find how many miles, &c. of the earth's circuit cor- refpond to the angle ACB. If any, part of the line AB be impaffable, and cannot be meafured di rectly, fuch part or parts muft be meafured trigo^; nqmetrically, by taking off ftation^ as is ufual in meafuring inacceffible diftances, 2 Way\ Chufe two places on the Earth fituated under- the fame meridian, in a level and even country, and as far off as poflible, fo that one may travel C 4. from. 24 GEOGRAPHY. Fig. from one to the other in a right line ; take the la- 2. titude of thefe two places feparately (Prop. V.) which note down ; and fubtracting one from the Other, gives the difference of latitude. Then mea- fure the diftance of the two places in a direct line ; and you will know what length in miles, &c. be longs to that arch, or difference of latitude. 3 Way. Take two places as before in a very level coun try, and in the fame meridian, and as far diftant as you can. At. thefe two places, let the meridian altitude of fome noted ftar be exactly taken, the difference of thefe altitudes fhews the diftance of thefe places in degrees, &c. Then if the diftance of the two places be exactly meafured,- it will be known how many miles^ &c. anfwer to that arch' on the earth ; and confequently how many to a degree. ' Cor. Hence at a medium, the length of a degree upon the earth is about 69.31 Englijh miles. In the year 1635 Mr. Norwood meafured the diftance between London and York, by which he found the length of a degree to be 367196 feet or 69.54 miles. Mr. Picard found a degree in the middle of France to be 57060 French toifes. And Mr. CaJ- ftni afterwards found the fame length. Mr. Mujfchenbroek found a degree in Holland to be 57033 toifes. Mr. Mauperiuis, afterwards correcting Picard's meafures, makes only 56926 toifes to a degree. In the year 1736 Mr. Maupertuis and his com pany were fent by the French king into the gulf of Bothnia, to meafure a degree of the meridian. And they found the length of a degree under the polar circle to be 57438 toifes. At Sect. I. GEOGRAPHY. 25 At the fame time Mr. Bouguer and CondamineF'ig. were fent into New Spain, and there they found 2. a degree under the equator to be S^753 toifes. But Bouguer in his Navigation ftates it thus, a degree at the equator 56748 toifes — in lat. 45 , 57000 — under the polar circle 57422. From thefe meafures, one may reckon the mean length of a degree to be about 57050 toifes, or 365586 feet = 6g 24 Englifb miles ; and the mean between this and Norwood and Picard is about 69.31 Englifh miles. . Note, a toife is reckoned equal to 6.4080 feet Englifh, or only to 6.4 by fome writers. PROP. IX. Prob. To find the circumference and diameter of the earth. Having the length of a degree by the laft Prop. 2 If that be multiplied by 360, the product gives the circumference; and this divided by 3.1 416 gives the diameter ; and half of the diameter is the radius. All this fuppofes the earth to be a fphere, or very near it. Cor. Hence the circumference of the earth is 24951.6 Englifh miles, the diameter 7942, and the radius 3971, or thereabouts. PROP. X. Suppefing the body of the earth to be every where of equal denftty ; its figure is nearly that of an oblate fpheroid. Let APEG be its figure, PG the axis, ACE 3 the equinoctial. And let FPHG be the inferibed fphere. 26 GEOGRAPHY. Fig. fphere. Draw BID parallel to AC. If it was not for the diurnal revolution, the equal gravitation on all fides, would caufe it to have a fpherical fi gure, fuch as FPHG. But by its rotation round its axis, a part of the gravity is deftroyed, by the centrifugal force ; and the diminution of the gravity in every place, will be proportional to the centrifugal force in each place; and therefore if we fuppofe the earth in form of a fluid, that cen trifugal force will caufe the fluid parts to rife high er about the equator, in order to keep an equi librium among all the parts of it. Now fince AF = AC — PC, the part AF is fuftained by the centrifugal force. And to find how much is fuftained at B thereby, we have (by '.¦ Cor. 3. Prop. III. Centrip. forces) the centrifugal force at A : centrifugal force at B in the plane DB : : AC : BD : : AF (the effect of the cent, force at A) : BI (the effect of it at B). And by divifion AC : BD : : FC or PC : ID. Therefore (by Prop. XIX. Ellipfis) GAPE is an ellipfis, and the earth a fpheroid. Cor. 1. The increafe of gravity, in going from the equator to the pole, is as the fquare of the cofine of the. latitude. Draw IC, and BL parallel to it or perp. to IL. Then fince BI reprefents the centrifugal force of B, acting in direction IB perpendicular to the axis PC ; if this be divided into the two forces IL, BL ; the force IL is no way oppofed to the force of gravity, but the force LB is directly oppofed to it. Therefore the diminution of gravity at A : to that at B : : as AF : to BL. But AF to BL is com- ' pounded of the ratio of A F to BI or IC to ID, and BI to BL or (by fimilar triangles) as IC to ID. Therefore AF : BL : : IC* or FC2 to ID*. There fore the diminution of gravity at A : is to the dimi- ; nution Sea. I. GEOGRAPHY. ay nution at B : : FC1 : ID*. "Whence the diminution Fig. at any place B being as ID1, it is continually dimi- 3. nifhed (in that ratio) from P to A. But a diminu tion from P to A, is the fame thing as an augmen tation in going from A to P. Cor. 2. The gravity of a body at the equinotlial A, is. to the gravity of the fame body at the pole P : : reciprocally as PC to AC. For the gravities of the whole columns AC, PC, are equal; and therefore a certain quantity Q, will weigh at A, ^ ; and at P, will weigh -^. AL lrv_< And thefe are as and , or as PC to AC. AC re Cor. 3. Inftead of fuppofing all the earth in the ft ate 4* of a fluid, if we fuppofe there is in the middle of it, a fphere of folid matter, or terrella, as GFPH, en compaffed by a fluid.. The figure of the earth will. then approach nearer to a fphere ; fuppofing the denfity eve ry where equal. For let AFPQ^ be a part of the water or fluid; then the diminution of gravity, in the columns CA, CB ; going directly to the center, arifes from the centrifugal force as before. But the centripe tal force does not affect the parts of the colurhns CF, CL, below ; hut only the upper parts AF, BL. Therefore only the parts AF, BL, are in- creafed in the ratio of the cofine fquare of the lati tude (by Cor. 1.), the parts CF, CL remaining equal ; and confequently the wholes AC, BC, muft be nearer a ratio of equality ; that is, the earth approaches nearer to a fphere. And the more fo, the lefs the fluid matter, or the greater the terrella at the center. Cor. 4. And therefore the earth is the moft oblate it can be, when the whole of it is in a fluid ft ate. PROP. 28 GEOGRAP H Y. Fig. PROP. XL Prob. Having the length of a degree in J two feveral lati tudes of We earth; to find the polar and equina clial diameters of it. Let AC be the radius of the equinoctial, CP the femiaxis, draw BD parallel to AC, and BR perpendicular to the S. lat. DBR = j. Cof. — c, BR = v,d= length of a degree at B. Then by the nature of the ellipfes BD* =yy zzaa aa _ — zz. ee . Then (by Cor. i. Prop. XI. Ellipfis) DR = ff?. AndBD:DR::c:.f, oraa— fizz: — zz ee ce e± a a _ .ee' ::cc: ss ; that is, ee — zz : _ zz : : cc : ss. Whence zz - — —- Again, s : DR ( f± z ) : : Rad. . ccaa-\-ssee * ,ee J aaz ( i ) : BR or tt — But the radius of curva- ees ture in. B is equal ¥—, b being the parameter. (See bb my Fluxions, Prob. V. Sect. .II. Schol.) Now in the fame figure, the length of a degree in any place, is as the radius of curvature in that place ; therefore d is as i^» or as -if_5! , that is, bb bbe6s% as — ; b, a and e being conftant. But ZZ = ^ & ss and — . = . . Therefore tcaa + ssec ' ji ¦ccaa + teefi d is Sea. I. GEOGRAPHY. 29 d is as ¦¦ or as ______ , e being con- &" ccaa + sseel* ccaa + ssee^ 5* ftant ; and/* OC ~^~ ot* putting ^> = d\. For the fame reafon, in any other latitude, P OC . Whence p : CCaa + SSee ccaa+ ssee Therefore PCCaa + PSS, — _ — — ?* we have r X ^ + ss\ * ¦ = $*, and £ = -5. X fc/" 4- jjj'". 'Whence azzfe zz ¦— ¦ X ^ + ^V- If the two latitudes be at the pole and the equa tor, j _: o, and C _= o ; and S r_ 1, and e zz i; then —or/ becomes z_ v — ; and ? r: , e ' -01745 . and, = £252.^'745 , Take 30 GEOGRAPHY. Fig. Take the two latitudes of 66° 31', and 48 ° 5°'» 5. and the meafures, of the degrees 57438 and 56926, according to Maupertuis, and Picard corrected. And there comes out — or/ = 1.0109, and e zz 1 e 3237300 toifes _: 3929 miles, and a =_ 3972, the difference is 43 miles, that the earth is higher at the equator than at the poles, which is — part of 92 the radius. Again, if the meafure of a degree at the equator and polar circle be taken, which are 56753 and 57438, as fet down in the VIII Prop. Then the difference of the femidiameters will be — the ra- 216 dius of the earth. If we take the meafures at the equator and in France, lat. o, and 48 50, which are 56753 and 56926 toifes, the difference of the femidiameters will be - — of the mean femidiaifieter. Or if Pl- 577 card's own meafure betaken, 57060 ; the difference will be - — i. 320 If we take Bouguer's meafures 56748, 57000 and 57.422 ; for the latitudes o, 450, 66^. The firft two give ; the firft and laft, ; and the 339 2I5 two laft, , for the difference of the radii. *39 In the Aftronomy (Sea, V. Prop. 25. Con) I have lhewn from the theory of gravity, that the difference of the femidiameters is — — . And Sir 285 J. Newton makes it And other people pur- : • „ 230 fuing Sea. I. GEOGRAPHY. gi fuing different methods and meafures, have found Fig. other numbers, all differing from one another. 5- Particularly Bouguer by ufing feveral numbers, finds T l — - for a mean. Whilft Juan (who accompanied them to the fouth) makes — for the difference of 236 the radii. Scholium. In a cafe of fuch uncertainty as this, what can we conclude on ? Only this in general, that the earth is flatter at the poles than at the equator, or higher at the equator than at the poles. For fo great is the difference refulting from thefe feveral obfervations, that fome of thefe numbers are twice or thrice or even four times as great as others. And who knows which is right, or which to truft to ? Sir I. Newton's, calculated from the theory of gra vity, feems to be about a mean among them. The meafures under the polar circle and in France, make the height at the equator more than twice as much as Sir Ifaac. Whilft the fame meafure under the polar circle, and that under the equinoc tial, makes the height about the fame as Sir Ifaac does. Now all this inconfiftency muft arife from fome of thefe caufes. 1 . From the defeas of meafu- ring. 2. The fhrinking or lengthening of their poles. 3. Errors in taking the latitude. 4. The attraaive force of mountains. Or 5. That the earth is not a true fpheroid. 1. As to the defeas of meafuring, according to the account Maupertuis gives, all imaginable care was taken about it. Yet fome errors might happen, by not laying the rods in a right line from right to left. Or fome might happen, by not laying them exaaiy horizontal. For if in any cafe they made any angles, tho' very, fmall, but being often repeated, they would a- mount 32 GEOGRAPHY. Fig. mount to fomething ; and then the diftance of their 5«. ftations would meafure to more than it'really was. And it would be difficult enough to lay the rods truly horizontal, among fo much fnow as they had to wade thro'. Add to this, that thefe rods might bend in fome places and fo become fhorter. 2. As to the alteration in the length of the rods, Mauper- tuis tells us they could difcover none. But I never yet met with any fort of bodies, that were not af- feaed with heat and cold. And 1 can never think that wood, efpecially fir, is fecure againft it. For how could they be fure of the contrary ; for if they brought their rods out of a cold into a warm place, where their thermometers were kept, they would immediately expand with the heat, and during the time of meafuring would be reduced to the fame ftate as their, ftandard. In like manner, if the rods were brought out of a warm place to be meafured in a cold one by their ftandard, the cold would certainly contraa them, and reduce them to the fize of their ftandard. And this is the more probable, becaufe they took a great deal of care to have them nicely adjufted, which would require the more time to perform the operations in ; and being all the time in one place, they muft partake of one ftate. So that if the rods did really lengthen and fhorten, yet I cannot fee that they could have any certain way to find it out. And 1 take this to be the principal caufe of a degree meafuring to fo many toifes, happening thro' the contraaion of the rods, by fuch an extreme degree of cold. But if Maupertuis will have it. that heat and cold have no effba upon them; then the curious mechanic need feek no longer for a metal to endure heat and cold without any alteration ; for here wood will do the bufinefs. However, we know that wood will be contraaed or dilated by drought or moifture ;. and then this quality may be as pernicious as the other. Sea- I. GEOGRAPHY. 33 other. 3. The errors that may happen in taking Fig. the latitude, may be the incorreanefs of,theinftru- 5. ment, or the want of care in ufing it. But neither of thefe has any place here', as they had the beft of inftruments for this ufe, and the ableft hands to ufe them. 4. The irregular attraaion of moun tains may have fome effca here ; for all their ope rations were performed among' mountains. And a like effea happened to the fouth mathematicians ; for they found the attraaion of a great mountain, had fo much influence on a plumb line, as to caufe it to deviate 7 or 8 feconds from the perpendicu lar. But fuch an effea was not expeaed in the north, and fo no experiments were made about it. What is faid of mountains is applicable to fuch parts of the earth as are denfer than the reft, which will have a like effea, tho' they are quite invifible to us. And a fmall matter will deftroy the accu racy of an experiment fb nice and curious as this is. We may obferve that for every error of 1 fecbnd, there will be an error of 16 toifes; and in 20", will amount to 320 toifes, which makes a fenfible * difference. Add to this, that the very fhort bafes that are made ufe of in fuch cafes, is another caufe of error. For a very fmall error made there, will be greatly multiplied at the far end. And yet fhort bafes will perform better than long bafes, for a great part of -thefe will be upon uneven parts of the ground, which cannot be truly meafured . 5. But if the earth be an irregular folid ; no meafures can ever confpire to make it regular. And what ground have we to think it is regular. The internal parts of it are unknown to us. It may happen to be denfer towards the poles, or it may happen to be denfer towards the equator. But thefe two diffe rent fuppofitions will caufe different figures. And as the internal conftitution of the earth is unknown to us, the true form of it will always remain a fe- cret ; tho' it cannot be far from that of a fpheroid. D And 34 GEOGRAPHY. Fig. And tho5 thefe eminent perfons that were fent to 5. meafure the earth, have made furprifing difcoveries, and have executed their defign as far as human art could go : yet an operation fp nice, fubtle and critical, can never by any art be performed to the required exaanefs. So that I think it better to ac- quiefce in Sir I. Newton's numbers, for the figure of the earth, which he found by the theory of gra vity; at leaft till we can get fomething better. Espe cially as it is a mean among the different figures, which thefe inconfiftent and various meafures have afforded us. And upon this footing I will put down the following table, after I have confidered the lengths of pendulums. Cor. 1 . "The length of a pendulum vibrating feconds, increafes from the equator to the pole, as the force of gravity increafes, that is, as the fquare of the cojine of the latitude. For (by Prop. XXVII. Mechanics), the length of the pendulum, is as the force of gravity ; and therefore the increafe of the length, is as the in creafe of gravity, that is (by Cor. 1. Prop. X.) as the fquare of the cofine of latitude. Cor. 2 . The length of a pendulum at the equator is to the length of a pendulum at the pole, as the axis of the earth to the equatorial diameter. For f by Cor. 2. Prop. X.) the gravity is in that ratio. Yet by experiments, it appears that the length of a pendulum is ftill lefs at the equator ; and there fore the gravity is lefs there than in that ratio ; and confequently the earth is higher at the equator than we fuppofed. For the .gravity in any place muft be reciprocally as the height of the earth in that place from. the center ; becaufe the feveral columns of the fluid, reaching to the earth's center, exaaiy ballance one another. But the lengthening of a pendulum in different latitudes is fo fmall a matter, ' (being Sea. I. GEOGRAPHY, 35 (being 'but the 230th part), that it will be very Fig. difficult to meafure it exactly ; and efpecially in different countries, where this experiment is diftur- bed by heat and cold. However all thefe experi ments tend to prove this, that the earth is higher at the equator than at the poles. But to find ex- aaiy how much higher, will never be known from fo nice an experiment as the different lengths of pendulums, no more than it can be from the different meafures of a degree upon the earths A table of the lengths of a pendulum to vibrate feconds ; and the length of a degree, to every 5th degree of latitude. degrees. 1 Jength degree. length 1 pendulum. O 5 10 15 miles. 68.723 ¦'¦': 68.730 68.J50 68.783 inches. 39.027 39.029 39.032 39.026 20 25 3° 35 66.830 68.882 68.950 69.020 39.04439-°57 39.070 39084 40 45 59 55 69.097 69.176 69.256 69.330 39,097 39Al139.126 39.142 606"5 7075-. 69.40169.467 69,52269.568 39-158 39.168 39-17739.185 bo ] 85 I 90 69.601 69.620 69.628 39^9 l 39^95 39-l97 V 2 PROP. 36 GEOGRAPHY. Fi°". PROP. XII. The ebbing and flowing of the fea is caufed by the attractions of the fun and moon. **' Let ABCD be the earth covered with water, L the fun or the moon. Then fince the attraaive force of the body L is reciprocally as the fquare of the diftance, the water under L being nearer than the center T of the earth, the water at A will be more attraaed than the parts about the center T ; and therefore the water will rife at A and fubfide at D. For the difturbing forces of the fun and moon (explained in Cor. 3 and 4, Prop. XXVII. Centr. forces) will accelerate the water from the quadra ture D to the fyzigy A (by Cor. 8. ib.)r and retard it from A to the quadrature B. Now as the earth turns round its axis from D thro' A and B ; the water, from D to A being continually accelerated, will move fafter than the earth at A. And being continually retarded from A to B, and drawn back; it will move flower at B than the earth. Therefore at fome intermediate point N, the water will be at reft in refpea of the earth. Therefore fince it moves on all fides towards N, it muft be accumulated there ; and therefore it will be full tide at N. Alfo at fome point Q in the quadrant DA, the water will alfo be at. reft; but as it moves on all fides from Q, therefore the water will be depreffed at Qj that is, it will be ebb or low wa ter at Q. Likewife in the oppofite hemifphere BCD, the attraaion being greater at T, than at C, by the body L ; T will endeavour to leave C behind, which comes-to the fame, as if C was attraaed from T in direaion TC, juft as A was attraaed from T ; by the difference of the attraaions. The con- fequence Sea. I. GEOGRAPHY. 37 fequence will be that the fame effeas will happen Fig. in the hemifphere BCD, as in the other DAB ; the 6. water will be accelerated from B to C, and retard ed from C to D ; and at fome point P (oppofite to N) it will be accumulated, where it will be flood ; and at fome point O (oppofite to Q^) it will be de- prefled, and there it is ebb. So that there are two floods and two ebbs in the fpace of 24 (lunar) hours. The point N where it is' flood is always paft the moon's meridian at A ; and the farther paft it, the fhallower the water is, or the quicker the earth's motion. ' Generally, it is about 3 hours paft. Cor. 1. At the time of the new and full moon, the tides are the greateft ; and in the quarters the leaft. The Jormer are called fpring tides, the latter neap tides. For at the new and full, both fun and moon are in the line AC, and both raife the tides at A and C by their joint forces. But in the quarters, one raifes the tide at A and C, and the other at B and D. In the firft cafe the height of the tide will be as the fum of the forces ; in the fecond cafe, as their difference. But as the moon's force is four times as great as the fun's ; the fun will always make fome tide tho' lefs. • Cor. 2. But the fpring tides do not happen on the day of the change or full. Nor the neap tides on the days of the quarters ; but about 3 days after. For the waters will retain the motions impreffed upon them for' fome time after ; tho' the forces of the luminaries fhould quite ceafe. But as their forces are very little diminifhed in 2 or 3 days, the tides will frill increafe for a time. Cor. 3. Thefe are the greateft tides, when the moon (as alfo- the fun) is near eft the earth. D 3 For 6. $ GEOGRAPHY. Fig. For then the difturbing force is greater. Cor. 4. The tides are higher when the moon (and alfo the fun) is in the equinotlial. For the -effea will be greater in a great circle, and lefs in a leffer. At the pole it would be no thing ; there would be no reciprocation. Cor. 5. The tides are greater in leffer latitudes, than in greater. For the motion is lefs in a leffer (or parallel) cir cle, than in a greater. At the pole it would be nothing. Cor. 6. The times of the tide happening in particU' far places, may be very different according to thefitua- tion of thefe places. And likewife the height. For the motion of the tide is propagated fwifter in the open fea, and flower thro' narrow channels, or fhallow places. And being retarded by fuch impediments, they cannot rife fd high. » Scholium. As the moon is the caufe of a tide in our fea, fo the earth will alfo raife a tide in the lunar fea, if there is any. But that tide will be greater in pro portion to the greater force of the earth (which is about 40 to 1 ) ; and lefs' in proportion to its lef fer diameter (which is 1 to 3.65). Therefore if the moon raifes our tide 8.6 feet, our earth will raife their tide about 93,, feet: But as fhe turns the fame face always towards the earth, that tide will ftand there, without any reciprocation. And the moon will put on the figure of an oblong fphe-» Eoid. - PROP. Sea. I. GEOGRAPHY. 39 > PROP. XIII. Prob. To found the depth of the fea. Take a narrow cylindrical tube ah, 2 feet long 12. or more, the longer the better ; clofe at the top a, and open at the bottom b. Fit a cork to the end b, to go pretty eafy in. Then immerfe the open end b into a veffel of treacle, fo that the treacle may afcend about half an inch in the tube. Then whilft it is in the veffel, put the cork in at b, which will force the treacle up to c, and keep it there. Then this tube muft be inclofed in a ftrong brafs - veffel or cover DE, of great weight, that it may fink far enough -, and full of holes in the fides, to let the fea water go freely in and out. After the tube is put into this brafs cover, fill it full of wool on all fides, to hinder the tube from breaking a- gainft fhe cover. Then the cap G muft be fcrewed on to keep all faft. At F is a ring fixed to the brafs cover, and a long cord is to be tied to this ring ; and fo the whole machine funk into the fea, till it reach the bottom, if the cord will reach fo far. Then it muft be drawn up by the cord ; and , obferving how far the infide of the tube is daubed with the treacle, as at r ; the preffure of the water ; and confequently the depth will be known : and the depth is — — 1 X 10, in yards. ar For 10 yards deep of the fea is about equal to the weight of the atmofphere ; and at all depths, the preffure, as alfo the depth, is reciprocally as the fpace ar, that the air is compreffed into. There fore — zz number of atmofpheres preffing upon- ar the cork at r. But one of thefe atmofpheres is D 4 fpent 40 GEOGRAPHY. Fig. fpent in keeping it at b, or keeping the included 12. air in the ftate ab, when the depth of the water is nothing. Therefore the number of atmofpheres preffing it from b to r is 1 ; and confequently ar Dfie depth = ^- — i x 10. ar Scholium. It may be fuppofed that the length of the cord is" fufficient for meafuring the depth, without this apparatus. But this will not hold, for thefe reafbns ; i. becaufe the fhip being in motion, the weight does not fink perpendicular ; and 2. where there is any under current, it will alfo carry it out of the perpendicular, tho' the fhip was to be at reft. PROP. XIV. Prob. To find fhe diftance of any two places upon the earth whofe latitudes and longitudes are known. If the places have the fame longitude, or be un der the fame meridian, there is no more to do but to fubtraa one latitude from the other, when they are both on one fide of the equinoaial ; or add them, if on different fides ; and the difference or the fum is the diftance in degrees. _ In general, let EPQ^ be the meridian of one ' " place, EQ the equinoaial, P the pole, A and D the two places whofe diftance is required, PB the- meridian of B. Thro' A and B draw the great cir cle AB. Then fince the longitudes are given, their difference of longitude or angle APB, is found by 'fub'tracting one from the other. And the latitudes being given, the complements thereof will be known, or their diftances from the fame pole P, that is AP and BP. Therefore in the triangle APB, Sea. I. GEOGRAPHY. 41 APB, there are given the two fides AP, BP, and Fig. the included angle APB ; to find the fide AB, by 7. Cafe 8 of Spherical Trigonometry. From the end of the leffer fide AP, let fall the perpendicular AD, upon the longer fide PB. Then, Radius : Cof. APB : : ' Tan. AP : Tan. DP. Then DP taken from BP, when the angle APB» is lefs than a right angle ; or added to it when greater, gives the fegment BD. Then, Cof. PD : Col. PA : : Cof. BD : Cof. BA, the diftance required in degrees ; which multiplied by the number of miles in a degree (found in the table under Prop. XI. for the middle latitude), gives the diftance in miles. If AP and PB happen to be equal, as when the places are in one parallel of latitude •, the foliation will be eafier. The proportion is this, As Radius : Cof. latitude of either place AP : : S. half the diff. longitude APB : S. half the diftance AB, which doubled gives the . diftance fought, in degrees, on the arch of a great circle. PROP. XV. Prob. Having the longitude of two places in one parallel ; to find their diftance on the parallel. Finding the diftance in the arch of a great cir cle, was folved in the laft Prob. And to find the diftance along the parallel ; fubtraa the longitude of 42 GEOGRAPHY. Fig. of one from that of the other place, to get the 7. difference of longitude. Then fay, As Radius : Cof. latitude : : Difference of longitude : Diftance, in degrees of the equinoaial ; which multiplied by 69.31, gives the diftance in miles , Englifh. Or multiplied by' 60 gives the diftance in geographical miles. Otherwife thus, ' From the following table take the length of a degree for the given latitude, which multiply by the difference of longitude in degrees, give the diftance in the parallel, in geographical miles. A Table Sea. 1. GEOGRAPHY. 43 A Table fhewing how many geographical miles Fig. are contained in one degree of any parallel, for all latitudes. degr. geog. , deg. geog. 1 ( leg- geog. deg. ge°g- lat. O miles. lat. miles. lat. 46 miles. 4.I.68 lat. miles. So.oo 23 55-23 69 21.50 I 59.99 24 54.81 47 40.92 70 20.52 2 59.96 25 54-38 4« 40.14 71 19-53 3 5991 26 53-93 49 39-36 72 18.54 4 5 59-*5 59-77 27 28 53-46 50 5> 38.57 73 17-54 52-97 37-76 74 l6-53 6 59-67 29 52-47 52 36.94 75 15-53 7 59-55 3° 51.9b 53 36.11 76 J45i 8 59-4159.26 31 5'-43 54 35-27 77 13.50 9 10 3_i 33 50.88 50.32 55 56 34.41 33-55 ¦71 79 12.47 59-09 11.45 11 58.89 34 49-74 57 32.68 80 10.42 12 58.68 35 49- 1 5 5» 31-79 81 9.38 13 58.46 36 48.54 59 30.90 82 8-34 t4 *5 58.22 57-95 37 38 47.92 47.28 60 61 30.00 83 84 7-3i 29.09 6.27 16 57-67 39 46.62 62 28.17 85 5.22 »7 57- 3 « 40 45-96 63 27.24 86 4.18 18 57.06 4i 45.28 64 26.30 87 3-x4 »9 56.73 42 43 44.58 65 66 25-35 88 89 2.09 20 56-39 43.88 24.40 1.05 21 56.01 44 43.16 °7 23.44 9° 0.00 22 55-63 45 42.42 | 68 22.47 PROP. XVI. The origin of fprings and rivers is entirly owing to rain and vapours. The ancients had very unaccountable notions about the caufe of fountains and rivers ; and many of 44 GEOGRAPHY. Fig. of them abfurd'and even impoffible. Some of them imagined large caverns and receptables of water within the earth, placed there from the beginning, which were to keep running for ever without any fupply. Others fuppofed vaft quantities of fleam" or vapour to be raifed within the earth by the force of fubterraneous fires, which perfpiring thro' the earth, and condenfed near the earth's furface, might be fufficieht for the produaion of fprings and ri vers ; as if there were a vaft number of boilers or natural alembicks, plaCed within the earth, on purpofe to generate fleam, to be converted into water for fprings. Others flicking by the words of fcripture, would have the water of the fea, con trary to its nature, to afcend to the tops of moun tains, and breaking out from thence, to form fprings and rivers ; as if there was no other way but this, for rivers to be derived from the fea. Others again would have water conveyed from the fea by fome miraculous and fupernatural powers ; as if God had not been able to do this in a natural way, but in the firft inftitution of nature, fhould impofe a perpetual violence upon nature. Others recur to the influence of the celeflial bodies, for attraaing the water to the higher parts of the earth. All which fuppofitions are too weak to endure any examination. But leaving fuch whims and chime ra's, let us affign the true caufe. The true caufe then, is this. The heat of the fun draws vaft quan tities of vapour from the fea, which being carried by the wind to all parts of the globe ; and being converted by the cold into rain and dew, it falls down upon the earth ; part of it runs down.into the lower places-, where it immediately forms ri vers -, and the earth drinks tip the reft ; part of which ferves for the purpofes of vegetation, and tht reft defcending into hollovV cavern and places within the earth, is lodged there for a while ; which, Sea. I. GEOGRAPHY. 45 which breaking out by the fides of the hills, formFio-. little fprings, which will continue to run till thefe fources be exhaufted, or more rain comes for a new fupply. Many of thefe fprings running into the vallies, form little rivulets -, and feveral of thefe meeting together make a river ; and feveral of thefe meeting again, if the way be long enough, form large rivers, which at laft empty themfelves into the fea. That the fun exhales as much vapour as is fufEci- ent for rain is paft difpute ; it has been proved by1 aaual experiments ; and befides it is the only procefs that nature affords. The only difficulty is to know certainly whether the rain falling in a year, be fuf- ficient to keep the rivers running for a year. In order to know this the area of fome particular coun try muft be meafured in fuperficial degrees. Then all the rivers running from thence into the fea muft be noted, and their tranfverfe feaions taken juft a- bove the tide mark, at a mean depth, and a mean velocity. Then it will eafily be known how much is difcharged in a year at all thefe feaions of thefe rivers. Then the depth of. the rain falling in a year being known % experiment, being in fome places 1 6 inches, in fome places 20, in others 30. Then a quantity of water whofe bafe is the whole area of the country and of this depth, is the fund for a year's confumption. Therefore comparing thefe two quantities together, the queftion will be decided. A French author has made a computa tion this way, for the country lying about the river Seyne in France, and finds that 6 times as much water falls in a year as is fufficient to keep that river always running. Five parts out of fix are employ ed in feeding of trees, herbs, and expence of va pours, and extraordinary floods, and other wafting of the water, by finking into the ground. Dr. 46 GEOGRAPHY. Fig. Dr. Halley fays, the vapours that are railed co* pjoufly from the fea, and carried by the winds to the ridges of the mountains, are conveyed to their tops by the current of the air ; where the water prefently precipitates, gleeting down by the clefts of the rocks ; and part of it entering the hills, is colleaed into bafons or receptacles ; which being filled, the water breaks out at the fides of the hills, where it can find vent ; and forms fprings ; feve ral of which form rivulets, and many of thefe make a river, fuch as the Rhine, the ' Rhone, the Da nube, &c. , Thus one part of the vapours, is returned again by rivers into the fea ; another part, by cold nights, fall again into the fea, in dew or rain,' which is much the greateft part : a third part falls on the lower lands, and is the nourifhment of plants. All the phenomena of rivers and fprings confirm this account. For in a long drought, rivers and fprings grow very low, and are fometimes dry; growing lefs and lefs as their magazine is more and more exhaufted. In mountainous places, the rivu- , lets at the beginning are very fmall, -but grow big ger the further they go, by fijae accefs of other fmall parcels of water. Therefore the heads of all rivers are at the . mountains. In places where it never rains, and yet have rivers ; thefe rivers come from places where it does rain, but run thro' thefe coun tries that have none, in their way to the fea ; but receive no augmentation in their way ; and, in fuch countries there are few rivers, only fuch as are ge nerated from dews and mift. That the fea is no bigger for all the rivers running into it, is becaufe the water returns back to the land, in vapours. It is found by digging into the ground, that all places at any great depth are full of water. If it was not fo, there would be no need of engines to draw the water out of mines. For the rains that fall upon the Sea. I. GEOGRAPHY. 47 the earth, not being immediately fent off by the ri- Fig. vers, a part of it lying upon the grodnd muft gra dually fink into it, defcending lower and lower, thro' narrow fiffures and crevifes ; from whence none of it can return, but what breaks out in fprings, according to this theory. Cor. 1. There is a continual circulation of water from the fea to the land, thro' the air in vapours and rain ; and from the land to the fea, upon the earth, by the rivers. Cor. 2. Hence, thofe are the largeft rivers that run , thro' the largeft trail of ground. For the longer the traa of ground, the more rain falls, and confequently the greater any river will be. The further they go, they are perpetually gathering. Such is the Danube running over a length of above 800 miles, and at laft falls into the Euxine fea. Cor. 3. Wherever we find a large river, in any country known or unknown, we may be Jure of this, that it runs over a large trail of land. But where on- ' ly fmall rivers are to be met with, it indicates a fmall country. This may be of fome ufe to thofe that are feek- ing after new countries. P R O P. XVII. Prob. To draw a map of the world. A map is a geometrical draught or reprefentation of any country, fhewing the true fituations of all places contained therein, as to longitude and lati tude. And likewife all mountains, rocks, rivers, lakes, woods, forefts, mines, fands, havens, and whatever is remarkable therein ; and alfo how it is bounded 48 GEOGRAPHY. Fig. bounded by fea or land, on the eaft, weft, north, and fouth. It is an eftablifhed rule among all Geographers, that the top or upper part of any map is the north, for Geographers always turn their faces towards the north pole, in viewing or defcribing the feve ral regions of the earth. On the contrary, Aftro- nomers, obferving the twelve conftellations of the zodiac, look towards the fouth. Priefts look to wards the eaft, and Poets towards the weft. There are feveral methods of drawing maps ufed by different people ; the principal whereof we fhall here defcribe. i Method. A map of the world muft reprefent two hemi- fpheres ; and both muft be drawn upon the plane of that circle which divides the two hemifpfieres. The firft way is to projea either hemifphere upon the plane of the firft meridian, by the rules of or thographic projeaion, laid down before. And then upon another equal circle, projea the other hemifphere by the fame rules. Here the meridians and parallels are to be projeaed to every ten de grees difference of latitude, and longitude ; or if you pleafe, to every 5 degrees. Upon the plane of the meridian, the meridians will be ellipfes, and the parallel circles all right lines. LJpon the plane of the equinoaial, the meridians will be right lines, and the parallels of latitude will be circles. The fault of this way of drawing maps, is, that near the outfide, ' the circles will be too near one another ; and therefore equal fpaces on the earth, will be re- prefented by unequal fpaces upon , the map. 2 Method. Another method is to projea all the meridians and parallels by the rules of ftereographic projeai on, likewife laid down before. And this is to be done Sea. I. GEOGRAPHY. 49 done for both hemifpheres ; and may be either Fig. upon the firft meridian or upon the equinoaial, 8. And by this method all the parallel circles will be reprefented by circles, and the meridians by circles or right lines. Here again equal fpaces on the earth are reprefented by unequal fpaces in the map. For on the outfide, the circles are too far diftant, and near the middle too near together. To remedy thefe tnings, another method may be ufed. » » 3 Method. This may be done either upon the firft meridian, or upon the equinoaial. For the firft meridian; draw the circle PENQ for the meridian, thro' the center C, draw the equinoaial EQ, and NP per pendicular to it -, then P and N are the poles. Divide the quadrants PE, EN, NQ, and QP in to 9 equal parts, each reprefenting 10 degrees, be ginning at the equinoaial EQ. Likewife divide CP and CN, into 9 equal parts ; beginning at EQ^ Thro' the correfpondent points, draw the parallels of latitude. Again divide EC and CQ into 9 equal parts ; and thro' the points of divifion, and the two poles N and P, draw circles or rather ellipfes for the meridians. And another like projeaion be ing made for the oppofite hemifphere, the map is ready for inferring the feveral places and countries of the earth. For the projeaion on the plane of the equinoc- g> rial, draw AQBE, for the equinoaial; divide1 it ' into the 4 quadrants EA, AQ, QB, and BE-, and each quadrant into 9 equal parts reprefenting 10 degrees of longitude. From the points of di vifion, draw lines to the center C, for the circles of longitude. Divide any circle of longitude, as the firft meridian EC, into 9 equal parts, and thro' thefe points, defcribe circles from the center C for the parallels of latitude. Then number them as in E the $ impoffible for any ark to live (be preferved) at fea, or the ftrongeft man of war that ever was built. The like vaft tides would alfo be raifed in the atmofphere, attended with the moft violent commotion of all the body of air ; the con-* fequence whereof would be continual rains. In this cafe no place of fafety could be found for any animals, except they had the good luck to be upoa fuch a high mountain as was without its reach. Thofe that fuppofe the water to be over all the face of the earth at once* are forced to call in the power of omnipotence to effea it ; for it is a thing impoffible by any natural caufe, to produce fuch an effea. Therefore in order to preferve mankind and other animals, God was pleafed to order an ark to be built, to contain a few of every fpecies of animals, which were afterwards to replenifh the earth. As , this ark, with its contained animals, was to fwim upon the water of the deluge, all the time of its overflowing the earth ; it was neceffary that this flood of water fhould be perfeaiy calm, and free from all ftorms and tempefts. For if the ark came to be toffed about in a tempeftuous and raging fea,, or deluge of water ; from its ftruaure and magni tude, it muft inevitably perifh, with all its freight of animals. But I believe it would be a perplexing affair to make out how fuch a tumultuous concourfo of water fhould be fo very quiet and ftill, fo clear of winds, ftorms and tempefts, as is here required; And if this was granted* it would ftill be equally difficult to account for another phenomenon ; that, is, how all fhells and marine bodies,, fhould be F 3 thrown; 7*0 GEOGRAPH Y. Fig. thrown upon the land, or even to the tops of moun- 1 1. tains by fuch a ftill water ; and many of them bu7 ried deep in the earth ; this effea is not at all re- concileable with fuch a fuppofition. Therefore it does not appear that both thefe hypothefes can be true. For the calm fea, neceffary for preferving the ark, could move none of the fhells -, and the rough fea, neceffary for tranfporting the fhells, would deftroy the ark. The reconciling thefe things is not eafy ; .and perhaps here, as in other fuch cafes, our only refuge is, when we are pinched with any difficulty^ , to cry out for a divine power. Yet the Almighty generally brings about his purpofeg by natural caufes. But be that as it will, it is certain, that fome time or other this earth has fuffered a moft vi olent fhock, concuffion or agitatidn by. fome caufe or other, unknown to us ; but which manifeftly appears by its effeas. _ The next flood we have an account of, is that of Ogyges ; this flood overflowed all Attica, a country in Greece joining to the Mediterranean. This flood happened in the time of Ogyges, who was king- thereof, and dwelt in Thebes, a city of his own building. This country is a part of what is now called Achaia, being a part of Greece^ The next remarkable flood. was that of Deucalion, celebrated by the poets, efpecially by Ovid. This flood overflowed all Theffaly, whe're Deucalion was king. This country of Theffaly is a part of Greece, and joins upon the fea. Befides thefe floods, there is no doubt but great floods have happened in other countries ; but the people being ignorant of the ufe of letters, no ac count thereof is tranfmitted to us. All thefe floods muft heeds make great havock and devaftation in all the countries where they happen ; by demolifh- ing towns, plants, trees-, by beating and wafhing. down mountains ; and levelling the low grounds ; by Sea. I. GEOGRAPHY. 71 by filling up the channels of rivers, and changing Fig. their courfes ; by covering all the country with 1 1 . flime and mud, and giving a new face to the earth ; by breaking off part of a country, and re ducing it to an ifland ; and by inundations breaking into the land and forming gulphs therein. Thus fome have fuppofed that Sicily has been divided from Italy -, Spain from Barbary ; and, as fome fay, England from France. The violence of the wind alfo much increafes the depth and turbulent motion of a flood, efpe cially in fuch inundations as proceed from the fea. Thus high winds and tempefts frequently caufe the fea to tranfcend its ordinary bounds, and trefpafs on the land, to the great lofs and danger of the inhabitants. Thus it happened in Zealand and Holland, where feveral towns with multitudes of people were fwallowed up by the fea ; occafioned by the violence of the north wefterly winds. Several inundations or floods have alfo happened in Egypt, by the overflowing of the Nile, as is reported by feveral hiftorians, and the like in a great many other places. 2. Earthquakes are another great caufe of the changes made in the earth, and the defolation of feveral places, to the ruin of the inhabitants. An earthquake is a violent making of fome large part of the earth, fuppofed to be caufed by a nitrous and fulphureous vapour included in the bowels of the earth, which by fome accident taking fire pro duces an explofion, which occafions thefe terrible motions and fhocks of the earth at that rime, The firing thefe fulphureous vapours, may be owing to. their fermentation, or to the falling of roeks and ftones within the hollow places of the earth, and ftriking fire againft one another. t- .Thunder and lightening arifes much from the fame caufe, for the nitro- fulphureous vapours* ex- V 4 h*linS n GEOGRAPHY. Pig' haling out of the earth, and floating about in the ) *' air, at laft by their fermentation take fire, and pro duces that dreadful explofion called Thunder. The violence of earthquakes is fometimes fo great as to fplit and tear the earth, and caufe it to open for many miles 5 gaping at the exit of the ignified vapour* and clofing again after. And fometimes tearing and disjoining the ground, fo that after the ftiock, it immediately finks ; and thus a hill or a mountain is fometimes reduced to a great pool of wa ter or a lake. After this manner feveral confiderable traas of laild, with the cities and towns, plants find trees, have been fwallowed up, and totally funic ' » The ftrength of this confined vapour is fo great,"' that it forces up ftones, water, earth, and all things ' that fall in its way, with a very great impetuofity, Caftirig them fometimes to immenfe diftances.' In ' this manner they are thrown out of the mouths of ' Volcano's, or any place where the inflamed vapour ' Can find vent, or break open. And from the heat of the inflamed vapour, which forces out thefe bo dies, they are all of them alfo heated and inflamed thereby; fo that flames of fire are fometimes feeri to proceed out of the earth, with great quantities of melted metal or metalline fubftan'ces. Great floods of this kind are generally thrown out of burning mountains. When this happens in a place feated Underneath " the fea, the motion of the earth below, puts the fea into horrible tumult, making it rage and roar, with a hideous noife, raifing its furface into prodigi- • ous Waves, which are rolled and toffed about in a furious manner, overfetting and finking fhips, or throwing them on the land. By this, the fea is ¦ faifed far above its ordinary height, and depreffed ' again as far below. And thus it continues in the* moft extreme agitation, till the hot vaporous mat ter- Sea. I. GEOGRAPHY. 73 ter be all fpent and difcharged ; and then the earth- Fig. quake ceafes, till a frefh colfeaion of this fulphu- 1 1. reous vapour produces another. And this fulphu reous vapour difcharged into the atmofphere may furriifh matter for thunder and lightening ; as they are both generated from the fame caufes, and re quire the fame matter for both to work upon. The fhock of an earthquake is always felt at the fame moment in all places, tho' it has fometimes been known to reach many hundred miles ; and al- tho' the feveral places be parted by the fea lying between them. Mountainous countries are the moft fubjea to earthquakes ; for mountains are commonly ftony and hollow within, and therefore fit to receive thefe vapours ; and particularly fuch places as contain a great deal of fulphur and nitre, are of all others the moft fubjea to earthquakes, and generally fuf- fer moft thereby. Such combuftible matter taking fire will burn for a long time, and fend out vaft flames with a horrid noife, as is frequent in mount Etna and Vifuvius, and all fuch places. And thefe Volcano's are of ufe for difcharging the matter, and preventing worfe confequences. Earthquakes are far more frequent than great floods ; not one age paffes without feveral inftan- ces of this fort, with their direful effeas, fuch" as finking of towns and mountains, as happened lately at Lifbon. The changing the land into fea ; and the fea into land, by throwing up new iflands where none appeared before. And by fuch events as thefe, the face of the earth muft be continually changing, and muft be quite altered from what it was many ages ago. Other fmaller changes I pafs by, fuch as the al teration made in the furface of the earth by culti vation, the cutting down woods, inclofing ground ; building of towns and cities ; as likewife the de- vaftation 74 GEOGRAPHY; Fig. vaftation made by armies, by the invafion of coun- j i . tries ; where many beautiful places have been laid wafte, towns burnt and depopulated, and innume rable fuch like accidents ; alj which help to give a different afpea to almoft all countries upon the earth. Befides the changes already mentioned, moft places have changed their original names ; fo that bur modern geographical defcriptions or maps hardly give us any region under the fame name, by which it was known in former times. And this has occafioned great controverfy and difpute about diverfe countries mentioned by ancient writers ; and not only countries, but cities and towns have loft their original names, and at prefent go under other names. And all this may be in great mea fure owing to the conquefts made by one natiop making war againft another. For by this means the boundaries of countries have been quite chan ged, , fometimes by tearing a part from one country .and throwing it to another, or throwing feveral of them into one, or dividing one country into feveral parts. And the conquerors, either not knowing or not liking the old names, of towns and countries, impofed other of their own. Thus the ancient Scythia is now called Tartary. What was Thrace is now Romania, what was Libya is now Tripoli, or a part of Barbary. Achaia is now Lividia. -Cyrene is Barc'a. As to towns, At hens, is become Set tines. Berenice is Suez. Leopolis is Lemberg. Olympia is Longinico. Pranefle is now Paleftrina, Sufa is Schonller. Sparta is Mifitra. Delphos is Livadia. Pelufium is Damietta, and numbers of others. PROP. Sea. I. GEOGRAPHY. 75 Fig.. PROP. XXII. Prob. if. To fhew the dlvifion of the earth into its' feveral kingdoms and countries.* The defcriptioit of the fevCral countries of the world is the leaft part of my bufinefs, as belong ing more to an hiftorian than a mathematical Wri ter ; yet I fttall here give a fhort account of each country, and how it is • fituated ; without which it is hardly poffible to underftand any thing of hifto ry. And to have a right notion of any place, of its fituation, and extent, the geographer ought to futnifh himfelf with maps of each country, which Will fhew him at once all thefe particulars, as alfo what cities, towns, caftles, rivers, bridges, moun tains, and other remarkable things, are contained in it ; and without fuch maps we have very faint ideas of any nation or Country. The earth is generally divided in four quarters, 1 Europe* 3 Africa. 2 Afia. 4 America. But this divifion is very imperfea ; foj- there are vaft traas of land tinder both fhe poles, not yet difcovered, and Which belong to none of thefe four divifions. Therefore taking in thefe two po lar regions, the earth is more properly divided in to fix general parts, Which will take in the whole earth whether larld or fea. I. EUROPE, is boUhded on the north partly or Wholly by the frozen fea ; oft the fouth, by the .Meditefanean fea", on the weft by the Atlantic 6ce- a*n, arid on the eaft by the rivers1 Tanais, Oby, afld Volga, Which laft falls into the Euxine fea. In defcribing the fevei'al countries, I fhall begin with the north parts, and proceed to the fouth. Sweden, 76 GEOGRAPHY. Fig. 1 1 . Sweden. This country is contained between the n. lati tudes 60 and 72J and longitudes 30 and 56 from Ferro. It is bounded on the eaft by Mufcovy, on the weft with Norway, on the north with the Fro zen fea, on the fouth by Denmark and the Baltic fea. It is divided into the provinces Finmark, Lap land, Sweedland, Finland, Gothland, Livonia^ and. the Swedifh iflands. Norway. It, is contained between the n. latitudes 60 and 68, and longitudes 30 and 38. It is bounded on the eaft by Sweden, on the weft by the weft fea, on the north by the frozen fea, and on, the fouth by an . arm of the weft Jea. Denmark. . Contained between 54 and gy degrees of n. lat. and between 30 and 35 degrees of longitude. It contains Jutland, and the iflands Zealand, Funen, &c. Jutland is, bounded on the fouth by Germany ; on the other fides by the fea. Mufcovy or Ruffia. This is a large country, extending from the, n.' lat. 45 to 71 ; and between the long, of 48 and 107. It is bounded on the eaft by Tartary, on the weft by Sweden, on the north by Lapland and the Frozen ocean, on the fouth by Poland and the Cafpian and Euxine feas. It ftands partly in Europe and partly in Alia ; it is divided into the provinces , of Trims, Kargapolia, Divina, Condora, Siberia,, Obdora, Vologda, Cafan, Mordowitz, Kijjnovogrodt, Volodimar, Mofcow, Aftracan, Novogrod, Weliku.^ Plefkow, Severia, and many more.. Great Sea. I. GEOGRAPHY. y7 Fig. Great Britain* llm This is an ifland, and contains England and Scotland ; it is contained between the n. lat. 50 and 59, and long. 14 and 22. It is divided into a great many counties or fhires ; and there are a great number of fmall iflands, lying to the north and north weft thereof. Ireland. This is alfo an ifland on the weft fide of Great Britain, and is contained, between the n. lat, 5 1 and 56; and long. 9 and 14; which is alfo divided in to feveral counties. Germany. This lies between latitude 45 and 55, and long. 26 and 39. It is a large country, and coniifts of many provinces, dukedoms, circles, &c. It is bounded on the eaft with Pruffia, Poland, and Hungary, on the weft with France, on the north by Denmark and the Baltic fea, on the fouth by Italy. It contains on the north, Holland, Flanders, Weft- phalia, Lower Saxony, Upper Saxony. In the mid dle, Lower Rhine, Upper Rhine, Franconia. On the fouth, Suabia, Bavaria, Aufiria. Ey fubdivifion Holland contains Zealand, Utrecht, Zutphen, Over-IJfel, Friefland, Groningen. And Flanders contains Gejderland, Brabant, Lux- embourge, Limbourge, Artefia, Hannonia, Namur. Weftphalia contains Oldenbourg, Hoya, Diepholt, Schomberg, Embden, Lingen, Tecklenburg, Ravenf- burg, Ben them, Munfter, Lip, Mark, Ber^e, Chves, Juliers. Lower Saxony contains Holftein, Lawen- burg, Mecklenburg, Bremen, Verden, Lunenburg, Brunfwick, Halberjlat, Magdeburg. Upper Saxony contains Mifnia, Thuring, Brandenburg, Pomer'a- nia, &c. but the fybdivifions of thefe countries are cndlefs. Pruffia. 78 GEOGRAPHY. Fig. ii.* Pruffia. This extejjds from lat.. 51, to 53, and long. 35 to 45 ; it is bounded on the eaft by Poland? on the weft by Germany, on the north by the Baltic, on the fouth by part of Poland. Saxony. Bounded on the north by Pruffia, on the fouth oy Bohemia, on the eaft by Poland, and w.eft by Germany. Bohemia. Bounded on the eaft by Poland, on the weft by Germany, on the north by Saxony, on the fouth by Hungary. Silejia. Bounded on the north by Brandenburg, on the fouth by Hungary, on the eaft by Poland^ pn the weft by Bohemia. Poland. Bounded on the eaft by Ruffia, on the weft by -Silejia, on the north by Prujfia, on the fouth by Hungary. Hungary. This country extends from 45 to 49 deg. n. lat. and from the long. 34 to 41. It is bounded on the eaft by Turkey, on the weft by Germany, pn the north by Poland, on the fouth by Turkey.. France. This country is contained between the n. lat. 43 and 51, and long. 14 and 28. It is bounded on the eaft by Germany, on the weft by the Atlantic Vcean, on the north by the Englifh Channel, on the fouth by Spain and the Mediterranean Sea. Spain. Sea. I. GEOGRAPHY. 79 Fig. Spain. 11. It is contained between the n. lat. 36 and 45, and long. 10 and 23. It is bounded on the eaft by the Mediterranean, on the weft by Portugal and the Atlantic Ocean, on the north by France and the Bay of Bifcay, on the fouth by the Atlantic Ocean and part of the Mediterranean. Portugal. Bounded on the eaft by France, on the weft by the Atlantic Ocean, on the north by Spain, on the fouth by the Atlantic Ocean. Italy. This is contained between the n. lat. 38 and 47, and long. 27 and 4.1. It is bounded on the fouth and eaft by the Mediterranean Sea, on the north by ¦ Germany and Switzerland, on the weft by France and the Mediterranean. Switzerland. Bounded on the north and eaft by Germany, on the fouth by Italy, on the weft by France. Turkey. This is contained between 37 and 49 deg. of n. lat. and 38 and 55 deg. of longitude. It contains Greece, Little Tartary, and part of Hungary, and many provinces befides. It is bounded on the eaft by the Euxine Sea, on the weft by the Gulph of Venice, pn the north by Hungary, on the fouth by the Mediterranean. Iflands in Europe. The principal are the Orkneys, the Shetland ifle§, and W.eftern iflands near Scotland. The Ifle of Man, .Anglefey, and ifle of Wight., near England. Jerfey 80 GEOGRAPHY. ^iS-jBfy* Guernfey, Alderney, near France. The 1 1. iflands Rugen, Bornholm, Oeland, Gothland, Oefal, Dago, Aland, Ween, Zealand, Funen, Langland, Laland, Fulfiar, Mona, Femeren, Alfen, near Den mark and Sweden. Alfo Carmen, Hitteren, Sanien, Suray, near Norway, and an infinite number, of fmafler ones. The Azores or weftem iflands, lying weft from Spain, being 9 in number ; they are St. Michael, St. Maria, Tercera, Gratiofa, St. George, Pico, Fyal, Flows* Cuervo. Iflands in the Mediterranean, are Toica, Majorca, Minorca, lying near Spain. Corjica, Sardinia, Si cily and Malta, near Italy. Candia near Greece, Cyprus near Syria in Alia ; and a vaft number of fmall ones near Greece in the Archipelago. 2. A S I A, is bounded on the eaft by the Eaftern fea, on the weft by Europe and the Red fea, on the north partly by the Frozen fea, on the fouth by the Indian Ocean. Siberia. This country reaches from n. latitude 50 to 68, and from the long. 105 to 170. It is bounded on the eaft by the Japan Ocean, on the weft by Ruffia, on the north by the Frozen fea, on the fouth by Tartary. This is a wild, barren, cold country, and thinly inhabited, and under the. emperor of Ruffia. . Tartary. This was anciently called Scythia ; . it reaches from 4P-to 60 deg. of n, latitude, and from 56 to 1 ho longitude. It is bounded on the north by 38n !7 Alcudia i4on 25 Aldborough Uzn 1 2I Aldenburg |6on 1 32 Alenzon, "Alencon, Aleffio Alexandria Algeri , Algiers Alhama AHcantAllendorfAH faints bay Altneria AlmiflaAlmunecarAltamont Altena Altenburg, Owar, Altorf AmadubatAmadan, Hamadan A madia Amalfa A map all a AmafiaAmbara, Ajnba- 1 marjam J Amberg AmbrunAmelia Amiens Amphipolis Amfterdam AnclamAncona Andernach - Andero AndreaSt. Andrews Andujar, Anduxar, Angelo AngelosAngerlburg Angermund Angers Angol Angolefme . Angoura, Anchira, lat 181'fan 1.0 ft U I) 57 r. "37 n 12 f 37n 44 » 37 n for, 54n5in49 n 24n 35 n 11 n 4.1 n [211 42 n "3n 49 n 44 n 42 n '56 n 42 n ;2n54 n 43 n 5on43 n 43 n 56n 38n41 D i9n 54n5'n 47 n 38n 46 n 41 n Ion. 19 40 5' 28 2324 '7 3° jS 4 39 3830 33 33 9267 6335 285 56 55 32 2633 21 60 24 34 35 26 »3 35 18 2436 275 43 26 18 300 18 531 Angra Sea. I. GEOGRAPHY. Angra AnnaAnnan Anapolis Annecy Anflo, Obfto, Asipaeh Antiqaera AntibesAntiochAntivaxi Antwerp Apenzel Aquilla Aracan Arauco Arbela Arbon Archangel ArcoArdenburg Ardra ArebonAremburg ArequippaArezzoArgentpn,Argun Arhnfen Arien, Aire, Arica Aries Anaagh " Arlon£rmen tiers ArnheimAronaAronche&Arras, Atrecht ArundelArziUa AfaphAfchjaflenburg AfcoliAfopb, Afow Afti Aitorga .lat. Ion. 39" 350 3-3" 62 55" 18 39 & 3°Q 46 n *6 59 n 29 4-9" 3P 37n »3 43" 26 36 n 57 38 n 44 5m 21 46 n 29 42 n 35 20 n "3 37 « 300 35 n % 48 n 29 64 n 65 4611 3° Sin- 23 5" 24 5n 25 5011 26 i7n 3°5 43 n 33 46 n 21 5211 124 56, n 29 Son 21 i8f 308 43 n ?4 54n H 49 n 25 Son 22 S2n 25 4&n 28 39" "4 50^ 21. 51" 18 36 n 15 53n '7' 50n 29 42n 36 47 n 64 45 " 28 42 n '3 9* Aftracan Ath, Aeth AthensAttockAvallonAubigneAuchAveiroAuglburg, Aulburg St. Auguftine. Auguftow Avignon,Avila AvilesAutengabad AurichAufburgh AutUn Auxej-ra Axim Axonne Axuma AyamonteAwlen Axel'Axipoli Babilon Baca, BaccafTerai BacherafeBachuBadajoxBaden (Swabia) Baden (Swifferland) BaezaBagdatBuguagarBagniajuekBahus * Baja BalbecBalcaBalifore BambergBancalis Ban cock lat. Ion. 47 n 7? 5in 23 38 n 44 53n 92 47 n 23 +7" 22 44 n 4P 40 n 29 48 n 3' 3on 297 53" 43 43" 24 41 n *35 43" 14 i9f 95 53 n 26 48 n 3i 47 n 33 47" 22 5" 16 42n 24 '5" 50 37" 12 49" 3Q 5m «3: 45" 5» 3?" 64 37" i6> 45 n 55 Son 27 40 n 69; 39" 10 49" 24 47" 28 38" '4 33" 63 l6n 97 44" 39 58n 3i 47" ;4Q 53" :5? 37a ;*5 21 n 105 SOn 3« jn 119 l3n IZI Bai idoia ,o 333333^3333 — ¦3->33333333333333^333B33333333333333P O w?ttwwWWWbdbdtdWWbdWodtdWWWWWWbJbdb3WW 5*1 SggggggSg £,12,^ g g g g S-2.^ g.8 cTo ErErg-^K^gl^'S-siS-B-ffSRS £,£,£?? S ^ M N 4. si o\^ i-n^»0«»OOi»0\0+I1 + +woii co^I N - -f* O -O O <->i 00W -O ~,T? ,SNN^MHlNO O^1-" >-"NNNt»t»Vl»NO(a-F»' OO-f* H*-U(d\Ot»NM00Ox.0 " OQ-I^M3 ->o eo-.ONWi CSJ O O^ OsVQ OO-vl -.^.fr-vl QWi I. O-E* - »mm" QN Q £ -fr £ £, 3 td Sea. I. Braganza Brandenburg Braflan Breda Bremen Breffcia BreflawBreftBrefte, Breffini Bridgetown BriegBrilBrinBrifacBriltolBrixen Bruges, Brugge BrunfwickBruffelsBncawBuchorn Buckereft Buckingham Buda Budoa BudweisBuenos Ayres Bugia BugieBulac Bulgar BurgosBurgowBuric.ByghofCabo Cabul' Cachao Cadiz,- Cales Caen Caffa Cagliari Cahors Cajana Caifurh Cairo grand ' GEOGRAPHY. 93 lat. Ion. 42 n 12 52" 33 S6n 47 52" 24 53n 28 45" 3i Sin 37 48 n H 52n 45 13 n 319 5'" 37 S2n 24 49" 36 48 n 27 5m 16 47" 31 5'" 22 52n 3° 5i" 24 48 n 29 47" 29 44 n 47 S2n i? 4811 40 42 n '4' 4911 35 36f 318 36n 24 22 n 56 3on 52 54" 7» 42 n •3 48 n 30 52n 26 53" 5° 45" 35 33" 89 23n 125 36 n 10 49 n 18 45" 57 39" 30 44" 20 6411 47 35 n 123 3on 5* lat. Ion. Calahorra 42 n 16 Calais 5in 21 Calcedon 41 n 59 Calenburg 52n 30 Calicut 1 1 n 95 Calmar 57" '36 Cambaia I2n 124 Cambra 5m 22 Cambridge S2n •9 Campagna 41 n 35 Campeachy i9n 285 Candahor 35" 87 Candia 3<;n 45 Candy 8n 99 Canterbury Sin 20 Canton 23 n 132 Cape of good hope 34 f 40 Cape bianco 21 n 0 Cape verd i5n 0 Capona 41 n 36 Cardonna 41 n 2] Carelfcroon ;6n 35 Carefen, Laffeen i6n 7Z Cargapol 6311 56 Carlftadt 45" 36 Carmona 37n '3 Carpentras 44" 25 Cars 42 n 64 Carthage 36" 29 Carthagena 37" l7 Carthagena ion 3oi Carwar i5n 93 Cafal, Cazal 45 n 28 Cafbin 3&n 68 Cam an 34" 7o Caffel sm 29 Caffimere 35" 95 Caftanowitz , 46 n 37 Caftel novo 43 n 4' Caftello branco 39" 28 Caftiglione 45" 3i Caftle town 54" '4 Caftro 43 n 296 Catanea 38n 35 Cataro 42 n 39 Catjierlou^e 53" 1 2 Caudcbec 49" >9 • C ?axa 4* S-D-S 9 9. 9.9rQ GGG GGGGGGGGGGGGGGGor> ^^ o oo o o^o-rKo-e-o^es-o-o-o 14 b; E) < < S 5 EC E" B^' B^ B^ """ B* ~ — ~T _¦ &* £r" B^ - B* a- ET* far" !"t* **-• <¦»* •— h— > Hi- hi- hr" *-i* w «¦** »¦»- m »-*. /* en «, i-n tu m O w .^t "fe -P- + ^ -!*¦ >* <** **! N 0*-f>- ¦+- +- HWUi •- CC<-" — -t^.^iv^tnf|i.kj11j1ji.v^wil^j^ji u lunfl*4. + WA*w " iCT" O N 5,N N10 o o>»(HM ow OWU »ow O MOOW O\ow B - » - o o* a " O -2 £ Co oo ov£ ^ 00<-« ~-> & .3 333333333333 3.33 3— ,33333333 3.3. 333333 3 33333333 33—,' O p o o o o o p o oo-o n -p o o o o on o n o o-o nnonnnnncn n-oo-o n oo n o o o o o £* a a a a s n | o g ft S S g.§ S S,g.§.o gggggggggfgog o'gggggg^gg^ 5- - ? tf ? 5 f S3^3-^33Bgn g-S o s. 3 3 » SJ p .5 ? p b 5f p 3 So m * " * f°<& * 5-g-S I - 3 g.gtfj'3 2. a - « 3 « «3 Sf g» ffi B ^ p *£=<• t>P * KJ t- P. «* Kl • ft ft o n> w 5 B>3 3 a n ona enti zna 5. p 3! r> B fo 3 3 3 P 3 P3 OWWIOW.O - N W.OJ VI N vl O Q. 00\O w CN»« o\00^O 00+. H* O Cl VO Oo 004- - OOVO ¦+>¦ O -5 o\\o o v» O N OsCS. w f\ 333 3. 3333 3 3- 33333 3.S— ,333333 3. 33 3333 3-3. 33333333p333B3t-_ h ^ p l^g £ '& p^g g vj v£ g ^j 3 Sed. I. GEOGRAPHY. Croia CronflatCroffenCnenca, Cuenza Culiacan CullembachCulmCufco Cuilrin Czernihow D. DabalBaca Dagno DalaborcDamafcus Dambea DamiettaDamvillars Dancala Daneburg Dantzig Darmftadt St. Davids Debrecin, Dubitza Delft Delly DelmonhurjlDendermondDenia Derbent Derpt DeflawDeventer Deux ponts Diarbeck * Die DieppeDiephok DigneDijon Dillenburg. DiHengenDinkenfpoil DiulDixmude.Doblin, Dobetin 'at. (¦zn 17" S2n 40 n 24 n Son 53" «3^ 52 n 52 n l8n 24 n 41 n 59" 33n 5" 3in 49" i7n 53" 54" 49 n I2n 47" 52n 28 n 53" Sin 39" 42 n S8n 52n S2n 49 u 47 n 45 n 50 n 53n44 n 47 n Sin 48 n 49 n 26 n 'son Ion. 4247 35 16 264 3'39 308 35 55 93 119 4> 335754 522553 3' 39 28 101 42 23 99 28 24 18 72 49 332524 6224 20 28 26 242830 3° 8923 45 Dol 48 n Dolk, Dole, 47 n Doltabad 20 n St. Domingo i8n Donawert 49 n Dort Sin Dovay Son Drefden Sin Dreux 49 n Drontheim 64 n Dublin S4n Duilburg Sin Dumfries 55" Dunbarton S6n Dundee S6n Dunkirk sm Durazzo, Brazzi, 42 n Durham 55 n Duffeldorp Sin E. Ecija 37 n Edam 5211 Edinburg 5611 Egra Son Ehenheim 48 n Ekrenford 55" Elbaffen 41 n Rlbin 54" Elcatif 2sn Elfimburg S§n Elfinore, Halfingore, S6n Elvas 39" Ely 52 n Emboli 42 n Emmeric 52n Emden 53" Euchuifen S2n Engets • Son Ens 48 n Enfilheim 48 n Ephefus 38n Eppingen 40 n Erfurt Sin Erivan 40 n Erkelens Son Erpach 49" Erzerum '40 n lat. 95 Ion. 16 25 95 309 3o 2423 33 20 3o 12 26 1?'4 '5 21 43 »5 26 12 25 16 32 27. 29 43 395933 32 10 18 46 25-272+ 273+27 47 2931 45 25 29 80! Eflingen, 96 GEO Eflingen, Elfmg, Efna Effen Eftella Eftremots EuEvora Exeter EyndhovenEyfenach F. Faenza Falkenburg Falmouth Famagufta Fano Farewel cape Faro FefantaFefanta de Bagota Ferden, Verden Ferentino FermoFerraraFerrolFelipourFez Finale FlenfburgFlorenceFlour. Flulhing, Uliffingen, Fogaras - Foix FokenFondi FontainbleauFonterabia Fontenoi Forbi St. Foy FragaFrancfort (Saxony) Francfort (Rhine) Frankendal Frafcati Fredqrica G R A lat. Ion 49 n 25 25n 5m 43 n 38n Son 39n 1 n 5in5in 44" 54" Son 35" 44" 5526 '5 ic 20 9 '5 2S 303236 56 . , 34 6on334 36 nj 11 36 n 269 4 53" 42 n 43"44" 43 n 27 n 33n 44" 55" 43"45" 1 n 47" 43" 27 n 41 n 48n 43" 5111 44" 44" 4m 2n on 49 n 4m n 305 29 343&33 12 98 '5 29 2932 23 23 48 21 139 36 22 16 23 33 18 18 34 27 28 34 207, PHY. Frederickftadt FredericfcdeFreiburg Freifengen Friburg FridburgFriedburg FrifechFritzler FuldFuligno FurnesFurftenfelt G. Gaeta Gallipoli Gandia GangeaGani GapGaveren GeertrudenburgGeldres Galenhaufen Gemund Geneva GengenbachGenoa, Genoua, GermenftieimGerumenha Gepping GevaliaGhent, Gant, GiawleGibraltar Gieflen Gingen GingeGiovenaz20 Giiace GirgeGironaGifors Giula GiufdendilGlandeves Glafcow lat. Ion. 56n •32 5011 3' 4811 2/ 49." 32 4611 37 48 n 3i Son 28 47" 35 5m 3° 5'" 29 42 n 34 5m 22 47 n 37 41 n 35 41 n 47 39" 17 41 n 67 1611 102 44" 26 5'" 23 52 n 24 S2n 26 Son 29 49" 3° 46 n 26 48 n 27 44" 29 49" 28 39" 29 48 n 3° 60 n 38 5in 23 60 n 29 36 n 11 5m 29 49" 3° 1 zn 99 41 n 37 39" 36 2on 52 42 n 22 49" 20 47" 43 4j" 41 43" 26 56 n '4 Glatz Seft. I. E O G R A P H Y. 97 Glatz Gloucefter Glogaw Gluckftadt _ Guefne, Guifen Goa. Goes Golconda GolnauGombronGor Gorcum Goree GoritiaGorlitz Goflar Gotha Gottenburg GottingenGottorp Gowde. Grace Gradifca Gran Granada (Spain) Granada (America) GrandentsGratzGrave Gravelin GravenecGravina Grenoble Grimborb Gripfwald GrodnoGroengenGroll GrofettoGrotlkaGrupenhagen GuadaiajaraGuadalaxaraGuadix, Acci, GualeorGuamanca lat. Ion. Jon 37 j2n 16 S'n 37 54n 29 5,2 n 38 16 n 94 5 44 n 45 45 n 39 3m 101 2h 97 49" z5 39" 42 6f 130 S8n 63 25 n 78 47" 49 Sin 3° 42 n 5° 45" 26 36 n 161 5J" 3Z S8n 34 32 n 52 27 n 94 54n 27 49 n 35 3 n 124 46 n 40 45 n 32 40 { 294 48 n 3» 47" 32 Joinville Ifenach, Eyfenach Ifernia Ifney Ifpahan JudenburgJudoign St. Julian JuliersJ urea K. Kaffa Kakenhaufen KalifliKalloKaminec Kanifca, Canifa Kargapol KaulburenKeiferbergKeiferfwert Kempten Kerman Kexholm KielKilia KimiKingfton (Jamaica) Kiovv KiattanKnyffenKola Komorra KongelKoninglburg, Re- 1 gimont, J Koningfgratz KownoKrai n flaw KrempenKrems KrimKufstain KuttenburgLabiau ' Ladenburg Ladogna lat. Ion. 49 n 26 S1" 31 42 n 36 48 n 30 33" 70 47" 35 Son 2 5 49, f 3>3 5i" 26 45" 28 47" 61 57" 47! 52n 38 48 n 42 48 n 47 47" 38, 62 n 60 48 n 3' 48 n 27 Sin 28 47" 3° 3°" 76 61 n 53 55" 3' 47 n 54 66 n 44 i8n 302 5.1 " 5Z 46 n 33 53" 45 69 n 54 49" 39 S8n 32 54 n 42 Son 36 55" 45 5i" 43 54" 29 48 n 36 48 n 61 48 n 32 Son 35 55" 42 50 n 29 41 n 36] Lagos rr,r'r*r*r*Kr*fc*tr*r*t*tHr*t^lr*t*t,«t-4r* W UPUPPPPPPPPPPPPPP P.P <^ 3 3 3 ^STyS Eb 5 _£-o o o jt* o 13 a s s r1 n = sks, or-a 5- n e n « EPW =$ 9- er 2 » Slo. » w to ^o-S-S-EtS-S-S'S-g-Ji-p'a o-o-SS 0 o p " O • g 3 o .oj HjiA^^wivfl + w^-f'Vi'f'^-f'^^ + ^-fWW^ Nwi^^^vflLn^-^^^i^vi^ NUi+,4.4.WviOl £-* .-vj Nij-+^ nnO On — *-*» N -^ O to 004* ON OO^vj O OV*J NO NO +¦ 00 O OO^ OOOa OSCOOnO »-vO O •- «-f Ow -4NN j; /-n 333333B33 3 3 3 3 3 3 4U)'<^ OOhW^^ mU w N ?- 4^ <*> <->i V* V0O4 NW WUl4-^ *- *<| w m nwuwWW to to M tO ?-< N •* 4>- N VQW 2 (4J; 2-j"l© ON O1 N^O-f- Qst^i ¦« vjn NQ Qv^i^-j — OnQ QO"^"^ Nvn^^Uiym m Q- Oj M4t»4 OO OQOn *¦ W^ 00 4>- NO In N OO r ^i' i_fi .f' -E- 4s- 4*- -fr- 4».Un wi/)^»inW4liw,^^^<-^^wi^*f,fl*'i -mi^i 'in w wi i^ wi ^ ui 4\ ^i t-n -fa. 4. -f*. <-o 4- 4. 4- tr" ^v„ .. ui n Ti *-" uo ^333333gBg3333 3333 m^.3 3333333333 3 3333333', »o - -' Oo <** c? V H M H n nU H N MU>^ <*<^^ H H 1JUJ^WW-MN».W OWNIJU»4-w4>.WKi«M U» W NW M n n N J^ & «~ ocNQUn o^ O ^ — On -¦ m 0\* Ob^i 4» O *o «-n oo oovo VO f VO too GEOGRAPHY. LonguevilleLpngwy, Longovy Loo Lorca Loredo loretto, Lauritana Loudun Lovitz Lovrebrander Lourde Louvain, Lovain Louvo Loxa Loya, Loja, Loxai Lubec Luben (Bohemia) Luben (Saxony) Lublin Lubow, Lublaw LucLucca, Luca Lucern LucingannoLugano L'Jgo LundenLunenburg Luneville Lure La (Ton Lufne, Lucko ' Lutzen Luxemburg Luzara, Luzzara Lyons M. MacerataMackeran Madenburg, Man- \ heim j Madre de pop a Madrid. MadrigalMaefeyck Maeitricht ; Magadoxa Magdeburg .MagHano lat. lion. Son 19 5011 25 Szn 26 38n 16 45 n 33 43" 35 47 n 26 5in 40 25 n 88 43" ¦18 5m 24 •5" 121 ,5f 301 17* '.'4 54" yc 5m 36 32n 31 em 43 49 n 4' 45" 25 44" 3' 44" 27 43" 33 46 n 28 43" 9 j6n 34 54 n 3° 4911 26 48 n 36 46 n '7 51 a 45 51 11 32 500 26 45" 3i 4O n 24 43" 35 26 h 60 49 n 28 1.1 n 302 40 n •4 1 11 3°3 51 n- 2 5 5in z5. 2 n 61' S2^ 32' 49 r-' 34 MajorcaMalaga Malda Malrnoe St. Malo Malta, Medina Malvafia ManrefaMansMansfield Mante Mantua, Mantoua MaracaiboMaranaMarans St. Marco MardikeMargeritheim Marge-reftMarienburg (Germ.) Marienburg(Poland) Marienburg, Sweden Marmande Maiognia MarpurgMarfala Marfalquieur . MarfeilsMarfico Martaban Martha MartocanoMafcon, Mazon MaffaMaflerano MafulipatanMaetera Materan Maubeuge Maulcon St. Maupa MayenneMazaganMazara Meaco MeauxMeccaMechlin lat. Ion. 39" 20 32" '3 25 n 108 .5 5" 33 48n 16 35" 34 36" 49 4m 20 48 n . 18 52n 3' 49 " 21 45" 3' iin 308 4> n 46 46 n '7 39" 39 51 n 22 49 n 32 47" 49 500 24 54" 40 S9 n 34 44 n 23 42 n 5' 5 in z9 37" 34 36n 18 43 n 25 40 n 38 i7n 99 i2n 3°3 39" 3« 46 n 24 44" 31 45 n 28 1.6 n 101 41 n 38 ,7f 130 Son 23 43" 17 3Sn 44 48 n '! 33" 35" 34 36 n ,;6 49" 23 21 n 64 Sin 24] Me- Seft. I. MedenblicMedina celi Medina talmari Medina fidonia Mednick Meiflen Meldorp Meliapour MelindaMelicoMelle Metros, Meurs Melum, Melun Memel Memmingen Menchou Menen, Menin Mentz Meppen Mequinez Meran, Moran MergentheimMerida Mero Merfburg Merfperg, Merfpurg MefchedMefeen Mefember MeflinaMetlingMetz MeursMiddleburg Milan Milland Mindelheim Mihden Minflc, Minflci Miranda de douro Miranda de ebrg MirandolaMirepoix Mittau Moco, Mocho-. ModenaModica Modoa. G E O G lat. 'Ion. 53« 25 4m '5 25 ri 61 36" 11 5611 44 Sin 33 54" 29 •3" 100 3< 39 28 n 38 52 n 28 56 n 16 48 n 22 56 n 43 48'n 3° 49" 24 Sin 23 49" 28 53" 27 34" 8 47" 31 49" 32 39" 11 i7n 116 Sin i32 urg 47." 29 36 n 77 66 n 66 44" 53 3»n 37 46 n .36 49" 26 5i" 26 5'ti 23 45 n 29 44" 22 48n 3° 52 a 2 a 54" 49 » . i j-zn 11 f3" H . !•?" 32 i 13" 20 "6n 4+ 3 n 65 ' K" 3' 6n 36 >5" 46 R A« P H Y. 101 1-1 la.t. Ion. Modrufch.Madrufcl '45r 37 Mohilow, Mogilof 54t 54 Mofette 41 r 39 Mola 41 1 37 Molina 41 r 16 . Molife 42 r 36 MoKheim 48 r 21 Monaco 44" 27 Moncaller, Monca!e,44r 27 Moncon 42 n 24 Monddnnedo 43" , 10 Mondovi 44 n 27 Monluffon 46 n 22 Monopoli 41 11 38 Mens, Bergen , Son 23 Montallo 43" 35 Montaugis 48 u 20 MontauBan 44" 19 Montbelliard 47" 26 Monthrifon 45 n 23 Monte verde 4m 36 Montpelier,Mom- \ pelier / 43 n 24 Montreal (Sicily) 38n 35 Montreal (Spain) 43" 16 Morlaix 48 n 14 Morocco 32 a 9 Mortaigne 49 n '9 Mortara, Montara 45 n 29 Mofambique 15I 42 Mofbaeh, Mofpach 49" 28 Mofcow 55" 6; Mofpurb 48 n 32 Moulins 46 n 23 Mouful, Moful 36n 43 Mucyflaw 54" 53 Mulhaufen 48 n 27 Mnlton 30 n 92 Mungats, MunkatS' 49 n 42 Munick, Munchen f8n 32 Munfter ' ;2n 27 ' Munfterburg ;m 36, Murcia j8n >7 Muro, l .in 37 Muftagar- 36 n 26 Muxara, Muxasra 13711 16 Muydcn. k 2fl 25 ' M 102 GEOGRAPHY. N. 1 at. ilon. Naerden ;2" 24 Najara fc2" «5 Naclivan 39" 65 Namur 5011 25 Nancy 49" 26 NangafaquiNankin 32 n 151 32 n 129 Wants, Condivineurn 47" 17 Naples 41 n 3» Napoli 36" "49 Narbonne 43" 22 Nardo 4m 39 Narenza, Narenta 43" 39 Narni 42 n 34 Narfinga '5" 98 Narva 59." 48 Naffaw Son 27 Nata 9" 296 Navereins' 43" l7 Naverino 35" 46 Naumburg 51" 32 Negapatun 11 n 99 Negrais 17 n 112 Negropont 38n 48 Nemours 48 n 2.1 Nepi 42n 34 Neuenburg, Nuberg 48 n 31 Nevers 41 n 22 Neufchattel 47" 26 Neuftadt 47" 33 Newbu'rg 49" 3' Newcaflle 55-" 16, Newmark 47" 45 Newport 5rn 21 Nice, Nizza 43" 27 Nicomedia 41 n 5« Nicopin 59 n 37 NicopoKs, Nigepoli 45." 47 Nimeguen, Nim-'| j2n 25 megen J Ninove 5.7" 23 Niort 4on 18 Nifmes, Nij".e& 43" 24 Nifna 56n ,65 Niffa 43" 43 Nitracht, Neytra 49"51 1 39z4 Nixabour, Nafabor Nocera Nola, Nole Nombre de dios NonaNorcia, Nurfia Norden Norkepin Northeim Northanfen Norwich Noto NovaraNovigrad Novogrod, Weliki Noyon NurenburgNuys NyburgNyenburgNyflot O. OberwefelOchridaOchenfurf, Oczacow Odenfee OdorOdenburg (Weftph.) Oldenburg (Den 1 mark) j Oleron Olika Olinda OliteOlivenzaOlmutz Olympea, Longinico St. Omers OmmenburgOnegliaOppelen Oppido OranOrange,Oratavia GrbitellQ lat 36m 43" 4m ion 45" 43" S4n S8n Sin 5anS3" 36n 45 n 28: Ion 3* 343& 300 37 26 3°3<> 3° 2037 45 n 5&n 49 n 49 n 5m55?53" 63 n Son 41 n 49" 47" 56n Son53"54" 43" 5'" 45 5522 26 3029 5° 27 43 3»55 3°38 27 31 '7 47 8(343 42 n 3»n49 n 38n Sin 51 n 44" 50 n 38n 37° 44 n zS.ri420 16 10 37 41 22 28 2838 38 18 24 1 3» Ordunna Seft. I. GEOGRAPHY. 103 lat. 43 « 4zn 3»n 39" zon 48 n 30 n 2Qf 42 n 43" 5zn Sin 41 n 44" 40 n 5m 4.3" 52n OrdunnaOircnfe Origuella OriftagniOrixa Orleans Orleans new OropezaOrta, Horta Orvietto OfnabrugOftendOftuni Qfwego Otranto OudenardOviedoOxford P. Padang PaderbornPadua, Padoua PaitaPalamboang Palamos FalancaPalazzuoli Palencia Palermo Paleflrina, Prasnefte, Palma PalmyraPamiers Pampeluna Panama i Panay Pancale Papa Papenheim Pacaiba Parenzo Paris Parma Parnaw PaffanPaftoPatan Patqhuca, Patioquej|*i " Ion H 9 16 29 >°5 20 2312 34 3327 2240 300 40 2 121 if 5zn 45 n s? 8f 41 n 48 n 30" 42 n 37" 42 n 38 n 33" 43"43" ion 11 n 44" 47 n 49" 7" 45" 49 n 44 n S-8n 48 n 2 n 27 n u 28 32 298 •34 21 40 36 «3 3534 8 5920 16 296 129 2737 32 323 35 22 s 34 301 109 1275 JH 1 at. Ion Patna i6n °5 47 Patras ;8n Patti ;8n 36 Pan r3n 18 Pavia 45 n 29 Paul, Pol de leon 49" '5 St. Paul 24 n 328 Paz i8n 312 Pazzi 4m 52 Pedir 5" 114 Pegu i7n 117 Pekin 40 n 121 Pella 4m 43 Pelufium S'o 51 Peniche 39-n 9 Pennon de velez 35" >4 Percaflaw Son 53 Perga 39" 4» Pergamus 39" 47 Pengux 45"62 n »9 Permavelec 81 Peronne 50 n 24 Perpignan 43" 22 Perfapolis 31 n 74 Perthamboy 41 n 300 Perugia 43" 34 Pefaro 44" 34 Pefcara 42 n 36 Fell 48 n 40 Peterborough 52n 18 Peterlburg Son 56 Feterwaradin 46 n 42 Fetricow Sin 4Q Pettaw, Petaw 47" 37 Pettipoly • 7n loo Pfortlheim 49 n 28 Pfulendorf 46 n 28 Phalilburg 49" 27 Pharnacia 41 n 58 Pharfalus, Pharfa, 39" 43 Philadelphia, (Afia) 38" 49 Philadelphia (A- 1 merica) ; Philippi 4.1U 304 41 B 45 Philippopoli 42 r 45 Philipfburg 49 r 1 28 Placenza, '451 1 30 Picnia. 43 ' '1 33 Pignewl 104 GEOGRAPHY. • l at. Ion. lat. lion. Pignerol t 14" 2'/ Prevefa # 38" 44 n8 Pilau -5" 41 Priaman 1 1 Pilfen ;on 33 Prifferan 43 " 42 Pilfno, Pilzow jon 4C ¦Priftina 43 » 4' Pinfk ;2 n 47 Procupia 14" 41 Piombfrio 42 n 32 Prom ign 114 Pipley 21 n 106 Ptolomais 3zn 56 Pifa 43 "1 31 Puy envelay 45 n 24 Pifca 14 f|302 Puzziiolo, Pozzuolo ^n 36 Pifcataway 44" 3o8 CL Placentia 43 n| '5 Quakenburg 52n 27 Plata 23 f 3d Quebeck 47 n 304 Platea 38n 46 Queda, Keda 7" IIS Plawen Son 32 Quedlinburn 52 n 3' Plefcow 57" 5C Queflin, Queviling 26'n 129 Pluviers ' 48 n 21 Que'fnpy Son 23 Poggio imperial 43" 32 Queyang 27 n 126 Foi&iers 46 n 19 Quillebeuf 49" 23 Pola 45" 35 Quiloa 10 f 59 Policaftro 4°» 37. Quimper . 48 n 14 Polignano 41 n 39 Quimperlay 47 n »5 Polockz 55" .51 Quinque ecclefias 46 n 40 PondicheTry 12 n 100 St. Quintin 50n 23 Pont de efprit 44 n! 24 Quito 13 f 300 Ponta moufon 49" 26 '. R. • Popa madre 10 n 301 Raab 48n 38 Popayan ¦ 3n 302 Rachelfburg 47 n 37 Porentru 47" 27 Ragufa 43 n 40 Portalegre 39" IC Rajamahal 24n 106 Pbrtrentru 47" 27 Rajapour 22n 97 Port 1 'orient 48n 16 Rain 49 n 3i Porto 4m 9 Rakaw 36 n 59 Porto bello 9" 296 Rakonic 50 n 33 Porto hercole 42n 32 Rambervillers 48 n 26 Port St. Mary '37 n 1 1 Ramillies 51" 25 Porto rico 1811 3J3 Ranchira izn 306 Port royal I7n 301 Ranetz 55 n 43 Pqfega 45" 39 Rantzow ¦ 5411 30 Pofna ;2n 37 Raorconda I7n 99 Potenza 40 n 37 Raperfwill 47* 28 Potofi 22f 3"1 Razeburg 60 r 44 Pourfelui 28n 1 2C Ratipor 5or 3« Prabat 1 6 n 121 Ratifbon 49r 32 Prague s 150 n 34 Ratzeburg 54 r 3i Precopia, Orciup 44" 44 Rava, Rawa 52r 40 Premiflaw 49 n 42 Ravello 1 41 r 36 Prelburg , 48 r 3* I Ravenna 44 r 1, 33 Ravenf- Sea. I. GEOGRAPHY. Ilat. |lon. lat. 1 Ion. Raven/berg (Weftp.) ;zn 27 Rofienne 55" 4'' Ravenfberg (Swab.) . (.7 n 29 Roffanno (.on 38 Real, Chiapa, I7n 281 Roftock 54" 33 Realejo 1211 3°7 Roftow, Roftof 57" 60 Rebuick 45 n 47 Rotenburg 49" 31 Reccandi 43 n 36 Rottenburg 53" 29 Rees 52n 25 Rotterdam 5zn 24 Reggio 44 n 31 Rotweil 48 n 26 Regio 38n 37 Roven, Roan • 49" 21 Rennes 48 n 1; Rovigo 45 n 3Z Refcht 38n 7J Roye, Roya 49 n 23 Rethel 49 n 24 RudelTwert 46 n 36 Rethigen 48 n 29 Rudifto 42 n 53 Revel 59n 46 Ruffacu 48 n 27 Reutlingen 48 n 29 Ruvo 4m 38 Rezan 55n 64 Rzeczyca 53n 51 Rheines 49 n 24 S. Rhineberg Sin 26 Sabionetta 45 n 31 Rhinfelden A7n 27 Saccai 36n '55 Rhodes 36n 40 Saderafapaton I2n 100 Richelieu 47 n 18 Sagan 52n 35 Ries 43 n 25 Said 27 n 52 Rieiix 43 n 19 - Saintes 4611 '7 Riga 57" 45 Salamanca (Spain) 41 n 11 Rimini 44" 34 Salamanca, America, i'7n 285 Rio, Riom 45 n 23 Salamis 47" 44 Ripatranfone 43 n 35 Salenkamen 45" 42 Ripen 56'n 29 Salem 42 n 308 Rifano, Rifne 43" 42 Salerno 41 n 37 Riva 46 n 31 Salins 47" ie Rivadec 43" 10 Salifbury Sin 16 Rivoli 45" 27 Salle 34" n Roanne 46 n 23 Salm 48 n 27 Rochelle 4611 17 Salona 44n 39 r Rochefter Sin i9 Salonichi 41 n 46 Rochfort 46 n 17 Salfes 43" 20 Rocroix Son 24 Salfona 42 n '9 Rodez, Rodes 44" 22 Saltfburg 48 n 33 Roermond Sin 25 Saluzzo 45 n 27 Roleduc 51" 26 Samafcand 40 n 86 Romans 45" 25 Samaria 33" 58 Rome 42 n 34 Sandec, Sandecz 49 n 4i Ronciglione 41 n 35 Saridomir Son 42 Ronda 36n ii Sangueffa 47" 17 Rofchid, Rofchild 56n 32 Santa cruz 23 n 293 Rofes 42 n 22 Santa maria 8n 298 Rofetto 3m 6l | Santafe 36n 269 Santa 105 o crt OW «°^r — "~ - 5 S^ g 5 5 g g"g 3 3*g 3 3ffBgt 3 3 3 g^B 3^3 3 333 3 3 3 3 3 r_ Q > K V ^. _e. 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S— o Vq^-~i r- P 5 o ^00 ^cr" 2 ^ » " 3 o X -r— lj Oj Oj v^j tj sj^ ^ o~» wt ^, ^n ^n _|_ — 1 -^.n ^1 s_ri s^n s^i^T^IriivT^ToTv^ToTo-i^ OV^> s7is/i £7* to -vl -v) v^j c^^ yQ w^_, ^ „_ q .^ >,£, «\o-vj o^4 W h- u «U,>-v4 «\O^J — VO "-^ -wMsflWi^'H 0^ 0\**J -O £ 32333^3-33 BBBB33B33BB3BBB3BB B3B3BB3333B3 3 3 3 3 3* O.Oj W tO OO 0\ tO N ON1-^ 4^. 4* OJ OO OO tO OO; On OJ OO N On 4»- n« 10 4* tO OO 4 Vj 4 m to to uo to Q-^0^ QQ4- " Uj — ui ?°~^ ^ O ^' 4> 4 O "O 4*>>-> 4- - ,OQh Q oj ^-j vq C^4* --J to C\^vj Q\ O ^O ^ 4- *-n o o no GEOGRAPHY. ZeitzZell ZemlinZeng, Segua ZemobizZenonis Ze'rpft ZersZeverinum Ziericze ZigetZiltzZintzheim Ziffera Zittaw ZnaimZolaritzoZollernZolnockZornajam Zug Zunck Zutphen Zweybrilck ZwickawZwifalen Zwingenburg Zvvol, Swol, Zwonik Zytomierz at. lion. Xuahbgrod 45 » 3* Y. Yenhe 45 A z5 Yefd 33» 74 Ylft 53 n 2$ Yoahgfu 3ini34. York 54n 17 New York 41 n 304 Ypres 5i n 23 Yfendic Sin 22 Yfondun 47 n 22 Yvica 39 n 19 Yvoix 5on 25 Z. Zabern, Saburn 49 n 28 Zabern elfas 48 n 27 Zabes 47.n 45 Zabola 47 n 47 Zagrab 46 n 37 Zamora 4211 12 Zamolki 5on 44 Zamora 4m 26 Zara 44 n 37 Zargona 45 n 46 Zarnaw 5jn 40 Zaflow ' 5on 5° Zator 5on 46 Zawolocze 57" 51 Zeby 57 " 3° Zeigenheim (5111 29 lat. Ion. 53n 48 n 45 n 45 n 48 n 5zn 45 n 46 n 5m46 n 44 n 49 n 57n 5'n ' 48 n 64. n 48 n 47 n 5&n 46 s 47 " 52s 49n 5on 48 rf 49 n S3n 44 n Sin 32 3°42 37 44 63 3o 48 45 25 52 4 2954 34 36' 60 28 41 6028 2926 273229 28 25 42 521 SECT. . > '- \ [ III ] Fig. SECT. II. The ufe of the Terreftrial Globe, in the Solution of Geographical Problems. THE parts of the terreftrial Globe are much the fame as in the celeftial ; thefe are the brazen meridian, the wooden horizon, the qua drant of altitude, the horary circle, the femicircle of pofition, and the mariners compafs. The eaft ern edge of the brazen meridian is divided into degrees, and reprefents the true meridian. The upper fide of the wooded horizon is divided into degrees, and reprefents the true horizon. The quadrant of altitude is a thin brafs plate, fixt to the meridian at any point of it required, by a nut and a fcrew. That edge of it which is divided into de grees is called the fiducial edge, and properly re prefents the correfponding great circle. The ho rary circle is divided into 24 hours, making twice 1 2 -, and 1 2 at noon is at the upper part of the me ridian, and 1 2 at night at the oppofite fide ; the morning hours to the eaft, the evening ones to the weft ; the end of the axis reprefenting the pole, carries round the hand, which fhews the hour. The femicircle of pofition has its extremities fixt to the north and fouth points; about which it moves freely, and may be fet to any pofition. The mariners compafs is placed on the foot of the globe, . and ferves for fetting the globe in the fame pofi tion as the heavens. The circles drawn on the terreftrial globe are, the equinoctial, and the parallels of latitude to every 10 degrees, the two tropics, and the two polar circles ; alfo the meridians at every 10 de grees diftance upon the equinoctial, beginning at the ii2 GEOGRAPHY. Fig. the firft meridian, which is or ought to be, that pafiing thro' the ifland' of Ferro, one of the Ca naries. The ecliptic is alfo drawn on the terreftri al Globe, tho' it belongs properly to the celeftial -, and is put upon the terreftrial for the fake of re- folving fome problems of the fphere. Alfo rumb lines are drawn upon fome vacant place of the terreftrial globe, fhewing the 32 winds or points of the compafs. The furface of the terreftrial Globe differs from that of the celeftial, in this particular ; that upon . the celeftial are deferibed the feveral conftellations of the heavens, with the true places of all the flat's. But upon the terreftrial, the feveral countries of the world are deferibed ; all lands and feas ad joining to them ; all iflands, capes, bays,, moun tains, rivers, towns, and other things upon the earth are delineated in their proper fituation, juft as the places themfelves, which they reprefent, are upon the earth. So that this globe or little earth will be exactly fimilar to the great globe of the earth ; and all their correfpondent parts will be in a fimilar pofition. And if this terreftrial globe be turned about its axis, it will likewife have a fimilar motion with the earth itfelf. And fo there is an exact agreement between, the natural pofition of the feveral parts of the earth and fea, and their artificial reprefentation upon the terreftrial globe -, by which means a great many, problems of the fphere may be very eafily refolved. I fhall here lay down fuch as belong properly to the terreftrial globe, referring the reft to the celeftial globe,, de livered in Sect. III. of the Aftronomy. PROB. Sea. II. u5e OF THE GLOBE.' 113 Fig. P R O B. I. To find the latitude', and longitude of a given place ' upon the globe. Turn the globe round its axis, till you bring the place under the brafs meridian, there let it reft. Then the arch of the meridian, comprehended be tween the given place and the equator, is the lati tude ; which is north, if it lie towards the north pole -, or fouth, if towards the fouth pole. And the arch of the equator contained between the firft meridian and the brafs meridian, is the longitude. If you would have this longitude in hours, bring the firft meridian under the brafs meridian, and fet the hour index to 12, then turn the globe towards the weft, till your place come under the brafs meri dian ; and then the index will fhew the hours corre- fponding to the longitude, or the longitude in time. PROB. II. To find upon the globe, any place whofie latitude and longitude is given. - Reckon the longitude 'along the equator, and bring that point of it to the brazen meridian ; then count the latitude along the brafs meridian north wards or fouthwards, and there you will find the place required, . for either north or fouth latitude. PROB. Ill, The latitude- of any place being given ; to find all thofe places that have the fame latitude. Bring the place to the brazen meridian, make a mark there ; then turn the globe about, and all I thofe ii4 GEOGRAPHY. Fig. thofe places that pafs under that mark, have the fame latitude as the place given. Likewife all. thofe places may be found that have the fame longitude as a given place; for if that place be brought to the meridian, all thofe places then lying under the meridian have the fame longi tude. PROB. IV. • To find the difference of latitude, and difference of longitude between any two given places. For the difference of latitude. Bring each of the. places to the brafs meridian, and mark the points of interfedion -, then count the degrees of the meridian contained between thefe marks ; and you have the difference of latitude. ¦ For the difference of longitude. Bring each of the places to the brafs meridian, and mark the two points of the equinoctial cut by the meridian ; count the degrees of the equinoctial between the marks, and that is the diff. longitude. Or for the cliff, longitude in time ; bring one place to the me ridian, and fet the hour circle to i 2 ; then bring the other place to the meridian ; and the number of hours between 12 and where the hand (lands, is the difr". longitude in time. PROB. V. To find the diftance of two places upon the globe. . - Apply the quadrant of altitude, or- brafs femi circle to the. globe, to. pafs thro' the -two places. Then the number of .degrees between them, count ed upon the quadrant or femicircle is the diftance. Or take the diftance between the two places with ?. pair of compafTes,\and apply it to the equator, and Sedt. II. USE OF THE GLOBE. 115 and it will fhew how many degrees are contained Fig. in that diftance. If you would know the diftance in miles, mul tiply the degrees by 60, gives the minutes or geo graphical miles. PROB. VI. To reclify the globe. Set the globe upon a horizontal plane, fo that the needle may point towards the north, allowing for the variation., Raife your elevated pole fo many degrees above the horizon as is equal to the latitude ; moving the meridian up and down, till that number of degrees touches the horizon, there let it reft. Then the globe is rectified for the latitude. Count the degrees of latitude from the equator towards the elevated pole, and where it ends, is the zenith. To this point fix the quadrant of alti tude, fo as the graduated edge of it may touch that point. Bring the fun's place in the ecliptic to the me ridian, and then fet the hour index to 12 at noon ; ' fo the globe is rectified for the fun's place. And if your place be brought to the brafs meridian, then the meridian of the globe will coincide with the meridian of the place, and the globe will be in the fame fituation as the earth at noon. PROB. VII. To find how any given place bears from your place, upon a great circle. Having rectified the globe for your latitude, and fixed the quadrant of altitude in the zenith, by the laft Prob. Turn the globe about till your place come to the brafs meridian, there ftop it ; then turn the I 2 quadrant 1 16 GEOGRAPHY. Fig. quadrant of altitude about, till it pafs thro' the other place, let it reft there ; then count the nutn-r ber of degrees upon the horizon intercepted be tween the brafs meridian and the quadrant of alj titude, and that will be the angle of pofition. By this means you may know how all the coun tries round about are fituated in refped of your dwelling. And alfo what places bear upon any point of the compafs you defire. PROB. VIII. To find the antaci, periaci, and antipodes to any given-place. Bring the .given place to the meridian, and find its latitude •, reckon the fame' number of degrees from the equator to the contrary pole, and where the reckoning ends is the antseci. The given place being ftill at the meridian, fet the hour index to 12 at noon, then turn the globe about till the index points to 12 at night ; then the place in the fame parallel which is under the me ridian, is the place of the periasci. The. globe having this laft fituation, that place which is in the oppofite parallel, and under the meridian, is the place of the antipodes. PROB. IX. , To find what a clock it is at any given place ; when it is noon, or any other hour at your placd Redify the globe, and bring the place of your habitation to the brazen meridian, and fet the in dex to 1 2 at noon, if it be noon at your place, or elfe to the given hour ; then turn the globe about, till .the given place comes to the brafs meridian.; and Sed.II. USE OF THE GLOBE. 117 and then the index will point to the given hour atFig. that place. PROB. X. The hour of the day being given at your place ; ta find the place or places where it is noon, or any other hour propofed. Redify the globe, and bring your place to the meridian, and then letting it reft, fet the index to the hour given -, then turn the globe round, till the index points to 12 at noon,, if it is noon at the other place, or elfe to the hour given at that place ; and then all the places lying under the brafs me ridian are thefe required ; or where they reckon that time of the day. And thus all thofe places are found, . where the fun is in their meridian at a given hour, for it is then j 2 a clock there. * PROB. XL The day and hour being given.; to find the place of the earth where the fun is vertical at that time^ Having redified the globe for your place ; get the fun's place for that time, and find it in ,the • ecliptic, and bring it to the brazen meridian, and mark the place of the meridian with chalk.. Then turn the globe about till your, place comes under the meridian,, there flop it, and fet the index to the given hour. Then turn the globe about till the index point to 1 2 at noon, there ftop it, and ob ferve what place is under the mark on the brazer* meridian, for that is the place where the fun is ver tical. You may obferve, as the globe turns round,, that the fun will be vertical fucceffively to all thefe places that lie in that parallel. I g PROE. n8 G E OG R A P H Y. Fig. PROB. XII. %, To find all the places where the fun is in the hori zon riflng or fetting, at a given hour. The globe being redified, bring the fun's place in thcecliptic to the brazen meridian, and mark the place on the meridian. Turn the globe about till your place comes under the meridian, then fet the index to the given hour ; then turning the globe till the index points to 12 at noon; ob ferve what place is under the mark. Bring this ' place into the zenith, then all places at the horizon will be the places where the fun is riflng or fetting, the rifing being to the weft fide, and the fetting to to the eaft. PROB. XIII. To find what a clock it is by the globe,^ when the funfhines. Redify the globe to the latitude of your, place, .and fet it in a true pofition by the compafs; then letting the fun's light fall upon it, obferve oh the daritfide, that point of the equinodial that fepa» . rates the light from the dark part ; reckon how - many times 1 5 degrees is contained between that point and the horizon,- and To many hours it wants of 12 a clock, if the dark fide is on the weft ; or it is fo many hours paft 12, if the dark fide be eaft- ward. PROB: XIV. '_ To find in what places it is day, and in what places night, at a given time. This is done by Prob. XII. For all thofe places that are above the horizon have day, and thofe under. Sed. II. USE OF THE GLOBE. «9 under it have night. And all thofe places under Fig. ¦ the meridian above the horizon have their noon, and all places under the meridian below the hori zon have their midnight. Otherwife. If the fun fhines it may be done thus ; redify the globe according to the latitude, and fet it due north and fouth by the compafs. Then bring your place to the meridian, and in this pofition let the fun fhine upon it, and it will be day to all the illu minated part, and night time to the dark part of the globe ; and fun rife or fun fet, at the circle fe- parating the light from the dark part. For the globe is now in a like pofition as the earth, and therefore it is illuminated in the very fame, manner that the earth itfelf is illuminated. PROB. XV. To find the time of fun rifling and fetting. Redify the globe according to the latitude, and bring the fun's place to the brafs meridian, and then fet the hour index to 12 at noon. Then turn the globe about till the fun's place, comes to the eaftern fide of the horizon ; then the index will, fhew the time of fun riling; and if the fun's place, by turning the globe, be brought to the weftern part of the horizon, the index will fhew the time of fun fet. Hence if the time of fun fet be doubled it gives the length of the day. And if the time of fun rife be doubled it gives the length of the nights I 4 PROB. 120 G E O G R A P H Y. Fig. PROB. XVI. To find the time of day break, or the end of twi light, on a given day. Redify the globe to the latitude, and bring the fun's place to the brafs, meridian ; and fet the hour index to 12 a clock at noon. Then turn the globe about towards the eaft, till the fun's place be 18 deg. below the horizon, or the oppofite point of the ecliptic 1 8 deg. above the horizon, and then it is day break, and the index fhews the time. This diftance of 18 deg. maybe meafured with a pair, of , compaffes opened to tha/ diftance, or by the divi sions upon Ae quadrant of altitude. Again, turn the globe weftward till the fun's place in the eclip tic, be 18 deg. below the horizon* and then the hour index will fhew the time when twilight ends. PROB. XVII. To find what countries have no dark night at any time propofed. Bring the fun's place at the given time to. the brafs meridian, and count his declination upon the meridian, which is always towards the elevated pole. Then elevate the globe fo, that the height of the equinodial above the horizon may be equal to the declination -f- 1 8 degrees. Then reckon what height the pole is above the horizon, which will be the latitude, where twilight begins to be perpetual. Therefore in all greater latitudes, fuch people as dwell tjiere, will have no dark night, but twilight. Or the morning twilight will begin before the even ing twilight ends. PROB. Sed.II. USE OF THE GLOBE. 121 ' •'¦> Fig. PROB. XVIII. To find [in what latitude the fun begins to Jhine con- ftantly without fetting, at a given time of the year. And alfo at what latitude in the oppofite hemifphere, he begins to be totally abfent. Bring the fun's place in the ecliptic to the brafs meridian, and count his declination upon the me ridian, and that will be the complement of the la titude. Therefore reckon the fame number of degrees from the pole towards the equator, upon the meridian, and mark the place. Then turn the globe round, and all the places palling under this mark, are thofe in which the fun begins to fhine conftantly without fetting, for that time. Mark the oppofite point of the brazen meridian, and .turning the globe round, all the places in that pa rallel under the mark are thofe where total darknefs begins ; for there the fun begins on that day to be quite abfent, or totally to difappear. PROB. XIX. To find the length of the longeft day or night in any given place. Rectify the globe to the latitude of the place, and mark any point in the tropic of cancer for north latitude, or in the tropic of Capricorn for fouth latitude. Then bring that mark to the eaft fide of the horizon, and fet the index of the ho rary circle to 1 2 a clock at noon. Turn the globe about till the mark in the tropic comes to the weft em fide of the horizon. Then obferve upon the hour circle, how many hours the hand or index has gone thro' from 12 at noon ; and that is the length of the longeft day, as alfo of the longeft night. And 122 G E O G R A P H Y. Fig. And this fubtraded from 24, leaves the length of the fhorteft night or fhorteft day, at the1 given place. PROB. XX. The latitude of the place in the frigid zone being given; to find how many days the fun Jhines eonftantly upon it. Redify the globe according to the latitude; then turn the glpbe about, and obferve what point of the firft quadrant of the ecliptic falls upon the interfedion of the horizon and meridian to the north, fuppofing the place lies north latitude. Find that point of the ecliptic upon the wooden horizon ; and againft it ftands the day of the month when the fun is in it. And then the longeft day begins at that place. Again, turn the globe about till fome point of the fecond quadrant of the ecliptic falls upon the north interfedion of the meridian and horizon as before ; obferve what point of the ecliptic that is, and find it on the wooden horizon, and againft it ftands the day of the month when the fun is there. And at that time the longeft day at that place, ends." Then count the number of natural days contain ed between the beginning and ending of this long eft day -, and you will have the length of- it. And ' the day firft found is the day of his appearance ; and the latter, the day of his departure. And the number of naturaldays in the longeft night is the fame; and begins and ends at the fame diftance from the winter folftice, as the day does from the fummer one. Other wife thus. The globe being elevated according to the lati tude,' bring the folfticial point to the north part oC the meridian. Then reckon the complement of the Seft." II. USE OF THE GLOBE. 123 the latitude from each fide of the equator upon the Fig. meridian, and mark the places where the reckoning ends. Then turn the globe carefully about, and obferve what two degrees of the ecliptic comes un der the northern mark ; then the intercepted arch of the ecliptic, reduced to time, gives the num ber of days that the fun eonftantly fhines, or re mains above the horizon of the place ; which is the length of the longeft day there. And the op pofite arch equal to it, gives the number of days he is abfent, or the length of the night. Or ob- ferving the points of the ecliptic pairing under the fouthern mark, the contained arch reduced to time, gives the length of the night. The elevation. of the pole Continuing the fame, obferve the two degrees of the ecliptic that come under the north mark, and find them on the wood en horizon, and you will have the days that the fun is in thefe points, which days are thofe of the firft and laft appearance of the fun. And if the points of the ecliptic, that come under the fouth mark, be obferved, you will find on the wooden horizon the correfpondent days when the long nights be gins and ends in that place; SECT. > I JH ] Fig. : :.:,;'. SECT. IN. The Theory of Navigation, Spherical, and Spheroidical. With the Solution of fome uncommon Cafes. The ac~ tual Solution in Numbers, of the common Cafes of S ailing y by the Sphe roid ; and likewife of thefe in parallel Sailing. Rules for correEling a Reckoning. Of taking Altitudes. HAVING formerly writ a praaical book of Navigation, calculated entirely for the ufe of fuch as frequent the fea, I fhall here only ex- traa from it, fo> much of the theory as is proper for a courfe of mathematics. To which J fhall add fome things that were omitted there, or but flightly touched, as being lefs neceffary for failors; but ferve for compleating the theory. But as to the praaical part, which is the moft neceffary thing- for feamen, I fhall meddle very little with it here. Yet if any perfon wants farther fatisfac- tion in that point, he may confult the other book at his leifure. DEFINITIONS. D E F. I. Navigation is the art of computing a fhip's way, and guiding her thro' the fea, in failing from one place to another. DEF. Sea. III. NAVIGATION. 125 Fig. D E F. II. Difference of latitude of two places, is an arch of the meridian contained between two parallels paffing thro' thefe places. DEF. III. Difference of longitude, is an arch of the equa- , tor contained between the meridians of any two places. DEF. IV. Meridional diftance, is the diftance in the parallel of the place you are in, from the meridian of the place you came from. D E F. V. Rumb, is the line or way a fhip defcribes while fhe fails upon any one point of the compafs, and cuts all the meridians at the fame angle. D E F. VI. Courfe, is the angle which the rumb or fhip's way makes with the meridian. DEF. VII. Departure, is the whole eafting or welling a fhip continually makes, in any fingie courfe. DEF.. VIII. Plain failing, is the art of computing a fhip's way by plain trigonometry, in regard to her dif ference of latitude and departure. D E F. IX. Middle latitude failing, is a method of compu'- tingthe place of a fhip as to longitude, by help of the middle latitude, which is the latitude lying in the 126 N A V I G A T I O N. Fig. the middle way between the two places of the fhip. ^_ D EF. X. Mercator's or Wright 's failing, is the method of computing a fhip's place as to longitude, by help of a table of meridional parts. Inftruments ufed in Navigation. i. Charts. Thefe are maps of the fea and Tea coafts, and are generally conftruaed by Mercator's projection, where the meridians are all parallel, and the degrees of longitude equal in all latitudes, and the degrees of the meridian increafe towards the poles, in the ratio of the cof. latitude to the radius. Thefe forts of charts are fitteft for the failors ufe. 2. The Compafs. This is divided info 32 points or rumbs ; by 'this the fhip is fteered, upon any courfe, For the needle always points to the north, except fo far as it deviates by the variation, which is the difference between the true and the magnetical meridian, eaft or weft. ' An .azimuth compafs has befides, an index with a thread and two fights, and is moveable about the center. Its ufe is, to take the azimuth of the fun or a ftar. 3. The Log lime -, which is a card divided into feveral parts called .knots, and the end is fattened to the log. Its ufe is to meafure the fhip's mo tion, and is heaved into the Tea every two hours ; and the length run off, is known by a half minute glafs. For, fo many knots as are run off, fo many miles the fhip fails an hour. 4. The Sea quadrant, commonly called Davi/s quadrant. This is an inftrument to take the fun's altitude. 5. Hadlefs quadrant, another inftrument for ta king Sea. III. N A V I O A T I O N. 127 king the altitude of the fun or a ftar ; and is thebeftFig. inftrument for that purpofe. 6. The Crcfs (laff or fore ftaff, this is alio ufed for taking altitudes. 7. The Lead, this is fattened to a line divided into fathoms, and is ufed for founding the depth of the fea. PROP. I. In plain failing, that is, where no longitude is con cerned ; as radius : to the diftance run : : fo the fine cf the tourfe : to the departure. Let P be the pole, HL the equinoaial, A the 13. place of a fhip, PAH a meridian paffing thro it, ADFQj:he rumb or way of the fhip. Divide the meridian AP into an infinite number of equal parts, at k, y, x, &c. and thro' thefe points draw the parallels of latitude kB, yC, xD, &c. to cut the rumb in B, C, D, &c. thro' which points draw the meridians PB, PC, PD, &c. cutting the equi noaial in t, f, r, &c. By the feveral interfedions of the parallels and meridians, and the rumb, there will be made an infinite number of fmall ele mentary triangles ABk, BCp, CDg, &c. all equal and fimilar. For all the fides Ak, Bp, Cg, &c. are equal by conftrudion, for Bp — ky, and 'Cg — yx, &c. and the angles £AB, pBC, ,g"CD, &c. are equal by Def, V. therefore (Geom. II. 7. ) all the parts of the rumb, AB, BC, CD, &c. are equal, and alfo the fides kB, pC, gD, &c. Now in any one of thefe triangles AB&, it will be,, by plain trigonometry, as rad : f. kAB : : AB : £B •: : BC : pC : : CD. : gD, &c. and by compofition, rad : f. MB : : AB + BC' + CD, &c. ': kB + pC -f gD, &c. But when the "fhip arrives at F, AB -!- BC + CD, &c. = AF the diftance run ; and.iB + pC ,+ %D, &c. is the departure bv Def. 128 NAVI'G AT 10 N. Fig. Def. VII. Therefore it is, rad : f. MB : : diftance 13. run : departure. And 'alternately rad : diftance run : : f. courfe (MB by Def. VI.) : departure. Cor. 1. As radius -.diftance run : : cof. courfe :,diff. latitude. For rad : f. -IB A : : AB : Ak : : BC : Bp : : CD : Cg, &c. : : (by compofition) AB + BC + CD, &c. : Ak + Bp + Cg, &c. or Ak + ky + yx, &c. But when the fhip comes to F, AB + BC + CD, &c. zz. AF the diftance run, and Ak -f» ky + yx, &c. zz AR the difference of latitude. Therefore rad : f. £BA or cof. courfe : : diftance run AF : dif. lat. AR. Or alternately, rad : diftance run AF, : : cof. courfe ; dif. lat. AK , Cor. 2. As rad : diff. latitude: : tan. courfe : depar ture. For by this Prop, rad : diftance : : f. courfe : de parture : : (Cor. i.) cof. courfe : dif. latitude. And alternately, dif. latitude : departure : : cof. courfe : f. courie : : (trigonometry) fad : tan. courfe. Cor. 3. Hence, if a fhip moves uniformly along its rumb line, it approaches the pole uniformly. For if all the lines AB, BC, CD, are equal ; then all the lines Ak, ky, yx are equal, and deferi bed in equal times. And hence appears the ab- furdity of" what fome people affert, that the rumb can never reach the pole. For if this" was true, then a fhip keeping always in the rumb, could ne ver reach the pole, tho' fhe approaches it uniform ly. "Which is a dired abfufdity. < Cor. 4. If two fhips fail from A at once, one along the meridian AP, with velocity AK ; the other in the .rumb AF, with velocity AB ; they will both meet in the pole at once. It cannot be denied, that the firft fhip will get to Sea. III. NAVIGATION. 129 to the pole in a finite time, which is eafily deter- Fig. mined from the velocity along AP. And in every 13* point of time, they will be both in the fame paral lel of latitude. Thus they will be at K and B, at y and C, at x and D, at R and F, at once. And therefore when the firft fhip is at P, the fecond cannot be fuppofed to be fhort of it without an ab- furdity. And it is the fame thing if you take two points inftead of the two fhips. Cor. 5. As cof. courfe : radius : :fo length of the meridian AP : to the whole length of the fpiral AQP. Cor. 6. Hence, the diftance run, the difference of latitude, and the departure being laid out in right lines; they will form a true right angled plain trian gle. For make RA = RA (fig. 13.) and angle RAF 14. zz RAF. Then fuppofe the rumb AF to be unbent, or drawn back into a ftraight line, and laid upon AF (fig. 14). The points B, C, D, will fall upon B, C, D, and the lines kB, pC, gD, upon kB, pC, gD, in the two figures ; becaufe the angles at B, C, D, being equal ; the lines Bk, Cp, "Dg, are all paral lel to FR. And therefore the perpendicular RF being equal to all the lines kB, pC, gD, &c. will be equal to the departure. And fo thefe three lines make the right angled triangle ARF. Cor. 7. Hence, the fquare of the diftance run, is - equal to the fum of the fquares of the departure and difference of latitude. Cor. 8. This Prop, and Corollaries will refolve all •the cafes of plane failing, being thofe where no longi tude is concerned. There are fix cafes of plane failing, and they are all folved by working "thefe proportions for ward or backward, according to the different data. K S c h o L. J30 NAVIGATION. Fig. ,. Scholium. Having made fome remarks, jn my book of Navigation, on the nature of plane failing ; I fhall here confider it more particularly. And I fay, 13* 1. That meridional diftaqce, departure, and dif ference of. longitude, are all effentially different, Let a fhip fail from A to F ; when fhe comes at F, fhe will have made the meridional diftance RF? and departure kB + pC + gD, &c. and difference of longitude HL. Now the meridional diftance RF'or'Riw.4- mn + no, Sec. is lefs than the depar ture kB + pC + £D, &c. and this departure is lefs than the longitude HL or Ht + ts + fir, &c. All this is plain from the converging of the meridi ans towards the pole. Again, if a fhip fails from F to A ; fhe makes the meridional diftance Az, which is greater than the former RF ; and is like- wife greater than the departure kB + pC *f- gD, &c. but lefs ftill than the difference of 'longitude. Here then in the fame courfe back and forward, the departure and difference of longitude continue the fame, but the meridional diftance is different, being lefs towards the pole, and greater towards the equinoaial. 2. If a plain chart be conftruaed, fo that the ¦meridians be parallel to one another, and the de grees of latitude every where equal, and alfo equal to the degrees of longitude at the equator ; then if the way of a fhip be laid down upon- fuch a chart, according to the courfe and diftance, t.he depar ture and difference of latitude will be truly. found, as to their magnitude-; but the place of the fhip will have a wrong fituation in refpea of the me ridians, by reafon of the parallelifm of the meridi ans, that ought to converge to the pole. And hence any traverfe or compound courfe laid down upon, fuch a chart will give all the places of the fhip in Sea. IIL NAVIGATION. 1 3-i ifi a wrong fituation* in refpea of the meridians ; Fig. and confequently in refped to* one another. For 14. that reafon the plane chart is erroneous, except in places neat the equator, where the meridians are nearly parallel. Yet in a fmall compafs, as in a day's run, the error will not be fenfible in any la titude. For in fo fmall a compafs, the meridians continue parallel as to fenfe ; and therefore a traverfe •may be wrought exaa enough for the length of 24 hours, as is the common pradice. But the error of the plane chart does not affea the prefent pro- pofition, fdf this will give the departure truly, for any courfe tho' never fo long. 3. Hence it follows, that the triangle ARF in 14. this chart will be fuch, that if AR be the true dif ference of latitude, AF will be the true diftance- (ADF) upon the globe, RAF the true courfe (MB,) and RF the true departure; and therefore this conftruaion is true, and gives us a true way of failing. It is no objeaion to fay, that it does not give the longitude truly, for that is inconfiftent with its nature. A man may as well fay that Mer cator's pfojedion is falfe, becaufe it does not give the diftance or difference of latitude truly. For by its nature and conftrudion it is only to give the longitude truly. In fhort, both conftrudions are true, according to their feveral natures. Plain fail ing is true fo far as courfe, diftance, difference of latitude and departure are concerned ; and Mer cator's is true fo far as the longitude is concerned j but one of them cannot anfwer for the other. 4. What has been faid relates to a fingle courfe 13. as ADF, whofe angle of the courfe is MB. But inftead of failing diredly from A to F, if a fhip fhould fail to R and then to F -, the departure a- long AR would be nothing ; and along RF, would be equal to RF ; for in any parallel the diftance and departure are the fame. And confequently K 2 the *32 NAVIGATION. Fig. the departure in thefe two . courfes, is lefs than in 13. the fingle courfe AF. Likewife a fhip failing back from F to A, will make the fame departure as from A to F ; and from F to R and A, the fame as from A to R and F. But if fhe fails from F to ^ 2, and then to A, fhe makes no departure along Fz -, and from z to A, fhe makes the departure" zA, which is greater than from F to. A. Therefore a fhip funning upon feveral courfes, makes a lefs departure when nearer the pole, and a greater when nearer the equinodial, than in a dired courfe to the fame place. But in fmall diftances, as a day's run, the difference will be infenfible. 5. Since the departure in the courfe AF, is greater than the meridional diftance RF, and lef fer than AL ; it will be very near a mean between them ; and therefore the departure will be very near the meridional diftance xw, lying in the mid dle latitude between A and F. 6. The departure in fome of the cafes, is the moft proper and natural medium, for finding the longitude. And therefore I cannot but be furpri- fed at thofe authors that exclude it from the prac tice of navigation, at the fame time they are for ced to make ufe of it, or what is equivalent to it, for the fame purpofe. For inftead "of RF, they 14. do not fcruple to take the ^AF2 — AR% as if that was not the fame thing ; when it might eafily be had by a fimple proportion in this propofition. And they are fometimes forced to ufe the enlarged diftance upon Mercator's chart ; a line which there is no manner of occafion for, but for fuch a fhift ; by which means, they give us the moft clumfy fo- lutions imaginable. Such is the unaccountable prejudice of fome people. But further, .13, 7.I fay, this Prop, is true, not only in the fphere, but in the fpheroid, or on the lurface of any folid, generated by revolving round an axis, and even up on Sea. III. NAVIGATION. 133 on a plane or cylindric furface. For fince all theEig. angles MB, pBC, gCD, &c. are equal, by the na- 13. ture of the rumb ; the elementary triangles are all fimilar and equal ; and therefore the fame conclu- fions will follow as before, viz. that rad : diftance run : : f. courfe : departure : : cof. courfe : diff. latitude. And rad : diff. latitude : : tan. courfe : departure, whatever fort of a folid it is. And the diftance, diff. latitude, and departure will make a right an gled triangle as before. So far is this Prop, from being falfe, that it is univerfally true, whatever fort of a line the meridian is ; and therefore it can never lead into any error. I fhall only add, that in a fingle day's run, where the latitude is little va ried, the (fum or difference) of all the depar tures made by different courfes, in that time, will be the fame as the fingle departure, made between the firft and laft place of the fhip ; which will come to the fame as the meridional diftanqe, for that latitude. And this is the very praaice of all feamen ; to find the eaftings and weftings for every feparate cOurfe, and then take the total amount (that is the fum or difference) for the departure. A method fo eafy, natural, and expeditious, that it is impoffible to invent any thing that can in this refpea exceed it ; and therefore the Navigator need make no fcruple of continuing his praaice. Asd fo much for plane failing, PROP. II. In parallel or eaft and weft failing ; cof. latitude t radius : : diftance of two places in any parallel : dif ference of longitude. Let RF be the diftance in that parallel, then HL will be the difference of longitude. The co- fine of latitude is the radius of the parallel, and K 3 the n- 134 NAVIGATION. Fig. the radius of the equinoaial is the radius of thje 13. fphere -, and the radii of all circles are as any fi milar pa.rts.of the circumferences, as a degree, Sec Therefore cof. latitude at R : radius : : diftqnee RF : diftqhee HL on the equinoMial, which is the difference of longitude. This has likewife been demonftrated in Prop. 15, Sea. I. Cor. 1 . The length of a degree in any parallel, is as the cojine of latitude. Cor. 2. This Prop, wittfojve all the cafes of pa rallel failing, which are 3 ; by. working the proportion backwards or forwards, as occafim requires. PROP. HI. By middle latitude, failing, as cof. middle latitude ': radius : : departure : difference of longitude.. Suppofe a fhip fails from A to F, making the difference of longitude HL. Then fince the de parture kB -f- pC -+- ^D, &c- is greater than the meridional diftance RF, and lefs than the meridion al diftance A'z ;' and if x be taken in the middle between A and R, then xw, which is- a mean be tween RF and Az, is nearly equal to the departure, or the fum of the lines kB + pC + ^D, &c. there fore if xw be taken for the departure -, then by the laft Prop, it is, as cof. of the lat. of x : radius : : fo is xw : HL. That is, as cof, middle latitude : radius : : departure : difference of longitude. Cor. 1 . As cof. middle latitude : f. courfe. : : dif tance run : difference of longitude. For cof. middle lat. : rad: : departure : dif. long. and (Pr. I.) rad : dift. : : f. courfe : departure. Therefore cof. mid. lat. : dift. : : f. courfe : dif. long. Cor't Sed. III. NAVIGATION 135 Cor. 2. Cojine of middle latitude : tan. courfe : : ,, difference of latitude : difference of longitude. For cof. mid. lat. : rad : : departure : dif. longi tude, and (Cor. 2. I.) rad : dif. latitude : : tan. courfe : departure. Therefore, Cof. mid. lat. : dif. lat. : : tan. courfe : dif. long. Cor. 3. This Prop, and Corollaries will refolve all the cafes of failing, where the longitude is concerned, Md very near the tfuih, if the cOurfe be not too long. Scholium. This method Of failing is very fhort and eafy, and alfo very exaa in fhort courfes ; and does not ' differ much in long courfes, and therefore is a ve ry ufeful approximation. And I believe 1 fhall ne ver live to fee a fhip fail to that exaanefs, and this Prop, can determine it. And as this method is fo eafy, and performed without any other tables than the common ones ; it is ftrange that any perfOn fhould condemn or depreciate it. Nothing will go down with fuch people, but rules formed to the utmoft accuracy, as if any fhip could fail by fuch rules. Sailing or guiding a fhip at fea, is but a mechanical art, and cannot be executed to any great degree of exadnefs. And therefore it is needlefs to infift upon rules made to the utmoft geometrical rigour •, and the difficulty that attends the computation of a fhip's way by fuch rules, is fufficient to caufe them to be laid afide, when eafier will do as welL But when navigation is fb far im proved, as that a fhip can fail to any degree of ex adnefs, it will be time then tO feek out fuch rules as are perfed and accurate, and adequate to that end. K 4 PROP. i36 NAVIGATION. Fig. PROP. IV. The length of a minute on the fphere : is to the length of a minute on the meridian of Mercator's chart, for any latitude : : as the cof. of latitude : to radius. \5- For fince the rumb cuts all the meridians at the fame angle, in order to have the rumb a right line, all the meridians muft be parallel ; whence every cofine will become equal to the radius. Therefore every particle on the fphere, muft be increafed pro portionally, that is, as the cofine of the latitude of that point, is to the radius. Therefore fince the radius of the parallel is made equal to the radius of the fphere, a minute on the globe muft alfo be increafed in that proportion. And therefore as cof. lat : rad : : fo a minute on the globe, to a minute in the meridian line. Cor. r. From hence, by the method of fluxions, may be calculated the length of the meridian line in the chart, or the meridional parts for any latitude. Let AC be the radius of the equinodial, AB the latitude, draw BD, CG perp. to AC, and BE parallel to it, and let Bb be a minute ; draw bn paral lel to BD. And put CA zz r, arch AB zz v, DB r j, BE - x, and z n length of the meri dian line for the lat. AB. The triangles BDC and Bnb are fimilar ; whence CD (x) -. CB (r) : : bn : Bb zz JLx bn. And by this Prop. BE (a?) : AC x (r) : ; Bb ( — X bn ) : a minute of the meridian K x ' line in the chart zz — x bn. Now, if for thefe n S/C DC '&* fmall lines we 'put the fluxions, we fhall have Sed. III. NAVIGATION. 137 z zzV-y zz -111 — -, and the fluent is z =FiS' xx rr — yy 15. £30>_5*5r. x L . r+, 2 r—y Cor. 2. Therefore z = 2. 30258 $r xinto log. : co tangent of half the complement of the latitude — log. : radius (10). For z zz 2^21.r X log. : L^ll zz 2.3027- X 2 r — y log. : Jr-+-1. But (by Schol. Prop. II. Trigon.) J- r+y_ zz cot. -^arch, that y is the cofine of r — y zz cot. ^BG zz cotang. half the complement of the latitude. Whence 2 zz 2¦302rxlog¦CO¦tan•^COm•lat• r Cor. 3. Since the meridional parts are expreffed in minutes, wejhallhave z zz 79 1 5.7 x log. tan. ^co. lat. *-. 2.30258 X 180 X 60 fc , For_i_2 , zz 7915.705, and 3-HI59 cot. 4- comp. lat. r . £ - zz . — , by trigono- r tan. f com. lat. metry. PROP. V. /a Mercator's Jailing, as proper difference of lati tude : / — — : tan. courfe :, : D : diff. longitude, 79l5 but .'¦ ~ ¦' = .00012633. 7915-7 Cor. 3. As in plain failing, the rumb, the diffe- 17. renee of latitude, and departure make a right angled triangle ; fo in the chart, the rumb, the meridional difference of latitude, and difference of longitude, alfo make a right angled, triangle, fimilar to the former. Thus, in the triangle ADE, if AB is the diff. lat. BC the departure, AC the diftance. Then AD is the mer. dif. lat. DE the diff. longitude. Cor. 4. This Prop, and Cor. r. will refolve all the cafes of Mercator's failing, or where the longitude is concerned, with the help of a table of meridional parts. For by the. table of meridional parts, the me ridional difference of latitude is had, which is only the difference of the meridional parts belonging to the two latitudes. And that table is eonftruc- ted by Cor. 3. Prop. IV. Cor. i4o NAVIGATION. Fig. Cor. 5. By Cor. 3. and Prop. I. all the cafes of failing may alfo be refolved. Scholium; 13. Altho' all the cafes of failing may be folved by help of Mercator's chart, yet this chart ,is not a true reprefentation of the earth. For though eve ry fingle particle, or elementary triangle on the chart is fimilar to its correfpondent part on the fphere, yet the whole is not fimilar to the whole; for the parts of the chart being bigger than thofe in the fphere, yet they do not increafe in the fame proportion -, being nearly equal at the equator, and infinitely greater at the poles; and at the interme diate places, in all imaginable proportions. So that the whole cannot be fimilar to the whole, but very much diftorted. Nor is it pofBble that any projeaion upon a plain can be fimilar to a curve iurface, fuch as a fphere or a fpheroid. The aaual folution of the feveral cafes of fail ing are eafily drawn from the foregoing propofi- tions, and I fhall not trouble the reader with them here, "but refer him to the book of Navigation. But the three following problems, not being com mon, I fhall here give their folutions. P R OP. VI. Prob. One latitude, departure,, and difference of longitude being given ; to find the other latitude. By Prop. III. fay, As diff, longitude : to departure : : So radius : to cofine of middle latitude. The difference between the middle latitude and the given latitude, added to, or fubtraaed from, the Sea.HI. NAVIGATION. 141 the middle latitude, as the cafe requires, gives theFig. other latitude. 13. If this latitude found, be not thought exad enough, it may be corrected by Mercator. For the latitude found muft be fuch, that the proper diff. latitude, may be to the meridional diff". lati tude ; in the given - ratio of the departure to the difference of longitude ; and is eafily found in a few trials. Or it may be found after two trials by the rule of falfe. PROP. VII. Prob. One latitude, diftance, and difference of longitude being given ; to find the other latitude. 1. Affume any angle for the courfe, as near as you cart guefs, by the circumftances of the quef-, tion. From this, find the difference of latitude, by Cor. 1. Prop. I. making, As radius : to the diftance : : So the cofine of the courfe : to the diff. latitude. 2. Both latitudes being had, find the longitude by Cor. 1. Prop. V. thus, As radius : to the tangent of the courfe : : So the mer. diff. latitude : to difference of longitude; 3. If the lopgitude found, differs from the lon gitude given, as it generally will ; corred the courfe, by making it greater or lefs as there, is occafion ; then find the latitude, and then the lon gitude again. ' Repeat the operation till the lon gitude agrees with that given. But when you come near the matter, you may find the courfe truly from two trials, by the rule of falfe. 142 NAV1GATIO N. Fig. 13. Or thus, 1. Say as rad : cof. given latitude : : diff. longi tude : to the departure. 2. Find the fum and difference Of the departure and diftance ; take the logarithms of the fum and difference, and add them together-, half the fum of thefe logarithms is the log : of the diff. of latitude. Then find the other latitude. Again (by Prop. V.) 3. Say, as meridional diff. latitude : proper diff. latitude : : diff. longitude : departure, more ex- ad. Repeat the 2d and 3d articles over and over, al ways with the laft departure and diff. latitude ; till at laft you get the fame diff. latitude twice ;, and this is the true diff. of latitude. Whence you will get the other latitude e£ad. PROP. VIII. Preb. The courfe, departure, and difference Of longitude, being given ; to find both latitudes. Find the diff. latitude by Cor. 2. Prop. I. and the meridional diff. latitude by Cor. 1. Prop. V. thus, As tan. courfe : to radius : : . So, the departure : to the difference of latitude. And fo the diff. longitude : to the meridional diff. latitude. The difference of latitude being had in degrees and minutes, feek in the table of meridional parts-, for two latitudes, fuch as may exceed one another by that difference •, and that fubtraaing the meridional parts of one from the other, the remainder may be equal Sea. III. NAVIGATION. 14? equal to the meridional diff. latitude before found : Fig. which is eafily done in a few trials. And thefe are . the two latitudes. If inftead of the departure, the diftance or diff. latitude be given j the fplution is the fame, after the diff. latitude is known. PROP. IX. Let ABP be a quadrant of the fphere, AHF a 1 6. quadrant of an ellipfis reprefenting the fpheroi- dical figure of the earth, AC the equinoaial ; P, F, the poles ; BE, HI, parallels of latitude of the places B, H ; BD, HL perpendiculars upon AC. And put AC zz r, FC zz a, AB zz v, LH zz y, CL or HI zz x, rr — aa zz dd, s, c, t, fizz fine, cofine, tangent, fecant of the latitude of any place B or H, radius zz 1. Then, The radius of a parallel in the fphere BE : is to the radius of the parallel, having the fame latitude, in the fpheroid, HI : : as f 1 — -^_ ss: to 1. rr If two places upon the fphere and fpheroid B and H, have the fame latitude ; then the tangents at B and H will be parallel ; for they muft point to the fame ftar. And the ordinate LH in the fpheroid, will be nearer the equinoaial A, than the ordinate BD in the fphere. For by the conic fedions, if both tangents be drawn from points in the fame line BD or LH, they will meet in fome point of the axis CA. And confequently, that they may be parallel, the point H muft be nearer A. Suppofe the tangents from H and B to meet the axis CA, in T and S -, then the triangles CBD, DBS, LHT will be fimilar. And by nature of the ellipfis LH or v zz —s/rr—rxx, and DB zz sr, r BE 144 NAVIGATION. Fif • BE zz cr. Alfo CT zz J31,' and LT zz 10. X X Whence by fimilar triangles, BE or CD (cr) : DB (sr) : : DB : DS : : LH / -^Ar — **) : LT /rr — xx\ ax : r\/rr — xx. Whence sax zz cr v/Vr — xx, and ssaaxx zz ccr* — ccrrxx. There- crr fore x zz — ¦ zz HI. Therefore BE : \/ccrr + aass err HI : : cr : ¦ = : : ^/ccrr + aass : r : : v ccrr + aass ; ^yrr — rrss ¦+¦ aass : r : : ^/rr — ddss : r : : 7 rr Cor. i. # The fame things fuppofed as in Prop. IX ; the length of a minute on the fphere ; is to the length, of a mi en the fpheroid at H::as rra : to aa + — yy\. aa ' Thefe lengths are as the radii of curvature ; and this radius in the fphere is. r, and in the, fpheroid is *abb + ASf-A*f$ (by Prob. 5< ex. a. sea, II. zbba* Fluxions), and putting 2r for a, arid ^f for b the r I 3 rr — aa latus reaum, it is reduced to a + aa yy- - VVl •ar There- Sea. III. NAVIGATION. i47 Therefore a minute in the circle : a minute in theFig. 10. dd ellipfis : :r: aa + aayy ar dd : : arr : aa+~yy ' aa Cor. i. A minute on the fphere : is to a minute on ihe fpheroid at H : : as ax i + — cc V : to r. aa FoKyyzzaa — ™xxzzaa-a±x—^— rr rr aa + ddcc = aa rraacc _ g* -f aaddcc — rraacc _ aa + ddcc aa + ddcc aass And aa — rr — dd.cc aa — o-acc I + — cc aa dd _ ddss i ddss 1 ¦$¦ cc _ . dd '""" , dd I + — cc I -j cc aa aa Air dd Alio aa -f- — yy zz aa -f- aa aa , . dd — ' I + — cc aa aa + ddcc + dd's , . dd 1 + — cc aa zz ( becaufe cc + ss zz i ) aa 4- dd . dd " dd I + — cc I + — « 4£ aa r* arr : rr , dd \l Ana arr :aa + — yy\ 1 aaJJ I + — cc aa , dd \i : : a X i + -«| aa yaa 77> Cor. 2. if K zz v/— , rim r zz ^/^-xk_, rr da the cofine of the latitude, where a minute of the meridian of the fpheroid, is equal to a minute of the equinoctial. --3_r ~Jd~z - , and i + — cc — a aa For then i + —cc aa rr , «si 5 /rr v ,dd — , and i + ™_« zz v — zz K, and — cc zz. aa aa aa aa L 2 K- i48 NAVIGATION. Fi6\ K ~* *' whence cc = 7d>< K-7^- Cor. 3. If V and V be the length of the meridian of the fphere and fpheroid, at the fame latitude, q zz dL. Then V zz r^zkl v — i2? ; where S is the aa a a fine of twice the latitude. For \ttp zz tnen a X 1 ¦¥ pec1 :r-.:v:V aa r _ rv _ rv rv . . - aXi + pccM a x 1 + \pcc a 1 — Ipcc + Sec. zz (becaufe q is fomething lefs thanp)li-^v. WhenceVzz^ - & a 2a a za X fluent of ccv, of Turn of all the cc in the arch. But fuppofing the radius zz 1 (fig. 15) by fimi lar triangles, v (Bb) : s (nb) : : CB : BE : : 1 : c, and cv zz s, and ccv zz cs. And the Flu : cs (or the fum of all the BE X nb) is zz area ABEC zz Fl : • V cs ccv, and the area ABEC zz r- — Therefore ' 22 V zzr— — 3Ilx 1 + - =r-^-¥$ v — ¥1 cs a 2a 2 2 a 4a zz r — ^ ^, — iliSj becaufe cs zz ±S. Cor. 4. If v and V be the meridians for the fame latitude, in the fphere and fpheroid, in minutes of the equator ; then V zz LZM v - Al^S. , a a For if the lengths V and v be reduced to mi nutes, they muft be multiplied by ° — — or 3.1416 343?, which is the minutes in the radius. But V and Sed. III. NAVIGATION. 149 and v, now denoting minutes, are already reduced; Fig. and reducing El S, we fhall have V zz r~^ri v _ 1 6- 7^^S=rSl£lv--rlxi2S9fS. a a a And hence it will be eafy to make a table of the arches of the fpheroid in minutes, exaa enough for the earth. For if the fpecies of the earth's fphe roid be known, q is known ; and then r "^ and a 2_? will be given quantities ; whofe logarithms, a added to the logarithms of v and S, will give two numbers, whofe difference is the fpheroidal meri dian. Thus if £zz.o22 according to Maupertuis, then V zz .9945V — 28.7 S ; orV zz v — .00551; — 28.7 S, where .0055V -f- 28.7 S is the reduc tion, or diminution of the fpherical arch, PROP. XII. The fame things being fuppofed, as in Prop. IX. a minute on the fpherical chart : is to a minute on the dd fpheroidal chart, at the fame latitude ::asi-\ Cc : to 1. For by Prop. IV. a minute on the \ . a minute on ) . t . q fpherical chart J " the fphere $ ' And by Cor. 1. Prop. XL a minute on ) a minute on ) %a + dd Jr . r# the fphere j" ' the fpheroid J ' aa L 3 Alfo Igo NAVIGATION. Fig. Alfo by Cor. 2, Prop. X. 1 °' a minute on 1 a minute on the \ I dd the foheroid f : fnheroidal chart \'-'rc-*sJ 1 + —a. aa the fpheroid J 'fpheroidal chart V Therefore ex equ6, a minute on the 1 > a minute on the 1 fpherical chart j : fpheroidal chart J : : are * T^Kdf: arc [7Z¥jc :ii+^-ccii. aa I v , 633 X 428 (S. 250 22') zz 4 32.4 zz » m zz '1606.7 — tne mer. parts for 25 52. If this had not been exaa enough, you muft have repeated it, with s for 25 52, which would redify it. Then to find the proper d. latitude; 54" = 324.0'. and 25" 52' zz 1552'. And 28.7 xS — fizz 28.7 x.166 zz 4.76 the reduc. v zz 3240 .9945 v zz 1552 1688 v— Vzzi688 ,679 — 4. -1.997605 3.227372 3.224977 7 1674-3 zz pr. dif. lat. Then i58 N A V I G AT I O N. Fig. Then eof. courfe 27" 9,902348 17. d. lat. 1674 • 3.223755 — 10. Rad. diftance 2096, 3.321407, CA. Cafe 5. One latitude, courfe and diftance being given; to find the other latitude and difference of longitude. Examp, Afihipfails upon a courfe of 23% from E. in n. lat. 45» for 'the diftance 3700 miles; to find the lat. come to at A, and difference of longitude, DE. 450 zz 2,700, then S zz 1. •9945 2700 —1.997605 3.431364. 2685 and 28.7 x 1 is the reduaiori — 28.7 2656.3 3.428^ zzF— G. Then rad. 10. Dift. 3700 3.568202 Cof. courfe 230 9.964026 d. lat. 3406 - 3.532228, BA. and this being greater than 2656, fhews that the fhip has run into fouth latitude. , 34062656 750 =¦/— j> And 750 would be 12 30 on the fphere; but the fpherical arch being greater than the elliptical, it muft be reduced ; here S zz .422, and-91 v -|- 1000 28.7 X/= 4.1 + 12.1 = 16.3, red. Sea. III. J I A M I G A T I O N. 159 M.P 75o red. -f- 16.2 Fig. 17- v = 766.2 zz iz" 46'. Lat. 45 — 12 46 303° 772 S. v — .707 S. v zz .221 1.878728 — 1.967548 reduc. 3802 70 .928 1.846276 1 N 4 » 70.2 M — N+m — », 3732 Then Rad. 10. Tan. C. 23° 9,627852 mer. d. kt. 3732, 3.571941 djf. lo ng- i584» 3-I99793» DE. Thus all the cafes of Mercator's failing are fol ved, by the common rules, with the additional trouble of converting the arches of the fpheroid into thofe of the fphere, and the contrary ; and without any tables made on purpofe. And in like manner, all the cafes of parallel failing are folved, by converting the radii, of the parallels in the fphe roid into thofe of, the fphere, and the contrary, and with the fame taqlesi And all the three cafes are folved by Prop; XIV. Where note, that in Maupertuis earth, i- 4 iqss zz 1 + .011 ss,s being the nat. fine of the latitude. Parallel Sailing. Cafe 1. Given the latitude and difference of longitude of two places- in one parallel; to find their diftance. Example. i6o Fig. N A V I G A T I O N. Example. Suppofe a fhip fails eaft in the parallel of 5 2" 12', till her difference of longitude be 857' ; to find the dif tance failed. ¦ .011 I —2.041393 s 9.897712 9.897712 00687 — 3.836817 r.0068 zz 1 4 \. qss. Rad. J I.0068 icof. lat. 52 12 d. long. 857 diftance 528.8 10 0.002943 9787394 2.932981 »¦'¦ « 2.723318 ; ,- ": '¦'-•'-'•¦¦ Cafe 2. Given the latitude, and diftance of two places in one parallel; to find their difference of longitude. Examp. A fhip fails eaft 528.8 miles in the latitude of 52* 1 21 ; required her dif. longitude. Find 1 4 iqss = 1.0068 as in cafe 1, then, 1.0068 Cof. lat. 52 12 Rad. Dift. 528.8 i /d- long. 856.9 0.002943 97*7394 10. 2.723291 979°33> 2-932954 Cafe Sea. III. NAVIGATION. 161 Cafe 3. FlS' Having given the diftance of two places in one pa rallel, and their difference of longitude ; to find the latitude. Examp. A fhip fails eaft 528.8 miles, and makes her dif » longitude 857' ; required the latitude. Dif. long. 857 — 2.932981 Dift. 528.8 — 2.723291 Rad. — 10. Cof. lat. — 9-79°3!° This anfwers to 51 54 if it was for the fphere, but as the parallel is lefs in the fphere, it muft be reduced. Therefore find the reduaion, viz. log. : 1 4 .011SS, putting s zz S. 510 54'. S. 5r 54 S. 51 54 .01 j 1 — 2.041393- 9^95939 9.895939 .0068 — — 3-833271 9.790310 .002943 Cof. reduc^ 51 54 1.0068 Cof. lat. 52 12 9787367 true. PROP. XVI. Prob. To correcl a Reckoning at Sea. The motion of a fhip is fubjea to feveral ine qualities, which muft be correaed before the place of the fhip . can be truly known. And therefore the true latitude of the fhip muft be got, by ob- ferving the fun's meridian altitude 5 by which and M the i6z NAVIGATION. Fig. the fun's declination being known, the fhip's lati tude becomes known, or inftead of the fun, any known ftar may be ufed. Then if the obferved latitude be the fame as the computed latitude, there is no room for any correaiom But when they dif fer, the error will be owing to fome of thefe caufes, the variation, the motion of a current, error in the courfe, or in the diftance. And which it is, muft be determined by judging of the feveral circum- ftances that attend the reckoning. i . If the error lies in the variation ; which will be known, when an amplitude is had ; take the dif ference of latitude and departure, all the time you have been wrong, and fay, As Rad : S. error of variation : : So departure : to correclio'n in latitude : : And fo diff. latitude : to correclion in departure. Then if the true courfe is greater, the departure muft be increafed, and the dif. lat- diminifhed, by thefe correaions. And the contrary when the true courfe is lefs. For let AC be the fhip's way, AR the fuppofed meridian, Am the magnetic meridian, AH the true meridian, AR and CR the dif. lat. and departure as computed. Let fall CS perp. to the true meri dian AH. Then inftead of AR and CR„ AS and CS will be the dif. lat. and departure. Therefore AR muft be diminifhed by the quantity R« or dS, to get the true dif. lat. AS ; and CR muft be in creafed by Ri or »S, to get the true departure CS. And in the right angled triangles AS», CRn, the angles at R and S are right, and the angles at n are equal ; therefore RC» zz SA» the error of variation. Therefore in the triangle CR», rad : Cw or CR : : S.RCw : R», the correction in latitude. And in the triangle ARd, rad : Ad or AR : : S.RAd : Rd or S», the correaion in departure. But Sea. III. NAVIGATION. 163 But if the obferved courfe RAC be diminifhed Fig. by the error of variation, the perp. CS would fall 1 8. on the other fide of CR, then AR muft be increafed by »R, and CR decreafed by Rd or »S. For when one adds, the other fiibtraas. 2. If the error happens from the motion of a cur rent. If its direaion be known, to find the quan- °* tity of the error ; fay, as rad : error in latitude : : tan. current's courfe : correction in departure. This correaion muft be added to the departure, when the true dif. latitude is greater than the ob ferved ; but fubtrafted when lefs. For if AR is the meridian, AQj:he fhip's way by the current, Q^the fhip's place by the reckoning, in the parallel PQj and C her true place by ob fervation, in the parallel RC ; then QC is the er ror. Draw QD parallel to AR, then, in the tri angle QDC, rad : DQJPR) : : Tan. DQC f RAC) : DC, the correaion in departure ; which adds, if R is beyond P, otherwife it fubtraas. 3. When the error is in the diflance; as is very 20,' likely when the courfe is near the meridian. To find the error, take the difference of latitude and departure from the time of the laft obfervation, and fay, As diff. latitude : departure : :fi> the error in lati tude : to the correclion in departure. Then the correaion is to be added to the depar ture, when the computed dif. latitude is lefs than the true, that is, when the fhip is before the reck oning ; otherwife fubtraaed. For if the fhip's way be near the meridian, as in AB, and if BC be her parallel by the reckoning, and DE her parallel by obfervation. From A, the place of the laft obfervarion, defcribe the arch BF to cut the parallel DE in F. Then if the error was in the courfe, it will be no lefs than the angle BAF, which is improbable -, for fuch a great mif- M 2 take 1 64 N A V I G A T.I O N. Fig. take can hardly be made, if any care has been ta- 20. ken ; therefore, it is more rational, to fuppofe it in the diftance, and then it will be the fmall quan tity BD. Produce AF to H, and draw HG parallel to the meridian AD. Then by fimilar triangles AB (dif. lat.) : BH (departure) : : HG : FG. Then if BH be her parallel by the reckoning, HG muft be taken from BH. But if DE be the parallel by the reckoning, or the fhip foremoft, then FG muft be added to DF to get her true departure. If the courfe AH is very near the meridian AB, the cor rection of the departure FG is fo fmall, that it may be omitted. 4. When the error is in the courfe, as is very like ly when it lies near the parallel. To find the er ror, fay, 21. As the departure : dif. latitude : : error in latitude : correction in departure, from the time of the laft ob fervation. Then the correaion muft be added to the departure, when the reckoning is before, the fhip ; otherwife fubtraaed. But when the fhip's way is very near eaft or weft, this correaion is fo fmall that it may be omitted. For let AB be the meridian, BC the fhip's pa rallel by the reckoning, DE her parallel by ob fervation, AF her way by the reckoning, produce AF to cut BC in C. Then if the error was in the diftance, it would be no lefs than the line FC, which is quite . improbable -, therefore it is. rather the fmall angle HAF in the courfe, which may very well happen. Draw HG parallel to the meri dian AB, and from A defcribe the arch HF. Then the angle AHF being right, the angle HFG zz AHG, zz HAD, and AHD zz FHG, therefore the triangles ABH, FGH are fimilar, whence BH (departure) : BA (dif. latitude,) : : HG : GF, the correaion in departure. And as F is her true place, 1> rig. , R G I I y^ J [ : // X' f> 13 /(> £— ^c , ' CfCfiijnwhi YlM.pa.jfa. Sea. III. NAVIGATION. 165 place, VGF muft be added to DG or BH to haveFig. her true departure DF. But if H had been her 2 1 . true place it muft have been fubtraaed. If the courfe was in or near the parallel AP, GF would be nothing. 5. If a fhip fails along AH near four points, and 22. you know the error to be in the diftance AH, then her true place will be at E, and the error of depar ture EG muft be computed by Art. 3. But if you know the error to be in the courfe, then making AF equal to AH, her true place will be at F ; and the error of departure FG muft be computed by Art. 4. Therefore if both be wrong, fhe will be in the middle of EF at G ; that is, there will be no error in departure. Hence if it be not known, whether the courfe or diftance is wrong, no error muft be reckoned in departure, but it muft remain as it was, for then BH is equal to DG. In general, failing near the meridian', or near a parallel, or near four points, the departure cannot be correaed. But it may be correaed when her courfe is within a point or two of the meridian, or any parallel. 5. The error in departure being found, the error in longitude lies the fame way, being only greater in quantity ; which is found thus, as error in lati tude : correilion in departure : : fo the meridional diff. lat. (between the computed and obferved lat.) : correc- tion in longitude. The correaing a reckoning, depends upon find ing the latitude, and that is had by obferving the fun's meridian altitude, or zenith diftance. And as the bufinefs of finding altitudes is ufeful upon many occafions at fea, I fhall conclude with the two following propofuions, for that purpofe. M 3 PROP* 1 66 NAVIGATIO N. Fig. 22. PROP. XVII. Prob. The times of two equal altitudes of the fun, taken in the forenoon and afternoon of the fame day, being obferved by a watch; to find by. the fame w#tch, the time of his being in the meridian. I. If the fun's declination continues nearly the fame, during the interval of the two obfervations, then the middle time between the obfervations, taken by the watch, is the time of his being in the meridian. But if he makes a fenfible difference of declina tion in that time, then this mean time muft be cor reaed as follows. II. 23. Let HO be the horizon, Z the zenith, P the pole, GDC a parallel of altitude, I AC a parallel of declination ; and let C, D be the two places of the fun, fuppofed on different fides of the meridian PZH. Then AD will be the difference of decli nation, between the times of obfervation. Then in the triangle ZPC there are given ZP the com plement of the latitude ; ZC the co-altitude, and angle ZPC anfwering the time, both by obfervation ; therefore S.ZC : S.ZPC : : S.ZP : S.ZCP. Then having the angle >C, let AD (the dif. declination) be taken in minutes of a degree. Then fay, as •cof. fun's declination : cotan. ZCP : : fo 2AD : to the correaion in feconds of time. Then the correaion muft be added to the mean time, to find the true time of his fouthing, when the fun is going from the elevated pole ; but fub- traaed, when coming towards it. For, produce the meridians PAD, and PC to cut the equinoaial in B and E. Then the angles -PCA and ZCD being right, by fubtraaing the common &• Sea. III. NAVIGATION. 167 common angle ZCA from both, DCA will beFig equal to ZCP. And in the fmall reailineal tri- 23 angle ACD right angled at A, rad : AD : : cotan. DCA or ZCP : AC zz ADxcotan.^ZCP ^ And rad. by the fpherical feaors PAC and PBE, S.PC : rad .n /ADxcot. ZCPX t.^. , ._,_ : : AC ( — ) : BE or angle APC zz v rad / - AD X cot. ZCP But BR and AD j. reck d S.PC ° in minutes, and 1 minute of a degree being equal to 4 feconds of time, therefore BE reduced to feconds will be zz 1 ' , and half the o.Jr \> DC 2AD X cot. ZCP. rt j -r , r time in BE = _ — And if the fun S.PC be going from A to D (from the pole), the morn ing obfervation will be at C, and 7' vibratory motion of the mercury. The two ends Fig. have their tops flopped with pieces of wood ce mented in them, and thro' thefe pieces of wood are made two very fmall holes, in which two pins are put, when the inftrument is at reft ; but taken out when ufed, to let the air enter, that the mercury may reduce itfelf to a level. The horizontal part of the tube jnuft be as long as the index. To adjuft the inftrument, let the index be fo pla ced, that the flit in the fight vane, and that in the horizontal vane may be exaaiy upon a level. Then pour mercury into the tube till it rife to the fame level in both ends of the tube, and no highe'f ; then is your inftrument correa. Therefore when at any time the index is fo placed, that the mercury in both legs comes juft to the line of vifion drawn be tween the two flits in the fight vane and horizontal vane ; then that line lies horizontal. And a vifible line ought to be drawn on the index to fhew this. To ufe the inftrument, the obferver muft chufe a convenient place, where there is the leaft mo tion or wind to difturb him. Then fitting down, and talcing hold of the inftrument as ufual, and looking thro' the flits in the two vanes, move the index till the top of the mercury in the tube be in the fame line, there keep it, whilft you move the inftrument up or down, till the fun through the flit of the fhadow vane fall on the flit of the horizontal vane ; when all thefe things agree, obferve the degrees cut, and you will have the zenith diftance, as ufual. Here, in looking thro' the two flits, you obferve the top of the mer cury in the tube, which ferves for the horizon -, fo that the end of the index is moved in the fame manner, as the fight vane ufually is. To obferve a ftar, fince it can eaft no fhadow ; to have it in a line with the horizon vane and fha dow vane ; another perfon muft look for the ftar thro* 172 NAVIGATION. Fig. thro' the two flits ; the index being kept in its pro per pofitiom By this, improvement, the want of a horizon is fupplied ; fo that, an obfervation may be made from any headland, or any harbour, or any place on fhore, where the fun can be feen, without any regard to the horizon ; for the true level will always be pre ferred, whether on the top of a mountain, or at the furface of the fea. , ... To compleat the art of Navigation, thefe three things are abfolutely neceffary; the variation of the compafs, the latitude of the fhip, and the lon gitude of it. The firft may be found by an am plitude or azimuth ; the fecond is known from the fun's meridian altitude, by the help of this Prop. the third is ftill a fecret, and likely to continue fo. For, tho' many thoufand pounds have been paid for the pretended difcovery thereof; I doubt we fhall ftill remain juft as wife as we were before the difcovery ; except the ill fucccefs of it happens to teach us fo much wit, as to take better care of our money for the future. And indeed all unlikely ways and means for this purpofe, have been propofed and profecuted ; whilft the only probable method is never thought of, or quite negkaed. FINIS. TLTSTfaEnd DIALLING. OR THE Art of drawing Dials, O N / All Sorts of Planes whatfoever. In THREE PARTS. Seft. I. The fundamental Principles of Di alling. Sed. II. The Practice of Dialling, i-lluC- trated on all kinds of Planes. Sea. III. Of defcribing the common Furni ture of Dials ; and the Conftruc- tion of fome ufeful Dials of other kinds. Temp or a labuntur, taciiifque fenefiimus annis ; Et fugiunt frceno non remorante dies. Ovid Faft. L. VI. [ in ] THE PREFACE. A S the meafuring of time exaclly is a matter of great *^ confequence in all human affairs ; without which we could not rightly know when to go about any particular bufi- nefs ; therefore Dialling becomes a neceffary art. For as cer tain times and feafons are fet apart as mojl proper for perform ing fuch and fuch ailions, in which we are eonftantly imploy- ed ; not only in the feveral parts of the year, but at different times of the day ; therefore we have need of fome fuch inftru ment as a dial, to direil us to thefe ailions, and to inform us when thefe feveral periods of time are come. And tho' we be furnifhed with fome forts of moving machines, which will do this, as clocks and watches ; yet thefe are often out of or der, apt to flop and go wrong, and therefore require frequently to be regulated and fet right, by fome unerring inftrument as a dial; which being rightly conflruiled, will always (when the fun Jhines) tell us truth. And therefore whether we have any clocks or not, we Jhould never be without a dial. The original of Dials feems to be this. When 3$pi atten**- tively obferved the fun's circular motion daily round the earth ; that in the morning he rofe in the eaft, moved about to the fouth, and from thence to the Weft, in the evening, where he fet. They would at the fame time, obferve the alteration of the Jhadows ; that they were projected firft one way and then another ; moving round as the fun did, but towards oppofite points ; and that they grew longer and fhorter as the fun af- cended higher or funk lower . This would give the moft ob vious hint of meafuring time by Jhadows, or for making of dials : tho' they could not be exaclly executed without the rides of art, which therefore were neceffary to be known, and are the Subjetl of this Treatife. For as the motion of the fun is the iv PREFACE. the meafure of time ; fo no apter method could be thought on, than to fhew the time of the day by the fhadow of fome index, properly fixt upon a plane. At the firft, only three parts of the day were dijlinffly" ob ferved, which were fun rife, noon, and fun fet ; but this di- viflon being too general; they afterwards divided, the natural day into 24 parts or hours, as a number very proper and com modious, for diftinguijhing the various times of the day, which were to be diftintlly fhewn by a dial. The foundation of Dialling is entirely depending upot\>Af- fronomy. For the lines on a dial which fhew the boursifare "the interfetlions of the feveral hour circles, with the plane of the dial. And to projeil thefe hour lines, is the fame thing as to projeSt the fphere, upon the dial plane. And' therefore the making of dials depends, upon projecting the fphere, par ticularly the gnomonic projetlion, which is naturally adapted to this purpofe ; and which we have treated on before. In the following Book, the firft feclion contains the grounds of this^art; by Jkewing, how the feveral requifites are to be found, by the interfeilions of the circles of the fphere, with the > plane of the dial, from the principles of fpherical trigoname-* try ; from which the prailical rules are deduced. The fecond feclion contains the prailice, and that three dif ferent ways, r. Geometrically, by rule and compafs, which depends upon the gnomonic projetlion "of the fphere, before deli-r vered. 2. By trigonometrical calculation, by the tables of fines and tangents, which i.s the moft exatl way. 3. By the lines upon Collin's Dialling fcale, which is a method extremely eafy and ready. The third feilion Jhews the way of making fome other forts of dialr'i-and drawing the furniture upon any common dial ; that is the projetlion of the feveral circles, of the fphere; and inferting therein, fuch hours as have been ufed by other nations. And tho' thefe things are not abfolutely neceffary^ they may ferve fometimes as an ornament for a dial. W. Eaierfon. [ I ] DIALLING. DEFINITIONS. D E F. I. DIAL L ING or Gnomonicks,^ is the art of making Dials, and, DEF. II. A Dial is an inftrument for fhewing the hour of the day, by the fun's fhining upon it. The moft common and ufeful fort of dials, are thofe where the hour lines are deferibed on fome plane furface, upon which the fhadow of an index falling, fhews the hour by the termination of its fhadow. D E F. III. Dial planes, are thofe planes on which dials are or may be deferibed. And therefore any plane up on which the fun can fhine, may be a dial plane. All dials are denominated from that great circle of the fphere, to which the dial plane is parallel. Hence, D E F. IV. . A horizontal Dial, is a dial drawn on a plane pa rallel to the horizon. DEF. V. An equinoclial Dial, is upon a plane parallel to the equinoctial. B DEF, D I A L LING. DEF. VI. Erecl or vertical Dials, are fuch dials as are drawn on upright planes, or thofe perpendicular to the horizon. D E. F. VII. A direil Dial, is a dial drawn on a plane facing the eaft, weft, north, or fouth ; and is accordingly called- an Eaft dial, a Weft dial, a North dial, and a South dial. , The eaft and weft dials are parallel to the meridian ; and the north and fouth dials are parallel to the prime vertical. X) E F. VIII. A declining Dial, is one that faces none of the cardinal points, but declines towards the eaft or weft. Thefe are a fouth-eajl decliner, a fouth-weft decliner, a north-eaft decliner, and a north-weft de cliner. DEF. IX. An inclining or oblique Dial, is one whofe plane viands at oblique angles upon the horizon. And thefe may be either direa ones or decliners. DEF. X. A reclining Dial,, is one whofe plane leans back wards, or from you. DEF. XL A proclining or acclining Dial (fometimes called an incliner), is one whofe plane leans forwards or to wards you. The planes on which thefe are drawn will be prbcliners (or incliners) on one fide, and fecliners on the other. D E F. XII. Declination of. a plane, is an arch of the horizon, contained between the plane and the prime verti*- ca.1 ; or between the meridian and a plane perpen- / dicular DIALLING. dicular to the dial plane ; and is always reckoned from the fouth or north. DEF. XIII. Reclination and proclination of a plane, is the an gle it ma-kes with a vertical plane ; or it is the num ber of degrees that the plane leans from you or to wards you, reckoned from the zenith ; being the plane's diftance from the zenith. But Inclination is properly the angle the plane makes with the horizon. DEF. XIV. The center of a Dial, is the point where all the hour lines meet, or towards which they tend. DEF. XV. The Stile, Gnomon or Cock of a Dial, is a pin, or piece of metal, &c. raifed perpendicular upon the plane of the dial ; by the fhadow of this, as an index, the hour of the day is known. D E F. XVI. The Subftile, is the line on which the ftile is ereaed, perpendicular to the plane of the dial. This always goes thro' the center of the dial. DEF. XVII. Stile's height, is the angle which the top edge of the ftile makes with the fubftile ; that is, when the ftile is in the form of a triangle. And then the an gular point is at the centre of the dial. But if the ftile is a pin, the height is the perpendicular length of it ; if a parallelogram, the center is at an infi nite diftance. B z DEF. DIALLING. DEF. XVIII. The fubftile's diftance from the meridian, is the angle which the fubftile makes with the 12 o'clock line. DEF. XIX. -Meridian of the plane, is the meridian perpendi cular to the plane of the dial ; and therefore it is the fame as the fubftile. This is quite different from the meridian of the place, which is that me ridian which is perpendicular to the horizon. ^ DEF. XX. Height of the meridian, is an arch of the great , circle or dial plane, comprehended between the ho rizon and meridian of the place. DEF. XXI. Plane's difference of longitude,, is the angle on the fphere contained between the meridian of the place and the meridian of the plane ; this is alfo called the inclination of meridians. DEF. XXII. . Hour arch, is an arch of the equinoaial, anfwer- ing to the time ; or the angle at the pole. Thus 1 5 degrees anfwer to 1 hour, 30 deg. for 2 hours, ' 45 degrees for. 3 hours, &c. This takes its be- • ginning at the fubftile, DEF. XXIII. Hour angle, is the angle which any hour line up on the plane of the dial makes with the fubftile. D E F. XXIV. The horizontal line, is a line drawn parallel to the plane of the horizon in any dial ; and is made by "the DIALLING. the horizontal plane cutting the dial plane, and pafling thro' that point of the ftile, whofe fhadow fhews the hours. In like manner, DEF. XXV. The equinoclial line, is the interfeaion of the plane of the equinoaial and dial plane ; an azi muth line, the interfeaion of any azimuth with the dial plane, &c. DEF. XXVI. The contingent line, is a line drawn thro' the foot of the ftile perpendicular to the fubftile ; and ferves inftead of the equinoaial for finding the hour points upon it, thro' which the hour lines are to be drawn. When the contingent does not pafs thro' the foot of the ftile, it reprefents the equinoaial. B3 SECT. [ 6 ] Fig. SECT. I. The foundation of Dialling, and the ge neral properties of Dials, and Dial Planes. The Rules for calculating all the Requiftes. To find the time of the funs fhining on any 'Plane, Explanation of the lines on the Dial~ ling Scale. P R Q P. I. JF a right line be fixt any where upon the earth, parallel to the earth's axis ; the Jhadow thereof by the fun, moves uniformly about it, defcribing equal an gles in equal times. It is matter, of obfervation that the fun apparent- _ ly moves about the earth in 24 hours, with a uni-> form motion, defcribing 15 degrees of the equi noaial every hour. And altho' this apparent mo tion of the fun is really to be afcribed to the earth, which aaually moves uniformly round its axis in 24 hours; yet we may refer this motion to the fun, as it makes no manner of difference in the appear- 1 ances, for the rifing, fouthing, fetting, and horary motions, will all be eXaaiy the fame whether the motion be in one ,or the other. , This fuppofi- tion of the fun's uniform diurnal motion, is the foundation of all the meafures we have of time'; and Seft. I.,. DIALLING. 7 and particularly the whole art of Dialling dependsFig. Upon that fuppofition. Now fuppofe the fun in motion round the earth's axis, and that this axis by the fun's rays cafts a fhadow ; this fhadow, being in a right line with the fun which is the luminous body, and the axis which is the opaque body, will always be on oppofite fides of the axis, from the fun -, and therefore if the fun moves uniformly, the fhadow will likewife move uniformly about this axis. Let AB be the earth's axis, CD any other line i. _on the earth, parallel to it. Let ER be drawn „ perpendicular to AB, CD, and in the plane of the ' \ fun's motion ; to cut them in the points F and L. Suppcfe MFI, NLK drawn from the fun. Then by reafon of the vaft diftance of the fun, and the fmall diftance of the points F and L upon the earth compared therewith ; the lines MF and NL are to be looked upon as parallels. Therefore the angle RFM is equal to the angle RLN ; and thefe being equal to the alternate angles GFI and HLK made by the fhadows, thefe alternate angles will alfo be equal. Hence, whilft the fun feems to' move thro' the angle RFM about F, it will alfo feem to move thro' the equal angle RLN about L. And at the fame time the fhadow of F will move thro' the angle GFI, whilft the fhadow of L will move thro' the equal angle HLK. And therefore both the fun and the fhadow of L move uniformly about CD. And thus the fhadow of any other point, and confequently the whole plane of the fhadow of CD, moves uniformly about CD. Cor. i . Hence if the place of the fhadow- of this line CD be marked, or drawn with a black line, at any given hour ; every day when the fun returns, its fhadow fhall fall upon the fame black line, at the fame- 'hour of the day. B 4 Cor. 8 DIALLING. Fig. Cor. 2. Therefore if a line be fixed parallel to the' i . earth's fixis in any fixt plane ; and the place of its fhadow be marked at 12 o'xlock ; and likewife where ' it cuts the plane at every 1 5 degrees of revolution of the fhadow, thefe marks will denote the hours of the : day, and the fhadow will meafure time as truly upon that plane, as the fun itfelf does in the heavens ; that is, .there will be confirucled a true fun dial. For as the fun moves thro' 15 degrees every hour, the' fhadow will likewife move thro' 1 5 de grees every hour, in its revolution round CD ; and therefore its interfeaion with the fixt plane, muft needs point out the feveral hours. Cor 3. Hence it is the very fame thing, whether a dial be drawn upon any given plane, or upon that great circle of the fphere which is parallel to it. PROP. IL If a line be eretted perpendicular to a plane, and the top of it def tribe any curve therein, by its fhadow in the fun. If the plane be removed to any other place on the earth, in a quite parallel fituation ; the Jhadow will defcribe ihe very fame curve as before, and at the fame lime. I call that fituation of a plane quite parallel, when it not only continues parallel to itfelf, or to fome original plane, to which it was parallel at firft; but alfo when any right line drawn in it, continues parallel to itfelf, or to fome original line, to which it was at firft parallel. When only the parallelifm of the plane is regarded, it may have an infinite number of pofitions, for any one fide of it may be up or down. But when a certain line drawn in it, is to have the fame pofition, and tend the fame way, this fixes its fituation, if the face of. the plane Sea. I. DIALLING. 9 plane looks the fame way as before. So that if Fig a plane is moved from one place to another, fo that 2. it continues always parallel to itfelf, and a right line drawn in it continues parallel to itfelf; that plane is always in the required fituation. This be ing explained. Let the line AB be ereaed perpendicular to the plane CD, and let A the top of it by its fhade in the fun, defcribe the curve FGHI. Then if it be removed to cd, in fuch a parallel fituation ; , and the curve fghi be deferibed by the fhadow of the top a. This curve will be the very fame as FGHI. For draw. FB, GB, and fb, gb. And let SAF, TAG be the fun's rays, when the fhadow is at F and G ; and faf, fag, his rays when the fhadow is at /and g. Then from the vaft diftance of the fun, and confequently from the parallelifm of the rays SF, sf at the fame moment, and of TG, tg, alfo at the fame moment, another time ; we fhall have the angle AFB zz afb, and AGB zz agb, alfo F AGzz fag. And fince the perpendicular ab zz AB, therefore a/zz AF, ag zz AG, whence fg zz FG. After the fame manner is proved that gh zz GH, and hi zz HI, &c. Therefore the whole curve fghi is the fame as the curve FGHI. And becaufe the plane is in a parallel fituation, the angle cbf zz CBF ; and therefore the curve is in the fame pofition in refpea of any line drawn upon the plane. It is to be noted, that when the fun is in the e- quinoaial, then FGHI' or fghi will be a right line. For the equinoaial being a great circle, its plane will cut any other plane in a right line, fuppofing "" it either at the center of the earth, or any way on its furface. For any point of the earth, as well as the center, may be taken for the center of the equinoaial, or of the fun's motion at that time. But at other times the earth is not in the center of the io DIALLING. Fig. the fun's motion. For the line drawn from the 2. earth to the fun continues not in one plane, but defcribes a conic furface, and therefore the curve FGHI will be fome conic feaion. Cor. i. Hence the motion of the fhadow of a point as A, upon a plane CD, is the very fame, whether the plane be placed in the center of the earth, or any way on its furface, provided it be in a fituation quite parallel. Cor. 2. The place of the jhadow at the fame mo ment, will be at the fame place G or g, whether the place be at CD or cd. PROP. III. If a line be elevated above a plane and fixt there, its fhadow will be the fame, at the fame moment of time, whatever part of the earth it is placed in, pro vided it be in a fituation quite parallel. 3. Let the line AO be raifed above the plane CD, and fixt there, interfeaing the plane in O. From any point in it as A, let fall the perpendicular AB upon the plane. From the fun at S, draw the line SAG interfeaing the plane in G. Then (by Cor. 2. of the laft Prop.), the fhadow of A at that inftant will fall upon G, wherever the plane CD is placed, fo as to be in a pofition quite parallel. But if a plane be drawn thro' the fun at S, and the line AO, it will cut the plane CD in the right line GO ; where AGO is the plane of the fhadow, and GO the line of fhadow. Confequently the fhadow of AO, will at the fame moment fall upon the fame line GO wherever the plane CD is placed, in a quite parallel fituation. And if the ftile AO be parallel to the plane CD, the fhadow will fall on the fame line, in all places ; Sea. I. DIALLING. 1 1 places; as will be evident by letting fall a per- Fig. pendicular from any other point of the line ; for 3. the fhadow of that point, will always fall upon the fame point of the plane ; and the line of fhadow drawn thro' thefe two points will always be the fame. PROP. IV. In every dial that has a ftile, the edge of it, that gives the Jhadow, muft always be parallel to the earth's axis, and point direclly to the two poles. It has been fhewn in Prop. I. that the fun moves uniformly about the earth's axis ; and confequently, by reafon of the fun's great diftance, it alfo moves uniformly about any line parallel thereto. That is, if a plane be fuppofed to be drawn perpendicu lar to the earth's axis, or to this line, and a circle be deferibed about the ftile, as the equinoaial cir cle is about the earth's axis -, then the fun, and confequently the fhadow, which is oppofite to it, will move uniformly in this circle, defcribing equal angles or equal arches in equal times. And this it will always do, from the parallelifm of the ftile to the earth's axis. But alter the pofition of the ftile, fo that it may not be parallel to the earth's axis, as before ; and then no fuch regular motion will be made about it by the fhadow. Therefore the po fition of the ftile to fhew this regular motion of the fun, muft be fuch, as to be parallel to the earth's axis, or to point direaiy to the two poles. Cor. Hence the ftile muft be fo fixt upon any dial plane, that its edge may point direclly to that pole which is elevated above the dial plane. PROP. i2 DIALLING. Fiff. 3- PRO P. V. In any dial, the angle of the ftile' s height af>ove the fubftile, is equal to the height of the pole dbove the plane of the dial. It has been proved in the laft Prop, that the ftile muft be parallel to the earth's axis. Now the an gle that the axis of the earth makes with a great circle of the fphere ^parallel to the dial plane, is 1 meafured by the arch of a great circle, which is perpendicular to the former, and paffing thro' the earth's axis, which arch is contained between the parallel great circle and the axis, or the angle which the axis makes with that parallel great circle. But this angle is the elevation of the pole above that great circle. And fince the ftile is parallel to the earth's axis, and the fubftile parallel to the inter feaion of the perpendicular circle with the paral lel circle ; therefore the contained angles will be 1 equal ; that is, the angle' of the ftile and fubftile is equal to the elevation of the pole above thedial plane, or above its parallel great circle. Cor. i. If the dial plane be parallel to a great cir cle paffing thro' the poles ; the ftile will have no angle of elevation at all above the dial plane, but will be parallel to the fubftile. Cor. 2. If the ftile be continued thro" the plane on the other fide ; its angle with the fubftile will be equal to the height of ihe other pole, above the other fide of the plane. For the alternate angles being equal, the height of the ftile above the fubftile, is the fame on the other fide of the plane. And the height of the pole being the fame on both fides its parallel great cir cle, Sea. I. DIALLING. 13 cle, it will alfo be the fame on both fides the dial Fig. plane ; that is, equal to the angle of the ftile and fubftile. P R O P. VI. The interfetlion of the meridian of the place and the plane of the dial, is always the hour line of 12 o'clock ; and all the other hour lines are the interfecli- ons of the feveral meridians or hour circles with the plane of the dial, all paffing thro' the ftile. It is plain, that when the fun is in the meridian of any place, that it is 12 o'clock in that place. But the plane of the meridian paffes thro' the ftile, and the fhadow of the ftile being in the fame plane with the ftile and the fun ; this fhadow will appear on the dial plane where the meridian interfeas it; and therefore this interfeaion is the hour line of 12. Likewife /ince all the meridians interfea one another in the earth's axis, they may be fuppofed to interfea one another in the ftile, which is pa rallel to the axis, fince all the motions and revo lutions are alike performed round both. And fince the fun, the ftile, and the fhadow, are all in one , plane ; the fhadow muft appear where this plane interfeas the dial plane. And therefore when the fun is in any hour circle, the fhadow will fall upon that interfeftion, which therefore will be the proper hour line at that time. Cor. 1. All the hour lines meet in one point, which is the center of the dial. For as all the meridians pafs thro' the ftile, they will all pafs thro' the point where the ftife cuts the dial plane; that is, thro' the center of the dial. And all the interfeaions with the dial plane, that is, all the hour lines, will meet in that point, which is the center. Cor. i4 DIALLING. Fisc. Cor. 2. When the dial plane paffes thro' the poles, the hour lines will be parallel to one another. For then the center, or the point where all the" hour lines concur, iaat an infinite diftance. PROP. VII. ¦ In any dial with a center, if the hour lines be pro duced thro' the center, fo as to appear on the other fide of the plane ; and if the file be alfo produced thro' the plane; you will then have a dial for the back fide of the plane ; where the fame hours belong to the fame lines produced. For fince the meridians or hour circles interfea the dial plane in right lines, and the ftile is in the plane of every hour circle ; its fhadow will al ways be in that interfeaion. Therefore in what ever part of the hour circle the fun is, its fhadow will always be in the interfeaion of it with the dial plane, and therefore in the fame right line. But it will likewife be the fame hour of the day, when the fun is in the fame hour circle, whether it fhines upon one fide of the plane or the other. Whence the fame right line continued thro' the center will always denote the fame hour. And if the fun was in different or in oppofite points of the fame hour circle ; any line will ftill denote the fame hour, tho* fometimes it may be the morn ing hour, and fometimes the afternoon hour. D PROP. VIII. In all upright dials, the hour line of 12 is perpen dicular to the horizon. For i Sea. I. DIALLING. 15 For the interfeaion of the meridian of the place Fig. with the dial plane, (by Prop. VI.; is the hour line of 12 o'clock. But the meridian of the place is perpendicular to the horizon ; and the dial plane is alfo perpendicular to the horizon, therefore (Geom. V. 15.) their intjerfeaion is perpendicular to the horizon ; that is, the 12 o'clock line is per pendicular to the horizon. PROP. IX. In a direct eaft or weft reclining dial ; the hour line cf 1 2 is parallel to the horizon. For the meridian of the place is perpendicular to the prime vertical ; and the dial plane is per pendicular to the prime vertical ; and therefore the common interfeaion of the meridian and dial plane is* perpendicular to the prime vertical. But the interfeaion of the meridian and dial plane (by Prop. VI.) is the hour line of 12. Therefore the hour line of 1 2 is perpendicular to the prime vertical ; and therefore is parallel to the horizon. PROP. X. A dial removed from its true place to any other, and placed in a fituation quite parallel, and thefunfhine- ing on it ; will always fhew what a clock it is, at the place it came from. For (by Prop. III.) the fhadow of the ftile of this dial, will fall upon the fame hour line, at the fame moment of time, wherever the dial is placed, in fuch a parallel fituation. Therefore at every parti cular hour of the day, at the original place, the fhadow will fall on the fame hour of the dial at the other place, juft the fame as if the dial had remained 16 DIALLING. Fig. remained at the firft place. And therefore it al ways fhews the hour of the day at the firft place. Cor. i. If any dial whatever be removed from its original place, to another place under the fame meri dian, and placed in a quite parallel fituation, it will (hew the hour of the day truly at this laft place ; that is, it will go truly there. For all places under the fame meridian have each hour of the day, at the fame moment of time. That is, it is 1 2 o'clock at one place, when it is 1 2 o'clock at the other ; and one o'clock at the firft place, when it is one at the other, &c. Cor. 2. But in other places, not under the fame meridian, it will not fhew the true time for this laft place, in this fituation, but 'will go f after or JloWer, proportional to the difference of longitude from the ori ginal place, according as it is removed eaftward jr weftward. P R O P. XL If a reclining plane be fet fo far back, {in a great circle perpendicular to that plane,) juft as many degrees as it reclines, and parallel to its firft pofition ; it will then be an eretl plane. 1 4» Let AB be the reclining plane, GB a great cir cle -of the fphere perpendicular to it, C the center ; remove the plane to G, and draw CBD, and CG ; then DB, being a vertical plane, the angle ABD will be the reclination of the plane AB. And AB being removed thro' the arch BG, into a parallel fituation at G -, fo that the angle ABD may be equal to GB, or the angle GCB ; then the line GC will be parallel to AB ; and GC being a radius of the fphere, it is perpendicular to the furface at G ; but FG being alfo parallel to AB, FGC is one right line, and FG a vertical plane. Cor. Sea. I. DIALLING. 17 Cor. 1. If a vertical plane FG be moved in a pa-V'xo. rallel direclion thro' any arch of the great circle GB, 4. perpendicular to it ; its reclination ABD will be equal to that arch GB. Cor. 2. Likewife in a proclining plane AB, if it be fet fo far forward, that the arch GB may be equal to the degrees of proclination, and be placed parallel \ ¦ to itfelf; it will then be a vertical plane FG. Cor. 3. If a reclining plane AB, continuing pa rallel to itfelf, be carry' d thro' the arch of a great circle perpendicular thereto BH, equal to the plane's inclination to the horizon, ABG ; it will then be a horizontal plane at H. For fince angle ABD zz GCB, and ABG zz BCH, GCH zz DBG zz a right angle, and AB or FG parallel to the horizon at H. PROP. XII. If any vertical plane ABID, whofe horizontal line r is BI, he removed, to any place b in the plane of its great circle, in a quite parallel pofition. Its horizon tal line bi, will be elevated above the horizon of the place b, the quantity of the arch Bb. Or angle hbi zz arch Bb. From the center C of the great circle Bb, draw BC, Cb, and draw bf parallel to BC, and the ho rizontal line bh, of the place b. Then fince the angles CBI, fbi, Cbh are right angles ; we have angle Cbf zz hbi. But from the parallels BC, bf, angle O/zz BCb; therefore /Wz zz BCb zz arch Bb. Cor. Hence if the plane BD be removed thro' an arch of 90 degrees, the horizontal line BI will become a vertical line. C PROP. i8 DIALLING. Fig. PROP. XIII. If an ereSl declining dial plane be removed along the plane of its great circle, till its difference of longitude be equal to the plane's difference of longitude ; it will become a full fouth or north erecl plane. 6. Let NESW be the horizon, Z the zenith, HZD the great circle wherein the dialling plane is placed, being at Z -, NS the meridian, P the pole. Draw the meridian PB, perpendicular to the great circle HD ; and B is the place where the dial plane is a full fovith or north plane, becaufe the meridian of the place B is perpendicular to it. And (by Def. 19.) PB will be the meridian of the plane. Therefore (Def. 2 1 ) the plane's difference of longitude is the angle ZPB, equal to that thro' which the plane was moved. * Cor. 1. The plane's difference of longitude is the dif ference between the longitude of the given place, and ihe longitude of the place where the dial plane is a di rt 'tl fouth or north plane. Cor. 2. And to find the new latitude ofB. In the fpherical triangle PZB ; rad : cof. latitude (S.ZP) ; ; cof. declination (S.Z) : cof. new latitude (S.PB.) PROP. XIV. A reclining eaft or weft dial plane will be an up right plane, under the Jame meridian, when placed 90 degrees from thence in a parallel fituation. g For let Z be the zenith as before, WE the prime *' vertical, NAS the great circle parallel to the dial plane. Then the circle NAS cuts the meridian NS in the points N and S. And (by Prop, XL), if the Sea. I b I A L L t N G. 19 the plane be moved from Z to A in a parallel fi-Fig. tuation, it will be erea at A. And (by Prop. XII.) 6; it will be erea in any place of its great circle NAS. And therefore at N and S, it will both be erea, and in the meridiah, where ZN and ZS are 90 de grees. Cor. The new latitude N or S, where an eaft or we/l recliner is eietl, will be the complement of the old latitude at Z. And the new declination is the ¦complement of the reclination. For NP zz complement of ZP* and ZP zz comp. NP zz'new lat. Alfo the angle WN A or the decli nation zz comp. ANZ or of AZ the declination. PROP. XV; Prob. The declination and reclination of a dial plane being given. To find the new latitude and declination, where itjhall be ah upright plane, upon the fame meridian. Let NASW reprefent the fphere; NESW the horizon, P the pole, Z the zenitJh, NS the meri dian, WQE the equinoaial, WZE the prime ver tical, ?\, ?2, P3-, &c. hour circles, HMA a great circle parallel to the dial plane. Draw the Vertical ZB perp. to HMA. Then M will be the new place, upon the meridian NS, where the plane is upright, and PM the complement of its lati tude. Therefore in the right angled fpherical triangle ZBM, there is given the angle BZM the declination j and the fide ZB the reclination ; to find the angle BMZ the complement of the decli nation at M ; and the hypothenufe ZM, which fubtradled from PZ the comp. of the latitude, when the plane runs between the zenith and pole, f otherwife PZ fubtraaed from it, when the plane is below the pole)-, leaves PM, the complement of the new latitude. C 2 Cor. 20 DIALLING. Fig. Cor. i. As. cofine of the old declination 7. Radius: So tangent, reclination : Tangent of an arch. And the difference between this arch and the complement of the latitude, is the complement of the new latitude. For (by Cafe 9. right angled fpherical triangles) cof. BZM : rad : : tan. BZ : tan. ZM, Cor. 2. Radius : Sine of the declination : : Cojine — reclination : Sine — new declination. For (by Cafe 8.) rad : S.BZM : : cof. BZ : cof. BMZ. PROP. XVI. Prob. The declination and reclination of a plane being gi ven; to find the new latitude, and longitude, where it jhall be a direcl north or fouth upright plane. Let NS be the meridian as before, Z the zenith, P the pole, and HMA the great circle, to which the dial plane is parallel. Draw the meridian PFi, perpendicular to the given circle HMA, then F is the place where the plane becomes a direa north or fouth vertical plane-. And drawing ZB perp. to HMA. In the triangle BMZ there is given the reclination BZ, and declination zz angle BZM, to find ZM and angle BMZ, as in the laft Prop. Then in the triangle PMF right angled at F ; there is given PM and angle PMF ; to find the comple ment of new lat. PF, and difference of longitude MPF. Cor. 1. Let B zz PM {the difference between MZ and the comp. of the latitude. Then) ip, Radius : Sine ofB:: Cojine Sea. I. DIALLING. 21 Cofine — declination at M : Fig. Cofine — new latitude. 7. For (by Cafe 2.) rad : S.PM : : S.PMF : S.PF. Cor. 2 . Radius : Cofine of B : : Tan. PMF : Cot an. MPF the diff. longitude, (by Cafe 3.) Or thus, Radius : Cotan. B : : Cotan. new latitude : Cofine — diff. longitude. For (by Cafe 4.) Rad : Cotan. PM : : Tan. PF : Cof. MPF. PROP. XVII. Any dial whatever, whether direcl, declining, or in clining, &V. made for any latitude -, will Jhew the time truly, in all places of the fame latitude ; provi ded it be placed in a like fituation, in regard to the meridian of the place ; that is, provided it have the fame declination, reclination, and horizontal pofition. For if the dial be placed alike in different places under the fame parallel, the fhadow of the gno mon, at the fame hour of the day* cannot but fall upon the fame hour lines. For imagine two dials to be made for two fuch different places ; there being exaaiy the fame data for both ; the fame hour lines and every thing elfe will be the fame in both. And that they may both go true in their refpeaive places, for the apparent time, there is nothing more required, than to give them both the fame fituation. Cor. 1. Any fort of dial made for one place; will go true in any other place, tho' in a different latitude, and longitude; provided it be Jet in a proper fituation. C 3 • For 22 DIALLING- Fig, For by this Prop, if it be removed to any place in the fame parallel, or of a different longitude ; and placed alike, , in regard to the meridian, ho rizon, &c. it will go truly there, for that place, And by Cor, i. Prop. X. if the fame dial be remo ved from this fecond place to a third place, under the fame meridian -, and fet. there in. a quite parallel pofition, to that it had in the fecOnd place ; it will go truly at this third place-. Therefore it would go alike at all the three places. Cor. 2, Any dial made for any place of the world^ ' will go truly at any other place of the world, being placed in a proper pofitioft. PROP. XVIII. Prob, To find the requifites for a horizontal dial. „ Here is nothing required, but the latitude of '' the place ; from which the angles of the feveral hour lines with the 1 2 o'clock line muft be found. Let NESW be the horizon, NS the meridian^ P the pole, Z the zenith -, Fc, ¥d, Yg, &zc. the feveral hour circles upon the fphere. Then we are to find the arches Nf, Nd, N^, &c. or the an gles at the center Z of the circle NESW. There fore in the right angled fpherical triangle PNr, there is given PN the latitude of the place, and angle NPc, which is 1 5 deg. for an hour, 30 deg. for 2 hours, 45 deg. for 3 hours, &c. to find the oppofite fide Nf, or Nd, pr Ng, &c. which are the Ifour angles. Cor. 1 . Radius : Sine of the latitude : : Tan. hour arch : Tan. hour" angle. For (by Cafe 7. right angled* fpherical triangles,) Rad : S.PN ; : tan. NPf, or the correfpondent arch of Sea. I. DIALLING. 23 of the equinoaial : tan. Nc, which is the meafure of Fig. the hour angle : : and fo tan. NP*/ : tan. Nd, &c. 7. Cor. 2. A horizontal dial made for fouth latitude, ' will ferve equally for the fame degree of north latitude ; turning the fouth end to the north, and reckoning the hours the contrary way from the meridian. PROP. XIX. Prob. To find the requifites for an erecl diretl fouth or north dial. Here the height of the pole above the plane muft 7." be had, and this is equal to the complement of the latitude. Therefore let NESW be the horizon, Z the zenith, P the pole ; WZE the plane of the dial. Then the pole is elevated above the plane of the dial, the arch PZ, which is the complement of NP, or the complement of the latitude. Then if the hour circles Pi, P2, P3, &c. be drawn, cut ting the right circle WZE in q, r, s, &c. Then to find the arches Zq, Zr, Zs, &c. In the right an gled fpherical triangle PZ^, there is given PZ the complement of the latitude, and the angle ZPq ; to find the arch Zq, or the angle at the center. And the like for the arches Zr, Zs, &c, in the tri angles ZPr, ZPj, &c. Cor. r. Radius : Cofine'latitude : : Tangent-hour arch : Tang, hour angle. For (by Cafe 7.) rad : S.PZ : : tan. ZPj -: tan-. Zq : : tan. ZPr : tan. Zr, &c. Cor. 2. In a diretl fouth or north inclining dial, the fame analogy muft be ufed, only taking for the fecond term, the fine of the pole's height above the dial plane 5. to be found by Prop. XL C 4 Cor. 24 DIALLING. Fig. Cor. 3. A diretl fouth dial; is a horizontal dial on' 7. the fouth fide of the globe; and a north dial, on the •north fide. PROP. XX. Prob. To find the requifites, for erecl declining dials. Here the latitude of the place, and the declina tion of the plane muft be given ; and what is fur ther wanted muft be found from thefe. And that is, 1 . the height of the pole above your dial plane. 2. The diftance of the fubftile from 12 o'clock. And 3. the plane's difference, of longitude. Let NESW be the horizon, C its center, NS the meridian, P the pole, and Z the zenith ; HA the declining vertical plane; Pi, P2, P3, &c. me ridians or hour circles, cutting the circle HA in /, v, &c. Let the meridian Pv2 be perpendicular to HA, then the plane of the meridian Pvi will be perpendicular to the plane of the dial ; and there fore the angle ZPv is the plane's difference of lon gitude (by Def. 21), and the arch Pv is the height of the pole above the plane. And the arch Zv is the fubftile diftance from the meridian. Therefore in the right angled fpherical triangle PZ?;, all the 3 requifites will be found. For we have PZ the complement of the latitude, and the angle PZA ¦the complement of the declination AZE ; from whence will be found Pv, Zv, and the angle ZPv. Cor. 1. As radius : Sine declination : : Cotan. latitude : Tan. fubftile' s dftance from 1 2 o'clock, and lies the contrary way as the declination. For (by Cafe 1. of right angled fpherical trian gles), rad : cof. PZv : : tan. PZ : tan. Zv. . Cor. lt>" Sea. I. DIALLING. 25 " Cor. 2. Radius : Fig Cofine declination : : Cofine-latitude : . Sine ftile' s height. For (by Cafe 2), rad : S.PZ : : S.PZv : S.P^. Cor. 3. S. latitude : Radius : : Tan. declination : Tan. plane's diff. longitude. For (by Cafe 3), rad : cof. PZ : : tan. PZt; : cotan. Z?v : : tan. ZPv : cotan. PZ-y. Cor. 4. For the hour angles, it will be, As radius : Sin. pole's height ': : Tan. hour arch from the meridian oj the plane : Tan. hour angle from the fubftile. For let P/i be any hour circle; in the triangle Ptv, there is given the pole's height Pv, and the hour arch, equal to the angle tPv; to find tv, (by Cafe 7), rad : S.Pv : : tan. tPv : tan. tv zz angle at the center, or the hour angle. That all the requifites are rightly found from thefe proportions, will appear thus. The planes of the two meridians PZ, Pv, interfea one another on the fphere, in an angle ZPz?, equal to the plane's difference of longitude. And the fame me ridians, interfea the plane of the dial in an angle equal to ZV, as it fhould be. For they interfea one another in the center of the fphere, which is the center of the dial. And they interfea the dial plane HA in Z and v. and Zv meafures that angle at the center. And for the fame reafon, tv meafures the hour angle from the fubftile. Again Pv meafures the angle of the ftile above the fubftile. For Pv is perpendicular to HA, and the arch Pv meafures the angle formed at the cen ter of the fphere, which is the center of the dial ; one 26 DIALLING. Fig. one fide being the axis drawn from P, the othe^ 7. the fubftile drawn from V. Alfo ZP-y has been fhewn to be the plane's dif ference of longitude. Cor. 5. The plane's dif. longitude on the fphere, is converted into the fubftile' s diflance from 12, by the interfetlion of the two meridians {of the place and of the plane), with the dial plane. For the angle ZPw expreffes one, and arch Zv the other. Cor. 6. And every hour angle on the fphere, is converted into the hour angle in the dial, by the in terfetlion of the fame two hour circles, with the plane of the dial. For the angle tPv exprefles the hour angle on the fphere, and tp the angle at the center. Scholium. Tho' the rules laid down in thefe Corollaries are fufficient for finding the feveral requifites ; yet by having other data, they may be found by other proportions, fome of which for variety, I fhall here let down. In Cor. 1. For finding the fubftile's diftance. Cof. pole's height above the plane (Pv) ; Sine-latitude {cof. PZ; : : Radius : Cof. fubftile' s dift. from 12, By Cafe 6. Or Radius : Sin. declination {cof PZv) : ; Tan. pole's height above the plane (Pv) : Sin. fubftile' s diftance (Lv). By Cafe 10. In Sea. I. DIALLING. 27 In Cor 2. For finding the ftile's height. Fig, Cof. jubftile's diftance (tv) : 1- Sin, latitude (cof. PZ) : : Radius : Cof. ftile's height (Pv), By Cafe 6. ; Or Radius Sin. fubftile 's diftance {Zv) : Cotan. declination (tan. PZv) : Tan, ftile's height (Pv). By Cafe 7. In Cor. 3. For the plane's diff. longitude, Cof. latitude (S.PZ) : Radius : : Sin. fubftile' s diftance [Zv) : Sin. plane's diff longitude (ZPt>). By Cafe 5. Or Radius : Tan. latitude {cotan. PZ) : : Tan. ftile's height {Pv) : Cof. plane's dip. longitude (ZPp). By Cafe 4, Or Radius : Cof. declination (S.pZi>) : ; Cof. Jubftile's diftance {Zv) : Cof. plane's dif. longitude {ZPii). By Cafe 8. Or Cof. ftile's height (Pv) : Sin. declination (cof. PZw) : : Radius : Sin. plane's dif. longitude (ZPi>;. By Cafe 1 i; Or Sin. ftile's height (Pv) : Radius : : Tan. Jubftile's diftance (Zv) : Tan. plane's dif. longitude (ZP^;. By Cafe 13, PROP, 28 DIALLING. Fig. 7. PROP. XXI. Prob. To find the requifites for a diretl eaft or weft incli ning or reclining dial. The latitude of the place and the reclination of the plane muft be given. And from thefe muft be found, 1. The height of the pole above the plane. 2. The fubftile's diftance from 1 2 o'cock. 3. The plane's difference of longitude. §_ Let HZON be the meridian, HO the horizon, ZN the prime vertical, P the pole, Z the zenith. HFO the inclining plane. Let the meridian PFD be perpendicular to HFO. Then the plane of the meridian PFD will be perpendicular to the plane of the dial, apd PF will be the meridian of the plane (Def. 19.), and OPF the plane's difference of longitude (Def. 21.) alfo PF (or angle PCF) will be the height of the pole above the plane. And FO (or angle FCO) the diftance of the fub ftile from the meridian. Therefore in the right an gled fpherical triangle PF"0, there is given PO the latitude of the place, and the angle POF the reclination or proclination of the plane ; to find FO, FP, and angle OPF. Cor. 1 . Radius : Cof. reclination (or proclination) : : Tan. latitude : : Tan. fubfiile's ' dift. from , 1 2 o'clock ; running upwards towards the north, or downwards towards the fouth. For (by Cafe 1), rad : cof. POF : : tan. OP : tan. OF zz angle OCF, which the. meridian FC (or fubftile) makes with CO the 12 o'clock line, (or meridian of the place;, at C die center of the dial. Cor. Sea. I. DIALLING. 29 Cor. 2. Radius : Fig. Sin. latitude : : 8. Sin. reclination {or proclination) : Sin. ftile's height. For (by Cafe 2.) rad : S.OP : : S.POF : S.PF zz angle PCF, which the ftile PC makes with the fubftile FC, at C the center of the dial. Cor 3. Radius : Cof. latitude : : Tan. reclination (or proclination) : Cotan. plane's dif. longitude. For (by Cafe 3.) rad : cof. OP : : tan. POF : co tan. OPF zz angle made by the meridian of the place, and that of the plane. Cor. 4. For the hour angles, it will be, As radius : Sin. pole's height above the plane : : Tan. hour arch, from the meridian of the plane : Tan. hour angle from the fubftile. For, drawing any hour circle P1D, in the right angled fpherical triangle PFI, there is given PF, and angle FPI, to find FI ; by Cafe 7. Rad : S.PF : : tan. FPI : tan. FI zz angle FCI, which the cor- refpondent hour line on the dial makes with the fubftile, at the center. The calculation in the triangle OPF is for an eaft dial ; but for a weft dial, let A be the pole, and draw the meridian AGB perpendicular to HGO; then there are the fame data in the triangle AGH, as in PFO ; whence the fame things will be found for a weft dial. Scholium. Other rules for finding the requifites may be de duced from the fame triangle CPF, as follows. Cor. 1. For the fubftile's diftance. As Cof. pole's height above the plane ( PF) : Cof. 30 DIALLING* Fig. Cof. latitude (P0) : : 8. Radius : Cofifubft. dift, from 12 o'clock (FO), By Cafe 6. Or Radius : Cotan. reclination (POF) i : Tan. ftile's height, or pole's height (PF} X Sin. fubftile' s diftance (FO). By Cafe 10. Cor. 2. For finding the ftile's heights As Cof jubftile's diftance (FO) ; Coj. latitude (PO) : : Radius : Cof ftile's height (PF). By Cafe 6. Or Radius : Sin. Jubftile's diftance (FO) : : Tan. reclination (POF; i Tan. ftile's height (PF;. By Cafe 7. Cor. 3. For the plane's diffl longitude. As Radius : 1 Cotan. latitude (PO) : i Tan. height oj the pole above the plane (PF) f Coj. plane's dif. longitude (FPO)» By Cafe 4. Or Cof. Pole's height above the plane (PF) t Cof. reclination (POF) : : Radius : Sin. plane's dif. longitude (FPO)» By Cafe 11. Or Sin. latitude (OP) : Radius : : Sin. Jubftile's diftance (FO) : Sin. plane's diff. longitude (FPO)» By Cafe 5. PROP. Sea. 1. DIALLING. 3* Fig. PROP. XXII. Prob. To find the requifites for fouth declining reclining dials. Here muft be given the latitude of the place, and the declination and reclination of the plane ; and from thefe muft be computed, i. The height of the meridian. 2. The height of the pole above the plane. 3. The diftance of the fubftile from the hour line of 12. And 4. The plane's difference of longitude. Let WPASH reprefent the fphere, C its center, 7. WNES the horizon, NS the meridian, WOE the equinoaial, P the pole, Z the zenith, and HMA the plane of the dial. Draw the meridian PFi perpendicular to the circle HBA ; then the plane of this meridian will be perpendicular to the plane of the dial, and therefore PF will be the meridian of the plane, and the angle MPF the plane's dif ference of longitude, and PF the height of the pole above the plane, and MF the fubftile's dif tance from 12, and MA the height of the meri dian. Therefore all the requifites will be found in the triangles NAM and PMF, right angled at N and F. Therefore in the triangle NAM, there is given NA the complement of the declination AE, and angle NAM the complement of the reclina tion MAZ ; to find AM, NM, and angle NMA. And the latitude NP being given, PM will be known. Then in the triangle MPF, there is gi ven PM, and the angle PMF, to find MF, PF, and angle MPF, as required. Cor. 1 . Sin. reclination :. Radius : : Cotan. declination : Tan. height of the meridian. For 32 DIALLING. Fig. For (by Cafe 9, right angled fpherical triangles), 7. cof. MAN : rad : : tan. NA : tan. AM zz angle ACM at the center of the fphere, or of the dial, contained between the horizontal line CA, and the meridian CM or 1 2 a clock. Cor. 2. Radius : Cof. declination : : Cotan. reclination : Tan. of an arch A. Then put B zz difference between A and the latitude. If A is lefs, your pole is elevated ; if greater, the op pofite pole. Then Cotangent oj B : Coj. reclination : : Sin. declination : Tan. Jubftile's diftance jrom 12. And the fubftile runs upwards towards the north in reclining S.planes, and their oppofite incliners. For in the triangle NAM (by Cafe 7), rad : S.NA : : tan. NAM : tan. NM zz A -, then NM — ¦7 NP zz PM zz B. Alfo (by Cafe 8), rad : S.NAM ' : : cof. NA : cof. NMA zz S-NAM x cof. NA rad And in the triangle MFP, (by Cafe 1;, cot. PM : rad : : rad:tan.PM : : cof.NMA ( = S-NAMxcofNA) v rad ' ivri? S.NAM x cof. NA , .,„., : tan. MF zz _ _ zz angle MCF, cotan. MP e at the center, and in the plane of the dial. Cor. 3. Cof. fubfiile's diflance : Cof. B : : Radius : Cof file's height. For (by Cafe 6), cof. MF : cof. MP (B) : : rad : cof. PF zz angle PCF at the center of the dial, CF being in its plane, is the fubftile. Cor. Sea. I. DIALLING. 33 Cor. 4. Sine of B : Fig* Radius : : 7. Sin. jubftile's diftance : Sin. plane's diff. longitude. For (by Cafe 5), S.PM (B) : rad : : S.MF : S.MPF zz angle on the fphere, between the meri dian of the plane, and the meridian of the place. Cor. 5. For the hour angles it will be As Radius : Sin. pole's height above the plane : : Tan. hour arch from the meridian of the plane : Tan. hour angle from the fubftile. For drawing any hour circle P03, interfeaing the circle HMA in 0. Then there will be given the angle FPo, contained between the meridian of the plane PFi, and the meridian P03. Therefore in the right angled triangle PFo, there is given PF the. height of the pole above the plane, and angle FPo, to find Fo by cafe 7. Rad : S.PF : : tan. FPo : tan. Fo zz angle FCo, at the center of the dial, made between the fubftile ZF, and the hour line Co correfponding to the hour circle Po. Cor. 6. The height of the meridian is equal to the angle made at the center of the dial, by the 1 2 o'clock line, and the horizontal line ; and jhews how much the 1 2 o'clock line is elevated above the horizon ; and it runs upwards in reclining planes, and downwards in proclining ones, towards the Jame hand as the declina tion is. For MA is the meafure of the angle MCA con tained between the 12 o'clock hour line MC, and the horizontal line CA ; being in the plane of the dial ACHM ; and the angular point at C, the cen ter of the dial. For the fame' reafon MF, or the angle MCF, fhews what angle the 1 2 o'clock line MC makes D with 34 DIALLING. Fig. with the fubftile CF, at the center C ; the triangle 7. MCF being in the plane of the dial. And like- wife Fo or angle FCo fhews the angle contained between the fubftile FC and the hour line Fo ; the triangle FCo being alfo in the plain of the dial ; and fo of others. Here the plane of the dial ABH falls between the zenith and the pole, but if it pafs between the pole and the horizon, the calculation will be the fame ; and then the pole P will be elevated on the other fide of the plane. But if the dial plane pafs thro' the pole, then B will be o ; and MF, PF, are o. Hence Cor. y. If A (NM) is greater than the latitude (NP), the pole is depreffed below the plane, and the op pofite pole elevated above it. If A is lefs than the latitude, the pole is elevated above it. If they be equal, the pole falls in the plane of the dial; and then MF, PF, are nothing. For when A zz latitude, B zz o. Cor. 8- When the reclining' plane falls in the pole ; then to find the plane' s difference of longitude, it will be, As radius : Sin. latitude : : Tan. declination : Tan. plane's dif. longitude. For when M approaches near to P, the triangle MPF will be a plane triangle, and the angle MPF (the plane's dif. longitnde) will be the complement of PMF or NMA, that is, when M falls upon P, the plane's dif. longitude will be the complement of NPA. Therefore in the triangle NPA, there are given NP the latitude, and NA the comp. de clination, to find the angle NPA (by Cafe 13), S.NP : rad : : tan. NA : tan. NPA : : cotan. NPA : cotan. NA. Other- Sea. I. DIALLING. 35 Otherwife thus. Fit*. Radius : *, S. .declination : : Cof. reclination : S. plane's dif. longitude. For (by Cafe 8), rad : S.NAP : : cof. NA : cof. NPA. And other proportions may eafily be found to do the fame thing, as this which follows. Cof. latitude (NP) : S. reclination {coj. NAP) : : Radius : Cof. dif. longitude (S.NPA). Cor. 9. The fame rules ferve equally for an inclining plane, ufing the proclination inftead of the reclination ; but here A will always be greater than 90 degrees -, and the oppofite pole always elevated above your pi me. For then the point M falls between Z and S. Scholium. From the fame two triangles NAM, and PMF, other rules for finding the jequifites may be dedu ced ; fuch are the following. In Cor. 2. For the fubftile's diftance. As Radius : Coj. reclination (S.NAM) : Sin. declination (cof. NA; : : Coj. an angle M (NMA;. Then Radius : Cof. M (PMF; : : Tan. B (PM) : Tan. fubftile's diftance (MF). By Cafe 1 . Or Coj. pole's height above the plane (PF; : Coj. B (PM) : : Radius : Cof. fubftile's diftance (MF;. By Cafe 2. D 2 Or 36 DIALLING. Fig. Or Radius : 7. Cotan. M (TMF) : : Tan. pole's height above the plane (PF) : Sin. fubftile's diftance (MF). By Cafe 10. Or Cotan. declination (tan. NA) : Tan. pole's height above the plane (PF) : : Sine oj A (NM) : Sin. fubftile's diftance (MF). This appears from Cor. 1. Prop. 27. Sea. III. Trigonometry. In Cor. 3. for the angle of the ftile's height. As Radius : Sine of B (PM) : : Sine of M (PMF) : Sin. ftile's height above the plane (PF). By Cafe 2. Or Cof. fubftile's diftance (MF) : Radius : : Cofine of B (PM) : Cof. ftile's height (PF). By Cafe 6. Or Radius : Tangent of M (PMF) : : Sin. fubftile's diftance (MF; : Tan. fitile's height (PF). By Cafe 7. Or Sine of A fNM) : Shi. fubftile's diftance (MF) : : Cotan. declination {tan. NA) : Tan. ftile's height (PF). By Cor. 1. Prop. 27. Sea. III. Trigonometry. Or Sin. height of the meridian (MA) : Sine of B (PM) : : Cof. Sea. I. DIALLING. 37 Cof .declination (S.NA) : Fig. Sin. ftile's height (PF). 7 By Cor. i. Prop. 26. Sea. III. Trigonometry. In Cor. 4. For the plane's dif. longitude. As Radius : Cojine of B (PM) : : Tangent of M (PMF) : Cotan. plane's dif. longitude (MPF). By Cafe 3. Or Radius : Cotan. B (PM) : : Tan. pale's height above the plane (PF) : Cof plane's dif. longitude (MPF). By Cafe 4. Or Cof. ftile's height (PF; : Coj. M f PMF) : : Radius : Sin. plane's dij. longitude ('MPF). By Cafe 11. Or Coj. pole's height above the plane- (PF) : Sin. declination {cof. NA) : : Cof. reclination (S.NAM) : Sin. plane's dif. longitude (MPF). For (by Cafe 8), rad : cof. NA : S.NAM : cof. ,...«. cof. NAx S.NAM , , .- „,, NMA zz — « - , and rad : col. PF : : r S.MPF : cof. PMF or NMA zz cof. PF x S.MPF . r therefore cof. NA x S.NAM zz-cof. PFx S.MPF. It may be obferved, that inftead of the two tri angles NAM, and MBF, the two triangles BZM and MBF may be made ufe of, the triangle BZM being complemental to the triangle NAM; from which two triangles the very fame conclufions will follow as before. D 3 PROP. Fig 3'8 D I A L L I tf G. PRO P. XXIII. Prob. To find the requifites jor a north declining, reclining dial. ¦ Here alfo the latitude of the place, artd the de clination and reclination of the plane muft be gi ven ; to find, i. the height of the meridian. 2. The height of the pole above the plane. 3. The diftance of the fubftile from the hour line of 12. 4. The plane's difference of longitude, as before. As in the laft Prob. Let WNES be the horizon, NS the meridian, P the pole, Z the zenith, and HMA the plane of the dial, and PF an hour cir cle perpendicular to it, which will be the meridian of the plane. Let ZB fje alfo perpendicular to HBA. Then all the requifites will be found in the right angled triangles ZBM and PFM. There fore, In the triangle BMZ, there is given the angle MZB the declination equal to EZA, and ZB the reclination of the plane ; to find BM, ZM, and angle ZMB ; and having ZP the complement of the latitude, PM will be zz PZ + ZM. Then in the triangle MPF, there is given PM and angle PMF; to find the fubftile's' diftance MF, the ftile's height PF, and plane's dif. longitude MPF or MP/. Cor. 1. Sin. declination : Radius : : Cotan. declination : Tan. height of the meridian, running up wards, towards the fame hand as the declination. But to the contrary hand, in procliners. For (by Cafe 7), rad : S.ZB : : tan. MZB : tan. BM : : cot. BM : cot. MZB. And here HM being the D- Sea. I. DIALLING. 39 the height of the meridian, lies now towards theFio- weft ; which was towards the eaft, when the recli- g° nation of the plane was northward. Cor. 2. Radius : Cof. declination : : Cotan. reclination : Cotan. of an arch A. Then put B zz A + complement of the latitude. And the north pole is always elevated in recliners. Then Cotan B : Cof. reclination : : Sin. declination : Tan. fubftile's diftance from 1 2 ; whicfy will be greater than 90% when B (PM) is greater than 90°. For in the triangle ZMB (by Cafe 9), cof. BZM : rad : : tan. ZB : tan. ZM (A) : : cot. ZM : cot. ZB. And in the triangle NAM (by Cafe 8), rad : S.NAM : : cof. NA : cof. NMA or PM/ zz S.NAM x cof. NA . , • „, • ¦, „.,„, , - . And in the triangle PM/ (by rad Cafe 1), rad : cof. PMF : : tan. B ."tan. FM. Or t> *a r -di\/tt? fS.NAM X cof. NA\ cotan. B : rad : : cof. PMF ^- . ) ; v rad J. tan. FM or /M zz tan. fubftile's diftance. Cor. 3. Cof fubftile's diftance : Coj. arch B : : Radius : Coj. ftile's height. And if B is greater than 90°, the jtile's height is greater than 900; or it. will appear elevated towards the north part of the plane. - For in the triangle MP/ (by Cafe 6), cof MF : cof. MP'(B; : : rad : cof. FP or /P. D 4 Coq 40 DIALLING Fig. Cor. 4. Sine of B : 9. Radius : : Sin. fubftile's diftance : Sin. plane's dif. longitude. For in the triangle MP/ (by Cafe 5), S.PM (B) : radius : : S.MF : S.MPF or MP/ the dif. longitude, or the angle on the globe between the meridian of the place and that of the plane. Cor. 5. Radius : Sin. pole's height above the plane '. : Tan. hour arch from mer id. of the plane : Tan. hour angle from the fubftile. This is demonftrated as in Cor. 5. Prop. XXII. Cor. 6. When A is equal to the latitude, then B is equal to 90 degrees ; and the Jubftile's diftance, and alfo the plane's difference of longitude, are each 90 degrees. For (by Cor. 2), cotan. B (o) : cof. reclination : : . fin. declination : infinity = tan. fubft. diftance zz 900. And (Cor. 4.) S.B (rad -. radius : : S. fubft. diftance (90) : S.plane's dif. longitude zz radius," the fine of 90. Cor. 7. The requifites may alfo be found by the ana logies laid down in the Schol. of the laft Prop, fir fit find ing A and B by this Prop. Cor. 8. When A zz the latitude, then the Jtile's height will be equal to the angle ZMB, and therejore Radius : Sin. declination : : Cof reclination : Cof. file's height. For when ZM zz latitude, M fajls upon Q, the interfeaion of the equinoaial and meridian, and then PM = 90° zz oppofite angle PFM. There fore the fide PF (or ftile's height) zz its oppofite angle PMF. Therefore in the triangle ZMB, to find Sect. I. DIALLING. 4< find the angle M (by Cafe 8), rad : S.MZB : : cof. Fig. ZB : cof. angle M zz PF. 9, Or thus. Sin. latitude : Radius : : Sin. reclination : Sin. Jtile's height. For (by Cafe 5), S.ZM : rad : : S.ZB, : S.ZMB = S.ftile's height PF. Or thus. Radius : Coj. latitude : : Tan. declination : Cotan. ftile's height. For (by Cafe 3 ), rad : cof. ZM : : tan. MZB : cot. ZMB or PMF zz PF. Cor. 9. In all theje dials, the jubftile lies to the contrary fide in regard to the declination. Or the fubftile lies always northward {in north latitude) jrom the upper part of the 1 2 o'clock line. For the ftile is a line fixt in pofition, and there fore a moveable plane being made to decline, to either hand, the fubftile (which is perpendicular under the ftile) muft needs lie to the other hand.. And fo the fubftile lies northward from the upper part of the meridian, or fouthward from the lower part. Cor. 10. The fame rules that ferve jor finding the requifites in a reclining jouth plane, will ferve equally for finding fhe requifites in its oppofite north incliner or procliner ; and lie on the jame hand, ij you face the plane. Put the contrary pole will be elevated. For by Prop. VII. The fame hours belong to both fides of the plane -, and therefore the fame requifites ; fince the inclination of the plane to the 4? DIALLING. Fig. the horizon continues the fame, and likewife the " 9. .declination ; only in one, they lie eaft, in the other weft. Cor. 11. Hence alfo the Jame requifites and the fame hours belong equally to two dials, that have equal declinations, one eaftward, the other weftward ; the inclination, ij any, remaining the Jame. But then they all lie towards contrary hands. For as there are the fame data in both, the quan tity of thefe requifites, and of the hour angles, muft v needs remain- the fame; and differ only in quality, or in their pofition towards the right and left. PROP. XXIVi In any dial whatever, if a line be drawn parallel to any hour line, to interfetl the other hour lines, and note the fixth hour line from this. Then any two hour lines' on each fide this fixth, which are equidifiant in hours, will alfo be equidifiant along this parallel line. 11. Let the plane CGLH be perpendicular to the. earth's axis paffing thro' C, CA any hour line, and CD another at fix hours diftance. Then fince in this cafe, all the hour angles about C, as BCD» DCF, are equal ; therefore ACD being fix hours, \ is a right angle. Now fuppofe GabdjE is any other plane, paffing alfo through GCH, cutting the planes of thefe hour circles, in the lines Ca, Cb, Cd, C/; which will be the correfpondent hour lines in that plane. Alfo let the plane BbfF be parallel to the plane of, the hour circle CA*, and therefore perpendicular to the plane GLH, cutting the former plane GabdfE, in the line bjE, and the planes of the hour circles, in the lines Bb, Dd, Fj; which confequently will ¦ be perpendicular to the plane GLH, and therefore parallel Sea. I. DIALLING. 43 parallel to orte another. Alfo bE is parallel to Gtf,'Fig. by Prop. n. B. V. Geometry. 11. Now fince CD is -perpendicular to BF, and the angles BCD, DCF, equal; therefore BD zz DF. Whehce in the triangle BbE, fince Bb* Dd, F/ are parallels, and BD zz DF, therefore bd = df. That is, in any dial plane CdbdjE*, if the line bE be drawn parallel to Ca, to cut the hour line Cd, fix hours from it, and feveral more Cb, C/, on each fide ; then thofe that ate equidifiant on each fide ia hours, are alfo equidifiant, in the line bdE, fet con trary ways from Cd. Cor. From hence you have a method oj drawing all the hour lines in the dial, if you have f even of them drawn. As fuppofe you have the hour lines C8, C9, 12. Cio, Cn, Ci2, Ci, and C2, drawn; and you want to draw C7, C6, C5, C4, To the hour line C2 draw the parallel kq, cutting the given hour lines in d, b, 0, p, q. From d in the hour line C8 (which is fix hours from C2 ;) make df — db, dg zz do, dh zz dp, and dk zz dq; and ¦ J '- thro' /, g, h, k, draw the hour lines C7, C6, Cg, Or if you want the hour lines on the right hand, it is but drawing a parallel to Cd, or C8, and pro-* ceeding as before. Alfo in any inclining or declining dial, when half the hour lines are drawn, the other half may be drawn, by drawing a parallel to the fixth houf line, to cut the reft ; and transferring the diftan ces of thefe hour lines, to the oppofite fide of the laft hour line ; which parallel ferves for a line of contingence. Having now fhewn how the requifites are found in all forts of dials, from the rules of fpherical trigo nometry, on which they depend; I fhall now refolve fome 44 DIALLING. Fig. fome problems relating to the time of the fun's 12. fhining on thefe dial planes, which likewife depend on fpherical trigonometry. In the foregoing problems, I have all along fup pofed the dial was to be drawn upon its parallel great circle, inftead of the plane of the dial ; fince it is the fame thing, by Cor. 3. Prop. I. For by reafon of the immenfe diftance of the fun, and the parallelifm of its rays, the motion of the fhadow of the ftile among the hour lines, muft be the very fame in both. In what follows I fhall alfo ufe the parallel great circle inftead of the dial plane ; for the fun will be in the plane of both; at the fame inftant ; and therefore will begin to fhine on both, ©r leave fhining on both, at the fame moment- of time. PROP. XXV. Prob. A diretl north reclining dial plane being given ; to find the time oj the year, when the fun will totally leave one fide of the plane, and fhine upon the other. 10. The fun is in the plane of any great circle, at the pointsrwhere his parallel of declination cuts that . great circle, and therefore the time muft be calcu lated when he is in thefe points. And at that time he is going off one fide of the plane to go upon the other fide. But when he wholly goes off one fide of a plane, the fun's parallel muft touch the other fide of the plane, in the meridian ; if the , plane is direa north or fouth. Let HZO be the meridian, EQjhe equinoctial, Z the zenith, P the pole. Take the difference between the latitude of the place, and the reclina tion of the plane ; and that will be the latitude of the place where the dial plane is perpendicular, to the horizon. And this will be north latitude, when the latitude, of the place is greater than the recli nation ; Sea: I. DIALLING. 45 nation ; but fouth, if lefs. And this new latitude Fig. of the plane will be the fun's declination, when it 10. quite leaves the plane; from whence the day of the year is known. For if ZA is the reclination, then the plane will be vertical at A, therefore AC is the dial plane, and the fun's parallel AB touches it in A. There fore whilft the fun advances from A towards P, it will fhine altogether on the fide of the plane to wards P, having quite left the other fide. There fore the fun's declination EA will fhew the time of leaving the fide towards H. Examp. Suppoje a north plane leans fiouthward 3 5 degrees, in , lat. 547 ; to find the time oj thejun's leaving thejouth fide. Here 35 fubtraaed from 54^ leaves 19^ for the fun's declination EA ; and this anfwers to May 1 8, when it quite leaves the fouth fide going northward. Alfo in its return towards the fouth, on July 26, it begins again to fhine on the fouth fide, leaving the fame declination as before. Cor. 1 . Hence if the difference between the latitude of the place and the reclination of the plane, exceeds the fun's greateft declination ; the jun Jhines on both fides of the plane, till he pajfies the equator ; and then he jhines only on that fide next the equator. For then the plane AC will be interfeaed by every parallel of declination ; till he come to the equinoaial. But beyond the equinoaial, the in terfeaion is below the horizon, when the fun fhines not. So, when the fun is between E and H, he fhines only on the fide AC towards E. Cor. 2. When the latitude is equal to the reclina tion, the Jun jhines halj a year on one fide, and halj a year on the other fide oj the plane. For 46 DIALLING. Fig. For then A falls upon E, and the plane EC be- 10. comes parallel to the circles of declination. Cor. 3. When the latitude oj the place is greater than 23.1 degrees, a diretl Jouth vertical or reclining dial plane, has the Jun jhining on both fides oj the plane, every day, whilft the jun is in the northern figns. But only on the Jouth Jide, when in the Jouth- ernjigns. Cor. 4. When a north recliner, reclines Jo much, • as that the point A jails more than 23T deg. beyond E towards H ; all the phcenomena will be contrary. Cor. 5. By the fame means, in any dial plane, how ever fituated, whether declining, inclining,. &c. the time of the year may be found, when the fun quite leaves one Jide of the plane. For if AC be the dial plane, which will be a di- rea fouth plane in forae longitude or other (by Prop. XVI.) Then by having the height of die pole above the plane, which is PA ; you will find its diftance from the equinoaial EA, by fubftrac- ting from 90, then will EA be the fun's declina tion, when the fun leaves the fouth fide. And the fun's declination being had, the time is known. And hence, Cor. 6. The fun will always leave Jhining upon one fide oj a plane, at the Jame time oj the year, when the pole's height above the plane is the fame, let its pofition, in other refpetls, be what it will. PROP. XXVI. Prob. Having the fituation of any diretl fouth inclining dial plane ; to find the hour of the day, when the fun leaves the north fide, to fhine on the fouth fide of it. The height of the pole above the plane muft firft be found ; for which add the proclination to the Fig.i WmMv. 7 Pll.pa, 46. Sea. I. DIALLING. 47 the complement of the latitude, when it leans fouth. Fig. Then fay, 1 3. ' As radius : Tan. fun's declination : : Tan. height of the pole above the plane : Cof. hour angle at the pole, from noon. Then that angle reduced to time, fhews how long before or after noon, the fun begins to fhine on the fouth fide. For let H AO be the meridian, EQ^ the equi noctial, Z the zenith, P the pole, AC the incli ned dial plane. Draw the fun's parallel of decli nation RL, to interfea the plane in S, and draw the hour circle SP. And when the fun is in S, it is juft going off the north fide upon the fouth fide of the plane. Therefore in the right angled fphe rical triangle APS, it is (by Cafe 4), rad : cotan. SP : : tan. AP : cof. angle APS ; which converted into time gives the hour before and after noon, when the fun is in S. Cor. 1. If the plane AC paffes thro' the pole P, or the 6 ^o'clock hour circle. The jun paffes from the north fide to the Jouth Jide always at 6 o'clock. And in the winter months, the Jun fhines not at all on the north fide. Cor. 2. .'If the plane fall below the pole, or lyeth between CP and CO ; the fun goes off the north fide bejore the hour oj fix in the morning -, and goes on again after fix in the evening. Cor. 3. If the plane fall in with the equator, the fun Jhines all the day on one fide ; being on that fide to wards which the declination is. For if EC be the plane, it fhines on the north fide only, when it has the north declination ER ; and on the fouth fide, when it has fouth declina tion. Cor. 4» DIALLING. Fig. Cbr. 4. In a north recliner, where the reclination is greater than the latitude ; the jun never (hines on the Jouth fide in the fummer months. Cor. In all dial planes, where the height of the pole above the plane is the fame ; the fun goes off at the fame hour of the day, reckoning time for each, at the place where it is a diretl fouth dial. PROP. XXVII. Prob. To find the hour of the day when the fun leaves one fide of a vertical declining plane, to Jhine upon ihe other. To do this we muft firft find the time from noon, when the fun leaves one fide of a direa north or fouth plane, to fhine on the other, by the laft Prop. And if the plane declines eaft ; to the time found, add the plane's diff. longitude in time, and the fum will be the diftance of time before noon, when the fun leaves the north fide. But fubtraa the plane's dif. longitude from the time found, and you will have the time of leaving the fouth fide. But if the plane declines weft, the faid difference of time will fhew the diftance of time before noon, when the fun begins to fhine, on the fouth fide ; and the fum will fhew the time of leaving the fouth fide. Examp. 6. Suppoje a plane declining eaft 25 dig. in lat. 54-j. Where the height oj the pole PB above the plane is 310 45', and the plane' s dij. longitude ZPB is 29 48. And the jun' s declination 180 north. As Radius — - — 10. Tan. Jun' s declination (18) 9-5l^l7 Tan. pole's height (31 45) q. 79 156 Cof hour angle at the pole 7824 9 .30 333 •JH Bifi J&filUind ' ~E\XL.f>a.4#. Sea. I. DIALLING. 49 Dif. longitude 29 48 Fig. fum 108 12 dif 48 35 14. Then 108 12 reduced to time gives yh 13™ be fore noon -, that is, at 411 47"" in the morning, or a little after fun rife, when the fun comes upon the fouth fide. And 48° 36' reduced to time is 3h 1 4m ; and fo it goes off the fouth fide again, at near a quarter paft three. N But if the plane had declined weft, the fun would have come On, 3 14 before 12, that is, at 8h 46" ; and gone off again at yh 13°'. Otherwife thus, Let PB be the meridian, P the pole, Z the ze nith, and SZA the dial plane. Let SBA be the fun's parallel, and draw PI perpendicular to SA. Then. A will be the place of the fun when it be gins tb fhine on the fouth fide of the plane, and S his place when he goes off it. Then in the oblique fpherical triangle ZPA we have given, PA the complement of the fun's de clination, PZ the complement of the latitude, and angle PZA the complement of the plane's decli nation. Alfo in the triangle SPZ, the fame things are given, viz. SP, PZ, and SZP ; from hence to find the arigles ZPA and ZPS, by Cafe 2, of ob^ lique triangles. Radius : .\ 10. S. latitude (cof PZ) 54" 30' 9.91068 Cotan. plane's declination {tan. PZ A) 2 50 \ o. 3 3 1 3 3 Cotan. ZPI, 29 48 — — 10.2420 s ' -¦* '¦¦ E Then $o DIALLING. Fig. Then 14. Tan. lat. 54 30 (cot. ZP) ¦— 10.14673 Tan.fun's'declin. 1^' (cot: PA) ^.guyy Cof. ZPI, 29 48 ¦ 9.93840 1-9-45017 Cof API or SPI, 78 24 9-30344 Then to API, 78 24. from SPI, 78 24 add ZPI, 29 48. fubt. ZPI, 29 48 APZ~zz 108 12, SPZ zz 48 36 which reduced to time, give the fame anfwer as before. PROP. XXVIII. Prob. To- find the hour of any -day, when the fun goes off the eaft fide of an eaft reclining dial plane, to jhine upon the wft fiie. The day being had, the fun's declination is known; therefore, jh Radius : Tan. Jim's declination : : Tan. pale's height above the plane' : Coj. hour angle from .1 2 at the pole, ,30, 45, &c. degrees, draw the lines Ci, C2, C3, C4, &c. to cut GI in 1, 2, 3, 4, 5, 6. Then GI Sea. I. DIALLING. 57 GI is the line of hours. But if lines be drawn Fig. thro' j, s, s, &c. at 10, 20, 30, &c. degrees, 19,, (and alfo thro' the intermediate degrees), to cut the line GI ; and numbered 10, 20, 30, &c. then you have the line, incil. mer. or the fecond line on the fcale, which is only the firft line reduced to de grees. 2. For the line of latitudes. Thro' s, s, s, &c. or the points of 10, 20, 30, &c. degrees, draw parallels to AC, as sm, sm, &c. cutting CD in m, m, &c. Then thro' m, m, m, &c. draw Am, Am, Am, &c. cutting the quadrant BD in n, n, n, &c. Draw BF parallel to CD; and transfer the lengths Bn, Bn, Bn, &c. upon the line BF, to 10, 20, 30, &c. And if the fame be donewith all the intermediate degrees, then BF will be a line of latitudes. And the whole- length of this line of latitudes, is equal to BD or AD the chord of 90°. 3. The line G. Pol. and L. Pol. are no more than two lines of tangents, reduced to hours. The radius of the greater, is the radius of the fmall line of chords, at the end of the line of latitudes, and this radius is half AC. Cor. 1. Put radius AC zz r, fin. latitude Cm zz s, irs then the length on the line oj latitude j zz - \Srr 4- ss Jor that latitude. For draw Bn. "Then the triangles ACm, ABn are fimilar ; whence Am (\/rr 4- ss) : Cm {s) : ; 2rs AB (2r) : B» or B20 zz — ; -. V rr + ss Cor. 2. Ij t zz tan. latitude; then the length on the line oj latitudes zz . — . , Jor any latitude. Vrr 4- 2tt For 5« DIALLING. Fig« For the fecant {s/rr+lt) : tangent (/).:: radius 19 ' (r) : the fine s zz ¦ tr . zz Cm. And Am, = Vrr+Tt (^/ACT&'z.)^^, whence vAr + # v ^/rr jp ttJ \Vrr -4- tt) tj 2/r : B» - Vrr -\- 2tt Cor. 3. Put y —fine oj the arch, whofis tangent is t\/2 ; theny\/2 zz length on the line, Bn. For y zz — - v zz fine of the arch whofe \/rr + 2# tangent is t^/2. The Ufe. ,gf To draw a dial by the fcale. Let C be the cen- "' ter of the dial, draw the fubftile CD, and ACB perpendicular to it. Then having the height of the pole above the plane, take it from the line of latitudes, and fet it from C to A and from C to B. Then take the whole length of the line of hours, and fetting one foot of the compaffes in A or B, with the other crofs the line CD in D. Draw the lines AD and BD. Then get the feveral hour arches from the meridian of the plane, and count ing them on the fcale of hours, fet them from D towards A, and from D towards B, in the lines DA, DB. And from C, draw lines thro* all thefe points, for the hour lines, as Qi, Cz, C3, C4, C5, and Cn, Cio, C9, C8, C7> 30# The ufe of the two polar lines is only for draw ing fuch dials as have no centers ; or where the dial plane goes thro' the poles. And the radius of each is the length of 3 hours on the fcale, to be ufed. To ihew the truth of this operation, Bifea T1W. pa. Sf Sea. I. DIALLING. 59 Bifea BD in O, then will BO or OD zz r, and Fig. fince BD zz 2r, and CB zz ^i£L_. and CD = 21, \/rr + ss 2rr ,X7, BC s VW _ CB- zz — ==r. Whence _ zz vrr 4- j* ^^ r Now by conftruaion DO is a fcale of tangents of the hour arches ; and the numbers on the line ex- prefs thefe hour arches. And as ODis the tangent of 45° (or radius), and OA the tangent of an other arch ; therefore AD is the difference of the tangents of thefe two arches ; which being known, the dif ference of the arches will be known ; and this is the hour arch correfponding to DA. For (by Cor. 2. Prop. 9. S. I. B. I. Trigon.) ~ x r, or r * & DO 4- OA Eli x r zz tan. this hour arch. But CD : CB : : BA r : s, and in the triangle CDB, CD : DB : : r : tan. CDB ; therefore s zz tan. CDB. Let fall AF per pendicular to- CD. Then in the triangles CFA and DFA, r : AF : : cotan. FDA : DF : cotan. FCA : CF ; and alternately, CF : DF : : cotan. FCA : cot. FDA : : tan. FDA : tan. FCA : : BA : DA, (be caufe DFA, DCB are fimilar triangles). Whence, DA tan. FCA zz _ x tan. FDA ; and r x tan. FCA BA zz r X s zz tan, hour arch X s. Therefore r : BA s : : tan. hour arch : tan. FCA the hour angle, which- is the fame proportion, as in Cor. Prop. XVI'II. and therefore is truly found. If Ca be an hour line, and Da greater than DO ; then Da is the fum of the tangents OD, Oa ; and (by Cor. Prop. VIII. ib.) the tan. fum of the arches DO + Oa Da , , — X r zz — ¦ r zz tan. hopr arch, corre- IpCT^-Oa Ba fponding €a DIALLING. Fig-fbondiflg to Da, on the line. Alfo tan. fCa zz 21. ^f.X£an,./D«, and r X tan./Ca zz — xn — t2B° Ba Ba hour arch x s. Whence as before, it will be, r : * : : tan* hour arch : tan. hour angle jCa. Alfo if A or a fall upon O, then r zz tan. hour arch, and tan. hour angle DCO zz s. Laftly, if O A zz Oa, then Ba zz DA ; and CA, Cs will be hour lines, whole hour arches ate equ-i- diftant from CO, or the third hour: Scholium. There are feveral other lines put upon fcales for dialling, but are only particular, as being made for fome particular latitude only. Thefe lines are, i. A line of chords. 2. A line of the fubftile's drftance from the meridian. 3. A line for the height of the ftife above the plane. 4. A line of the angle of the hours of 12 and 6. 5. A line of the plane's diff. longitude, or inclination of meridians. All thefe are calculated for, every degree of declination, by Prop. XX. for fome particular latitude, as that of London ; and put on a fcale together. And thus the feveral requifites for any declining dial are had by infpeaion. Its ufe is no more than this, count tfie plane's declination in the line of chords ; and a fine drawn dire&ly crofs, will cut all the other lines in their proper points, which the numbers will fhew ; and fo give all the requifites without calculation. And thefe requifites being had, the dial may be drawn by this Prop. When a fcale of this fort is not to be had, the requifites may be found by G timer's fcale; extend ing upon the feveral lines, according to the feveral rules and proportions laid down for that purpofe. SECT. [6t 3 SECT. II. The PraSiical Part of Dialling, or the Art of drawing Dials upon all forts of Planes. How to find the Situation of a Plane ; and how to place any Dial truly. PROB. I. To find the meridian line oj any plane. IF the plane be even, erea a pin on it perpen- 22. dicular to the plane, but if not, procure a plain board GH of an equal thicknefs, and faften it to the plane ; upon which erea an iron pin CF about the middle, nearly perpendicular, and of a pro per length; and let it be fharp at the top ; then take a wooden ruler, with a fharp iron pin in one end ; and lay the other end upon the fharp end of the iron pin F, which is faft in the board ; then carry the other end fteadily about ; the fmall pin in the ruler will defcribe a circle ADB, upon the board. Obferve two times in the day, when the fhadow of the top of the iron pin F, falls upon the circumference of the circle, and mark the two points as at A and B. Bifea the arch AB, in the point D ; and from D, draw a line DE thro* the center of the circle, which is eafily found. Then DE is the meridian of the plane. Or you may defcribe a circle round the center C, and raife the pin CF upon that point C, to eaft a fhadow ; ©V DIALLING. Fig. fhadow ; but then the pin CF muft be exaaiy per* 22. pendicular to the plane at C. If the day be cloudy, the fhadow may happen not to fall twice upon one circle ; therefore it may be proper to defcribe feveral circles, that it may fall upon fome of them twice. And if two points in feveral circles be marked, they will one torrea another, and the operation will be more exaa. To know whether the pin is perpendicular to the plane or not ; fet one point of the compaffes in fome point of the circle, and extend the other to the top of the pin F ; then if you fet it in any point of the circle, the fame extent will, always, reach to the point F, when the pin is truly per pendicular. It is certain, when the fun does not change his declination, (as he does not fenfibly in 1 2 hours), that he will have equal altitudes at equal diftances on each fide the meridian. And therefore if CD be the meridian of the plane, the arches AD and BD muft be equal. PROB. II. To find the. inclination oj any plane to the horizon. ; 23. Let ABC, be the plane, apply one fide of a qua drant to the plane, fo that the plummet may fall exaaiy on the other fide ; the firft fide will be ho rizontal ; therefore draw a line DF along that fide, which will be the horizontal line of the pl*ne. If you cannot apply a quadrant, then ufe a carpen ter's fquare or a level. ' To the line EF, draw the perpendicular EL, which will be the vertical line of the plane. To this vertical line EL, apply A ftreight ruler EK ; and to the end JC, beyond the plane, apply the quadrant GH, on the under fide ; and obferve what number of degrees are cut by the Sett. II. DIALLING. -63 the line and plummet at I, reckoning from the ru-Fig. ler; as GI, and that will be the reclination of the 23. plane ; or its proclination, if it lean forward. For inclining planes, a quadrant may be applied di- reaiy, to the vertical line. Or thus. Apply the fide of a fquare to the line EL on the plane ; and apply the fide of a quadrant to the o- ther iide as KH ; then fee what angle your line and plummet makes with the firft fide EK, as LKI, and that is the reclination or proclination of the plane. If the line KI hang along the line KH,' then your plane is horizontal ; if along the fide KL, it is vertical. If by turning the rule and quadrant all manner of ways on the plane, the perp. line KI always falls on the fide KH ; then the plane is truly hori zontal. And the like for the fquare and quadrant^ PROB. III. To find the declination oj any plane. Place a. board DM with one fide of it clofe to the wall or plane EF, and to lie with its upper fur face exaaiy horizontal ; there fix it* Upon this board, as it lies, find a meridian line MC (by Prob. I.), and draw a line on the board to reprefent it. Then lay a quadrant A MB flat upon the board, with one fide clofe to the plane, and fo, that the center C may be in the meridian line, and the bo dy of the quadrant, to that hand where the acute angle lies. Then note the degrees of the quadrant, cut by the meridian line, reckoning from the per pendicular fide of the quadrant CB, then that, viz. the angle MCB will be the declination. And the 24. 64 DIALLING. Fig. the declination will be to the right hand, if the 24. 'quadrant lie to the left ; and the contrary. If you know when it is juft noon, you1 may get a meridian line thus, hold up a plumb line, fo as the fhadow of it may fall upon your plane ; mark two points of the fhadow, and draw a line thro* them, and that fhall be the meridian line of the place. And if the plane be horizontal or direa north or fouth ; it will be the meridian of the plane too. Otherwife thus. This may alfo be done by a compafs and mag- netical needle. This is fixt in a box, fquare on all fides ; and to ufe it, apply the north fide of the box to a fouth wall or plane ; and the fouth fide to a north wall, fo as the box may lie horizontally. Then obferve how many degrees the needle is from the flower-de-luce when it refts, ( having due .regard to the variation'! ; and fo much will the declination be. And if the needle points towards the right hand, the declination is to the right hand, and the contrary, in fouth planes. But in . north planes, if the needle points toward the right hand, the de clination is to the left, and the contrary. You muft mind that no iron be near the compafs. The va riation is 1 yi- deg. weft in England. Or thus. This method is aftronomical, and the moft ex aa ; but it is proper to have an affiftant ; for two obfervations are required to be made at the fame time; the firft to find the diftance of riie fun's azimuth, from the vertical of the plane ; and the fecond, to find his altitude. For the firft, draw a horizontal line, upon the plane, and apply one edge of a quadrant to it, fo as the limb of the quadrant be towards the fun ; and Sea. II. DIALLING. 65 and holding it horizontal, let an affiftant hold up a Fig. thread and plummet in the fun, fo that- the fhadow of the thread may pafs thro' the center of the qua- drant; then obferve the degrees which the fhadow of the quadrant falls upon, counting them from the perpendicular fide, and you have the fun's azimuth, in refpea of the plane's vertical circle. Again to find his azimuth in refpea of the me ridian, his altitude muft immediately be taken ; and his declination being known, from the day of the month; his azimuth will be found by Cafe 11, of oblique fpherical triangles. ' Thefe two azimuths being had, it is eafy to know, whether the vertical of the plane lies to the eaft or weft of the meridian. For if the fun's azi muth from the meridian be greater, the plane de clines towards the fun ; but if lefs, it declines from the fun ; that is, when both azimuths lye one way. And then the declination is the difference of thefe azimuths. . , But if the azimuths lie contrary ways, the plane declines from the fun ; and the fum of thefe azi- • muths is the declination. If you know the exaa time of noon, and hold up a line and plummet, before the quadrant at that time, you will have the plane's declination at once, by obferving the fhadow, which muft pafs thro1 the center of the quadrant. If you obferve the fun's altitude, juft when he comes into that plane ; then the fun's azimuth be ing found, gives you the pofition of that plane. If your plane be a coarfe wall, nail a Jong ruler to it, to which. apply your inftruments. PROB. 66 DIALLING. Fig. PROB. IV. To draw a dial geometrically upon any plane. Projea the fphere by any method, upon that great circle of the fphere which is parallel to the dial plane ; and defcribe the feveral hour circles, and note the places where they interfea the primi tive circle. Draw lines from the center thro' all thefe points of interfeaion, and they will be the hour lines for your dial. For let NWSE be that great circle, which will ' bifea the globe, paffing thro' the center C. Let fig. 7. reprefent the globe ; P the pole ; then CP will be the earth's axis, elevated above the plane of the circle, the quantity of the angle or arch NP- Then fince the hour circles do all pafs thro' the axis PQ they all pafs thro' C, which is in the plane of that circle ; and the points of interfeaion with the primitive, 1, 2, 3, c, d, g, &c. being alfo in the plane of that circle ; therefore the hour cir cles interfea the primitive in lines which pafs thro' the, center C, and thro' thefe feveral interfeaions. And fince the fhadow will be in any line bf inter feaion, when the fun is in that hour circle ; there fore thefe lines of interfeaion will be the hour lines ; that is, lines drawn from C to 1, 2, 3, &c. will be the feveral hour lines of the dial. As all the hour circles in the orthographic pro jeaion of the fphere will be ellipfes ; and as thefe are difficult to defcribe, their interfecfions with the primitive will not be got exaa enough. Therefore this fort of projeaion is no way commodious for the drawing of dials, and therefore is never ufed. , In the ftereographic projeaion, all the circles of the fphere are circles in the projeaion, which are far more eafily deferibed than ellipfes, and far more correa 1 Sea. II. DIALLING. 67 correa ; therefore the interfeaions with the primi-Fig. tive will be more exaaiy obtained ; which makes 7. this method more commodious for drawing dials ; and therefore it is often ufed for this purpofe. But in the gnomonical projeaion, all the great circles are projeaed into right lines, which are ftill eafier to draw than circles ; therefore dials are beft and eafieft drawn by this fort of projeaion ; for which reafon, in what follows, I fhall make ufe of it in the geometrical conftruaion of dials. And the reafon of the feveral operations, will be obvi ous to any body that underftands the nature of this projeaion ; and needs no other demonftration. And the rules of this fort, of projeaion, you have in the laft feaion of my Treatife on Projeaion ; and therefore the dialift ought to be acquainted there- with, as it is the foundation of all thefe operations. To proceed in the eafieft method, imagine the perpendicular height of the ftile to be the radius of the fphere, and that the dial plane touches the. fphere at the foot of the ftile ; then if the fphere be projeaed on that plane, by the eye (or projea- ing point) at the top of the ftile, which is the cen ter of the fphere ; all the hour circles will be pro jeaed into their proper hour lines, which will tend to the projeaed pole, which will be the center of the dial. Arid "the projeaions of the leffer circles will make the reft of the furniture ; and will be conic feaions. And a line drawn from the top of the perpendicular ftile to "the center of the dial, will reprefent the earth's axis, and be the ftile or gnomon. In thefe conftruaiorts, the ftile is all along fup pofed to be a- perpendicular pin ; which afterwards is eafily changed into a triangular form, by draw ing a line from the top of it, to the center of the dial. F 2 PROB. 68 D I A L L I N G. Fig. PRO B. V, To draw an equinotlial dial, or an horizontal dial under the poles. 2$. Since the ftile here muft be a perpendicular pin-, from C the center of the dial, affumed about the middle of the plane, defcribe the fmall circle abd, of the fame diameter as the pin, which muft be cylindrical ; and that circle is to be the foot of the ftile. Alfo from the fame center C, defcribe ano ther circle i, 2, 3, &c. Draw Co for the meridian, and parallel thereto, draw 12 a for the 1 2 o'clock line, touching the circle abd in a. Then divide the outer circle into 24 equal parts, beginning at 12; and to the points of divifion, 1, 2, 3, &c. draw the hour lines, to touch the inner circle to wards the fame hand as 1 2 a was drawn. Then mark the hour lines with r, 2, 3, 4, &c. on the right hand, and 1 r, 10, 9, &c. on the left hand ; and draw another circle including the whole. Upon the circle abd eretX the pin or ftile, which muft be perpendicular to the dial plane, and of any length'. Then this will fhew the hour, by the fha dow on the left fide. To fet up this dial in any other latitude, it muft be placed with the 6 o'clock hour line horizontal, the 1 2 o'clock line pointing to the north, and the fouth edge S raifed above the horizon, to make an angle with it equal to the complement of the lati tude, or the height of the equinoaial ; and then the plane of the dial will be parallel to the equi noaial, as it ought to be. If you draw the hour lines from the center C„ the pin muft diminifh to a point at the top. And the divifion of the hours muft then begin at O ; and the fhade of the top will fhew the hour of the day, Sea. II. DIALLING. 69 day, among the hour lines. In this dial you need Fig. draw no more hour lines, than the fun fhines on in 25,. fummer, if the dial is drawn on the north fide. But if it is drawn on the fouth fide, you need but draw 1 2 hours, which is the longeft that the. fun can fhine upon it. If you have a mind to note the half hours and quarters, you muft divide each hour, 12 1, 12, 23, &c. into two equal parts for half hours, or into four equal parts for quarters, PROB. VL To draw a horizontal dial under the equinotlial. 1. Geometrically. Draw, two parallel lines upon your plane, AB 26. and DE, at a convenient diftance,. for infcribing the hour lines ; make MCF perpendicular thereto. Take CF for the height of the ftile, and Cc for the thicknefs thereof ; and draw ncf parallel to MCF. With the center F, defcribe the quadrant CH ; and with the center/, the quadrant *G. Divide each quadrant into fix equal parts at n, n, &c. and draw lines from the refpeaive centers thro' all the points of divifion, to interfea the line DE in, the points 1, 2, 3, &c. Then thro' thefe points 1, 2,' 3, 11, 10, 9, &c-. draw lines parallel to MC or mc, and thefe Will be the hour lines of the dial. If you divide each hour nn, nn, &c. into two or four equal parts ; and. draw lines thro' thefe points from the two centers, to interfea DE as before, then thefe interfeaions will fhew the half hours or quarters. Make MK z? CF, then will, MK3C be the- fi gure of the ftile, and Cc or, N[m its. thicknefs.; which ftile muft be ereaed upon the bafe CMwc, F 3 perpen- 70 D I A L L IN G. Fig. perpendicular to the plane of the dial ; and the 26. fhadow of the top will fhew the hour of the day. If inftead of leaving a fpace Cc for the ftile, you defcribe a femicircle round F, and divide all from the center F ; then fince C and c coincide, your ftile muft be exceeding thin, or at leaft it muft be brought to an edge at the top. This is the readieft way, but then your dial will not be fo exaa in the middle of the day. Or inftead of plate for the ftile, it may be ah iron pin, ereaed perpendicular upon any point of CM, and its height muft be equal to CF. Or you may have a plate of any height for -the ftile, with a fmall hole made thro' it at the height of CF, for the fun to fhine through among the hour lines. The hours muft be numbered on the weft fide 11, 10, 9, 8, &c. and on the eaft fide 1, 2, 3, 4, &c. This dial is to be placed fo, as its plane may be parallel to the horizon, and DE pointing eaft and weft, or CM north and fouth. Or to fix it in any other latitude, the north fide at M muft be, raifed to make an angle above the horizon equal to the latitude of the place ; and CM pointing to the north. 2. By Calculation. As radius : Stile's height CF : ; So tan, hour arch ( 1 50) : ' Diftance of the hour from MC (Ci). So tan. 30d : Diftance C2, &c. Then the diftances Cr, C2, C3, &c. fet off from C fo 1, 2, 3, &c gives the points thro' which the afternoon hours are to pais ; and the fame fet off from ; to 11, 10, 9, &c. give the points Sea. II. D I AL L I N G. 71 points where the forenoon hours are to pafs. Fig.1 Therefore lines drawn thro' all thefe points parallel 26. to MC will be the hour lines. ' If you would calculate for the half hours and quarters, you muft fay, as rad : ftile's height CF : : fo tan. 30 45' : to the diftance from C, on the line CE for a quartet : : and fo tan. 70 30' : diftance for half an hour : : and fo tan. 11° 15' : diftance for three quarters : : tan. 180 45' : dif tance for an hour and a quarter, and fo on. 3. By the Scale. Having drawn the meridian CM, and cm paral lel to it at a diftance equal to the thicknefs of the ftile ; from either polar line, which fuits beft, take the extent from the beginning of the 'line to I the firft hour, and fet from C to 1 , and from C to 1 1 on the line DE ; then take the extent from the be ginning, to II, the fecond hour, and fet it from C to 2, and from c to 10 ; likewife take the ex tent to III, the third hour, and fet from C and c to 3 and 9. Do the like with the reft of the hours. Then lines drawn thro' thefe points 1, 2, 3, 11, 10, &c parallel to CM, will be the hour lines of the dial. And the extent to III, or C3, is the height of the ftile. If you would have the hour lines of 7 in the morning, and 5 at night to come in ; you muft find the height of your ftile for that purpofe, thus. 5 hours zz 750. Then fay, as Radius : CE : : Co tan 750 : CF the height of the ftile; and to this. height the dial muft be made. F 4 PROB. 7* DIALLING. Fig. ., PROB. VII. To draw a horizontal dialjor any latitude. i. Geometrically. 2*,, Take any convenient point P, for the center of your dial, thro' P draw the meridian DE ; at P make the angle DPR equal to the latitude. Draw a line parallel to DE, at a diftance equal to the perpendicular height of the ftile, to cut PR in F. Otherwife take any point F you pleafe in the line PR, and draw 'FC perpendicular to DE, then FC is the height, and C the foot of the ftile. Draw FG perpendicular to FP. Thro' G draw the con tingent line AGB perpendicular to DE, which here reprefents the equinoaial. Let Gg be the thicknefs of the ftile, and draw dge parallel to DGE. Or you may proceed thus, having made the an gle DPR equal to the latitude, take any point G at pleafure, ,thro' which draw the contingent line AGB, perpendicular to DE. From G take GF, the neareft diftance to PR, and fet from G to D, and from gtod; arid from the centers G and g, with any radius, defcribe two quadrants, which divide into fix equal parts, or hours, beginning at G and g. And thro' the points of divifion, draw lines from the centers D and d to interfea AB in the' hour point S, m, m ¦, n, n, &c. Then from the cen ters P and^, draw lines thro' the points m, n, &c. as Pm, Pm -, and Pn, Pn, &c. and thefe will be the hour lines. The 6 o'clock hour lines muft be drawn parallel to AB ; and the hours before and after fix, are had by producing the oppofite hour lines thro' the center. Then PFC will, be the form of the ftile, which may be made longer or fhorter by producing PF. And A m E Ifl J I I n D 26' -t-A IF. L _ ^iallmtj _B. 14 -r ' V- PLir.yPfl.70. Sea. II. D I A L L I N G. 73 And the ftile is to be placed perpendicular upon Fig. the plane of the dial; that is, one fide upon the 27. fubftile PG, and the other upon the fubftile pg ; each fide of the ftile being perpendicular upon the fubftile ; for the plane of the ftile ought al ways to-be fet perpendicular on the fubftile, in any dial. And then the top' edge Will fhew the hour. If you_ allow no fpace5 G^ or Pp for the thick nefs of the ftile, it muft. be made exceeding thin to take up no room; or elfe it muft be made fharp on the' upper edge. And that edge muft tend direaiy to the center. Inftead of the triangular gnomon PFC, you may have a perpendicular pin FC, and then the fhadow of the top F, will fhew the hour. But the other is better, becaufe fome part of the fhadow will always be upon the dial ; but the fhadow of the top F, foon goes off. You may alfo have the ftile to be a plate of any form, with a hole in it for the fun to fhine through ; which hole .rnuft be at the height CF, perpendicu larly over C. If you would have the half hours and quarters ; you muft divide every hour in the quadrant into two or four parts, and draw lines from the cen ters D, d, as before, to interfea AB, and thro' thefe points of interfeaion, draw lines from the center of the dial, which will give thefe halves or quarters. But the quarters may be had near e- nough, by dividing each half hour into two equal parts. And to fave labour, after you have drawn all the hour., lines on one fide of the dial, or for one half of the day; you may transfer them to the other fide, for the other half of the day. As to numbering the hours, thofe on the eaft fide muft be numbered 1, 2, 3, &c. from PD 01- 12 74 DIALLING. 1 2 for the afternoon hours. And thofe on the 27. weft fide 11, 10, 9, &c. which are the forenoon bours. And the figures are mpft commodioufly placed with their tops towards the center ; becaufe we ftand oppofite to the fun, to fee the hour. A horizontal dial is the only one, that fhews the hours quite thro' the day, at all feafons. of the year. For if the fun is up, or above the horizon, he - muft ;needs fhine upon a horizontal plane, if nothing Hands in the way. The outfide of the dial terminating the hour lines, may be round or fquare or of any form you will, for it makes no difference whether the hour lines be long ox fhort. 2. By Calculation. As radius : Sin. latitude : : Tan. hour arch : Tan. hour angle. The hour arches are 150, 30% 450, 60°, Ji?, 90*; and if the quarters are taken in, they are 30 45', 70 30', li° 15', i5°o"; 1 8° 45', &c. Suppofe the latitude 54^; then the hour angles, found by the foregoing rule, for every half hour, will be as in the following table. A Table Sea. II. DIALLING. 7£ A Table of the hour arches and hour angles. °* ° 27. Hours. hour arches. hour angles. XII o° o' 0 o' Il4r I2t 7 3° 6 7 XI I J5 ° 12 18 ioi if 22 30 18 38 X II 30 0 25 10 97 2f 37 3° 32 00 IX III 45 0 39 9 87 3f 52 3° 46 42 VIII IV 60 0 54 39 7f 47 6y 30 63 02 VII V 75 0 7i 47 6f 57 82 30 80 49 VI 90 0 90 00 Conftrutlion. Take any point P for the center, and draw PD for the meridian. Take Pp zz thicknefs of the ftile, and draw pd parallel to PD. Make the an gles DP Xi, and ^Pi zz 12° 18'. The angles DPX and dpll zz 25 jo. And fo of the reft, as you find them in the table. Laftly, make the angle DPR zz the latitude, for the flile. 3. By the Scale. Affume any convenient point P for the center of the dial, thro' which draw the line VI P VI for the fix a clock hour lines. Make Pp the thicknefs of the ftile ; and thro' P and p draw the lines PD, pd perpendicular to VI VI, for the meridian or 1 2 a clock line. Then on the line of latitudes on the fcale, fetting one foot at the be ginning, 76 D I A L L I N G Fig.' ginning, extend, the other to the latitude of the 27. place ; and fet that extent from P to b and from p to d, in the fix o'clock line. Then take in your compaffes , the whole length of the line of hours,, and fettitig one foot in b, with the other cut the meridian PR in O ; likewife fet one foot in d, and with the other cut the meridian pd'in 0. ; and draw the lines bO, do. Then extend your compaffes. from the beginning of the line of hours to I (or 1 50), and fet. that extent from Q to 11, and from 0 to 1 ; and if you will, from; b to. y, and from d to 5 ; in the lines bO, do. Again extend frpm the beginning, to II (or 30°) ; and fet it from O to 10, and from 0 to 2 ; and alfo (if you pleafej from b to 8, and from d to 4. Likewife take the extent to III (or 45°), and fet fr6m O to 9, and from 0 to 3. After this manner go thro' all the hours. Then lines drawn from P, thro' the points .7, 8, 9, 10, 11, give the forenoon hours. And lines drawn fromp, thro' 1, 2, 3, 4, 5, will be the afternoon hour lines. The hour lines before and after 6, are had by pro ducing the reft thro' the center. ' If you would have the half hours and quarters fet Off by the fcale ; you muft extend to thefe half hours and quarters oh the line of hours, and fet them on the lines Ob and od, as you did for the whole hours. ¦ The ftile DPR muft be raifed at right angles., to the dial, upon the fubftile which here is the fpace' POop ; and muft make an angle with the fubftile equal to the latitude, as was faid before. There is alfo a way to conftrua dials, by the help of dialling tables, fitted to a certain latitude; which you may fee in fome books of dialling. Scholium. Sea. IL DIALLING. y? Scholium. 2y. It maybe obferved here, that if a horizontal dial be made for a place in the torrid zone, to fhew the hour by the top of the perpendicular ftile CF. And when the fun is in the neareft tropic, or between the parallel -of the place and that tropic ADL ; the fhadow of the ftile CF, upon the dial NAdgl, will go back twice in the day ; once in the forenoon, and once in the afternoon. For whilft the fun in the tropic, or parallel circle, moves thro' AGBD1KL, the fhadow will move thro' agadklk, and then fets ; fo that he goes back thro' ag, and thro' kl. In this cafe the fun is twice upon one azimuth in the morning, and twice in the evening ; and then the fhadow falls twice upon one line. And the arch ag or kl may be found, by calculating the fun's azimuth at A and G, by right angled fpherical tri angles. The like happens to any ftar here, whofe decli nation is greater than our latitude. For fuch a ftar will be twice upon one azimuth in the eaftern part of its orbit, and twice in the weftern part. In , great latitudes fuch a ftar never fets ; but in its re volution, whilft it is afcendingin the eaft, it moves towards the fouth, till fuch time as it rifes per pendicularly, and then its azimuth is the greateft. Then it returns again towards the north ; fo that it never gets to the fouth of us. PROB. VIII. To draw an eretl diretl jouth dial. i. Geometrically. Take a convenient point C for the foot of the 2%. ftile, thro* which draw the meridian CDE. Raife CF 78 DIALLING. Fig. CF perpendicular to CE, and equal to the perpen- 28. dicular height of the ftile. Make the angle CFP upwards equal to the latitude of the place, to in terfea the meridian in P, then is P the center of the dial, Draw FG perpendicular to FP, and AGB perp. to PG. Or proceed thus, take the point P near the top of the plane, for the center of the dial ; thro' P draw the meridian PE ; at P make the angle EPR equal to the complement of the latitude. Take any point G in the line PE, and draw GF per pendicular to PR ; or take the neareft, diftance from. G to the line PR ; thro' G draw the contingent line AB, perpendicular to PE, reprefenting here the equinoaial. Take Gg equal to the thicknefs of the ftile, and draw pgq parallel to PGE. Make GD and£2 zz GF, and from the centers D and q, with any ra dius, defcribe two quadrants LG, Mg. Divide each quadrant into fix equal parts or hours, each 150, beginning at G and g. From the centers D, q, draw lines thro' the points of divifion, to cut AB in the hour points m, m -, and n, n, &c. Then lines drawn from the centers P and p, thro' all the points m, n, &c. will be the hour lines. The half hours, &c. are drawn the fame way. And the perpendicular pin FC will be the ftile, or which is better, the triangle RPG ; which muft be fet perpendicular upon the hour lines of 12, PG and pg ; the ftile PR pointing downwards to wards the fouth pole. Note, all the lines except the hour lines, fhould be obfcure lines, or fuch as may be rubbed out. When the lines do not interfea within the plane, as at the firft divifion towards L and M. Take half the line qg in your compaffes, and running one foot along the line ^B, till the other foot fall upon the line' qc at c, (keeping the feet of the compaffes in Sea. II. DIALLING. 79 in a parallel pofition), thro1 c draw eha parallel toFig. pq ; and fet off ha zz half gp ; then cfraw pa for the 28. hour line. Set the forenoon hours on the left hand and the afternoon hours on the right, with the tops of the figures upwards, in which pofition only, they can be read. 2. By Calculation. As radius : Cof latitude : : Tan. hour arch : , Tan. hour angle. Here the height of the pole above the plane is the complement of the latitude. Here as in the laft Prob. the hour arches are 150', 30°, 450, &c. for whole hours; and 3 ° 45', 7° 30', 11° 15', 18* 45', &c. for quarters. Then having found the . hour angles, by the foregoing rule, put them or derly in a table, as in the laft Prob. which for the lat. 544- will be as follows, for whole hours. A Table of hour arches and angles. Hours. hour hour arches. angles. XII 0° o' o° o' XI I 15 0 8 51 X II 30 0 18 32 IX III 45 0 3° 9 VIII IV 60 0 45 IO VII V 75 0 65 14 VI 90 0 QO O Conftrutlion. Take any point P near- the top of the plane, for the center of the dial ; draw the meridian or 12 o'clock r8o DIALLING. Fig. o'clock line PE* and pe parallel to it, at a diftance 2 8.,equal to the thicknefs of the ftile. Then with ex tent of 60 from a line of chords, defcribe two quadrants, from the centers P and p. Then fet off the feveral hour angles as you find them in the , table, both ways from the lines PE and pe, upon thefe quadrants. And thro' the points, draw lines, from the centers P, p, for the hour lines. Thus EPXI, and ep\, is the firft hour angle. FPX, and epW the fecond hour angle, &c. Laftly, make the angle EPR zz complement of the latitude, for the ftile; to be fet upright upon the hour of 12. 3. By the Scale. Take the point P near the top of the plane for ' the center, and draw the meridian PE, and pe pa rallel to it, at a diftance equal, to the thicknefs of the ftile ; and thro' P draw the 6 o'clock lines per pendicular to PE. Then from the line of lati tudes on the fcale, take off the extent from the beginning to the complement of latitude, and fet that extent from P to b, and from p tod. Then take the whole length of the line of hours, or in clination of meridians-, and fetting one foot in b, crofs the meridian with the other foot at O. Like- wife with one foot in d, crofs the line pe in 0 ; and draw bO, do. Then take the extent from the be ginning, to I, II, III, &c. hours (or to 15", 30", 45% &c.) and fet from O to 1 1, 10, 9, &c. and from 0 to 1, 2, 3, &c. refpeaively. Then draw lines from P thro' all the points 11, 10, 9, &c. and from p thro' 1, 2, 3, &c. and thefe will be the hour lines of the dial. The forenoon hours are to be numbered towards the weft ; and the afternoon hours towards the eaft. Then make the angle EPF equal to the complement of the latitude; for the ftile. PROB. Sea. II. DIALLING. 81 Fig. PROB. IX. 28; To draw a diretl north Dial. 1. Geometrically. Draw a direa fouth dial for the fame latitude, and turn it upfide down, and fettingits face north ward, it will be a direa north dial ; and the ftile will point upwards towards the north pole. But the hours muft be numbered the contrary way from the fubftile ; thofe to weft, are the morning hours, and thofe to the eaft, the evening hours. And the midnight hours muft be left out, becaufe the fun fhines not then; and the hour lines of 7 and 8 in the morning, and 4 and 5 at night muft be drawn thro' the center, as in the horizontal dial. It is evident the fun never fhines upon this dial, but only in the fummer half year. 2. By Calculation. As radius : Cof. latitude : : Tan. hour arch : Tan. hour angle. The fame as for a direa fouth dial. And the. conftruaion the fame. 3. By the Scale. Here the center H muft be taken near the mid dle of the plane. And all the reft of the work muft be done, as in the fouth dial. Only the hour lines after fix in the morning, and before fix in the evening, are had by producing the oppofite ones thro' the center. You may if you pleafe, make a horizontal dial for the complement of the latitude -, and turn the north end up, and fet it to face the north, and it G "will 82 DIALLING. Fig. will be a true north dial. And the hour lines will 28. be rightly numbered ; but the mid-day hours muft be left out, which now become the midnight hours. A direa north dial is only the back fide of a fouth direa one ; for both dials lie in the prime vertical ; the meridian in the fouth dial is the hour line for 1 2 at noon ; but the fame meridian con tinued thro' the plane, is the hour line for 12- at night, in the north dial. And all the hour lines in the fouth dial being produced thro' the plane, will be the hour lines in the north dial, for then the fun is in the oppofite part of the meridian. And the ftile of the fouth dial continued thro' the plane. will be the ftile of the north dial. PRO B. X. To draw an erecl diretl eaft Dial. 1. Geometrically. 29- Draw the horizontal line AB, and in AB take a point Q, on the left hand. Thro' Q draw the contingent line EQ, making an angle with AB, as AQE, equal to the complement of the latitude. Thro' any point E of the line EQ, draw FEG per pendicular to EQ, for the fix o'clock hour line, and the fubftile. Take any length EF for the height of the ftile, and with the center F, taken in the line FG, and * any radius, defcribe the femicircle LEM. Divide each quadrant EL, EM into fix equal parts for hours, beginning at E. Thro' thefe points draw lines from the center F, to cut the line EQ_(which reprefents the equinoaial) in the hour points n, iu n, &c. Thro' the points n, n, &c. draw lines parallel to GEF, and thefe will be the hour lines. On ¦Fig. 27". Jforblv ik i a a i ti n v\y.Fn.s2 Sea. II. DIALLING. , 83 On GF raife the perpendicular plane GHMR,Fig. whofe height RM or GH is equal to EF; and that 29« plane, which is a parallelogram, will be the ftile, which muft be made very thin on the top HM. Or you may have a perpendicular pin fixt at E, whofe height is EF, for the ftile, But it is better to leave a fpace at EG equal to the thicknefs of the gnomon ; and defcribe two quadrants LEF, MEF, from two centers, and fo find the hour lines on each fide, feverally by thefe centers, as was done in the former Problems. And then the ftile muft be of a thicknefs equal to that fpace. Then the hours muft be marked towards the fouth, 5, 4, 3, leaving out the hours before fun rife; and towards the north 7, 8, 9, 10, 11 ; the fubftile being 6. If you would have half hours and quarters, di vide each hour of the femicircle into two or four parts, and proceed as in the whole hours. If you would have your dial to take in 1 1 o'clock, you muft find the ftile's height to anfwer that, af ter this manner. If 1 1 be the point for 1 1 o'clock, draw 1 1 F to make an angle of 150 with 11 E, and to interfea FEG in F. Then EF is the height of the ftile. 2. By Calculation. Radius : Height of the ftile EF in inches : : Tan. hour arch jrom 6 : Diflance jrom EG the 6 o'clock line. Thefe diftances being, found ' for all the hours, muft be put regularly in a table. But if you would have 1 1 o'clock juft to come in, you muft find the height of the ftile by this proportion ; as radius : length 1 1 E : :. tan. 1 50 : FE, the height of the ftile. Suppofe FE to be 3 inches, then the hour diftances from E, will be as in the following table. G 2 Hours 84 Fig. 29. DIALLING. Hours. diftances. Ill 3.00 IV 1-73 V 0.80 Subft. VI 0. VII , 0.80 VIII J-73 IX 3.00 X 5.20 XI 1 1.20 Conftrutlion. Make the angle AQE equal to the complement of the latitude, AB being parallel to the horizon. Draw the 6 o'clock line EG perp. to QE. Then fet off the feveral diftances from E as in the table ; that is, make E 5 and E7 zz 0.S0 ; E4 and E8 zz 1.-73 ; E3 andE9 zz 3 ; E10 zz 5.20 ; and Ei 1 zz 1 1.20 ; and thro' the points 7, 8, 9, &c. draw the hour lines parallel to EG. Laftly, make RM the height of the ftile zz E9 zz 3. 3. By the Scale. Having drawn the horizontal line ABQ, and QE making an angle with it of the complement of the latitude ; near the top, draw EG perpendicular to EQ for the fubftile and 6 o'clock line. Then in either of the polar lines, fet one foot of the com paffes at the beginning, and extend the other to I, fet that extent from E to 5, and from E to 7, on the line EQ^ Then extend from the beginning to II, and fet it from E to 4, and from E to 8. In like manner extend from the beginning to III, IV and V ; and fet thefe extents, from E (to 3 and) 9, 10, 11, refpeaively. Then thro' all thefe points draw lines parallel to EG, for the hour lines. And Sea. II. DIALLING. 85 And the diftance from the beginning to III on theFig. line, gives the height of the ftile ; to be fet perp. 29. upon the hour line of fix. PROB. XI. To make an eretl diretl weft Dial. This dial is made by the fame rules as the eaft dial, only changing the pofition of it. For in a weft dial, the point Q^ and angle AQE muft be taken on the, left hand. And when the hour lines are drawn, they muft be numbered the contrary way. The hours from the fubftile or 6 o'clock, muft be numbered towards the fouth 7, 8, 9 ; and from the fubftile towards the north, 5, 4, 3, 2, 1. If you look at the back fide of the paper, turning it to the light ; you will fee the true figure of a weft dial thro' the paper. Therefore if all the lines in an eaft dial were to pafs thro' the plane, they will make a weft dial, on the back fide of it. PROB. XII. To draw a Jouth eretl declining Dial. Example 1. Suppoje a dial declines weft 3 6 degrees, in the lat. 54i- 1. Geometrically. Take the point C about the middle of the plane 30. for the foot of the ftile, thro' which draw the ho rizontal line AB, and CF perpendicular to it, and equal to the perpendicular height of the ftile. Make the angle CFZ equal to the declination (36°), to the right hand if it decline eaft; or to the left, if weft ; to cut the horizontal line in Z. Thro' Z draw PZE perpendicular to AB, for the meridian. G 3 Take 86 DIALLING. Fig. Take ZF and fet it from Z to X, in the line AB. 30. Make the angle ZXP equal to the latitude (54f )» to cut the meridian in P. Then P is the center of the dial. Thro' C and P draw PCD for the fubftile. Thro' C draw QQ^perpendicular to PD, cutting the meridian in N, for the contingent line. In CQ make CT equal to CF, and draw PT. Take the neareft diftance from C to PT, and fet it from C to D, in the line PD, and draw DN. With the center D, and any radius as DC, def cribe the femicircle SCW. Divide the femicircle SCW into parts of 15 de grees for hours, beginning at the line DN, of where it cuts the circle. Thro' thefe points draw lines from the center D» to interfea the equinoaial QQv, in the hour points 11, 12, i, 2, 3, 4, &c. From P the center of the dial, draw lines thro' the hour points, as Pn, P12, Pi, P2, P3, &c. for the hour lines. Then the ftile may either be the perpendicular pin FC, or the. triangle PCT fet upright on the fubftile PC. Here the edge of the ftile PT muft be made very thin ; or which is better, a fpace muft be left at PC, equal to the thicknefs of the ftile, for it to ftand on, juft as if this dial was cut in two, thro' the line PCD, and one part feparated from thg other to that diftance. If you would have half hours and quarters, di vide each hour of the femicircle into twp or four parts, and draw lines thro' thefe points and the center, to cut the equinoaial, as before. The meridian PE muft be numbered 12, and the hours on the left muft be 11, 10, 9, &c. and thofe on the right 1, 2, 3, 4, &c. If you had rather begin at the center of the dial, you may proceed thus. Take the point P near the top, Sea. II. DIALLING. 87 top, for the center ; thro' which draw the meridian Fig. PE ; in PE take any point Z, thro' which draw 30. the horizontal line AZB. Make the angle ZPX equal to the complement of the latitude, cutting the horizontal line in X. Make the angle BZF equal to the complement of the declination, to wards the right hand, if the declination be weft ; but to the left, if eaft. Make ZF equal to ZX ; from F draw FC parallel to ZP, to cut the hori zontal line in C ; then CF is the height of the ftile, and C the foot of it. Thro' P and C draw PCD for the fubftile. Then draw the contingent QC|» and finifh the reft, as before. 2. By Calculation. Here we have given the latitude of the. place (54°i.), and the declination of the plane (360); to find the fubftile's diftance, the height of the pole above the plane, and the plane's dif. longitude. Radius — — 10. Sin. declination (36) 9.76921 Cotan. latitude (547) — 9.85326 Tan. Jubftile's diftance 22 45, 9.62247 Again, Radius - 10. Cof declination (36) 9.90795 Coj. latitude (547) — 9-7^395 Sin. file's height, 28 1 9.67190 And, S. latitude (547) — 9.91068 Radius ¦ 10. Tan. declination (36) 9^86126 Tan. dij. longit. 41 45, 9 95058 G 4 Then 88 DIALLING. Fig. Then add 15 degrees continually to 41 45 (the 30. diff. longitude), for the hour arches before 12. And fubftraa 15 deg. continually, till you come at the fubftile ; beyond which you muft add 15 deg. continually, to the remainder of the hour where the fubftile ftands ; for the hour arches after 12. And thefe hour arches are all reckoned from the fubftile. All thefe muft be regularly put down in a table againft the refpeaive hours. If you cal culate for quarters, you muft add and fubftraa 30 45' continually. Then you muft calculate for the hour angles, by the following analogy, and place them againft the feveral hours, as in the fol lowing table. As radius — — S. ftile's height (28 1) Tan. hour arch (86 45) Tan. hour angle (83 6) Tan. 71 45 Tan. 54 55 And fo on. 10. 9.67185 11.24.577 10.91762 10.48 1 81 10.15366 Hours. hour arches./ 86° 45' hour j angles. IX 83° 6' X 7r 45 54. 55 XI 5^ 45 35 37 XII 4i 45 22 45 I 26 45 *3 19 II " 45 fubftile 5 35 III 3 15 1 32 IV 18 15 8 49 V 33 15 17 7 VI 48 15 27 45 VII 63 15 42 59 VIII 75 15 66 7 Con* Sea. II. DIALLING. 89 , . Fig- Conftrutlion. 30, Take P near the top, for the center of the dial ; and draw the meridian PE, for the 1 2 o'clock line. By the line of chords make the angle EPC, 22° 45' the fubftile's diftance, on the right hand, as the de clination is weft ; and draw the fubftilar line PCD. Make the hour angles from the fubftile, as you find them in the table, viz. CPIX zz 83 6 ; CPX zz 54 55; CP11 zz 35.37; CP12 zz 22 45; CPi zz 13 19; CP2 zz 5 35 ; and CP3 zz 1 32; CP4ZZ8 49, &c. andPIX, PX, Pu, P12, Pi, P2, P3, &c. will be the hour lines. Make the angle CPN zz 28 1, for the ftile. 3. By the Scale. If you have a fcale fitted to your latitude ; find the declination on the line of chords ; againft which, on the other lines, you will find all the requifites, viz. the fubftile's diftance, the ftile's height, and plane's dif. longitude. If you have no fuch fcale, you muft find the requifites by Gunter's fcale, if you would work the whole inftrumentally ; other- wife by calculation, as before. Having made a table of the horary diftances from the fubftile,' choofe any point H near the top for the center, thro' which draw the meridian or 1 2 o' clock line perpendicular to the horizon. Make the angle EPC zz 22° 45', the fubftile's diftance from the meridian ; thro' C, draw PCD for the fubftile, on the right hand, becaufe the declination is weft. Draw bPd perpendicular to PC. Then from the line of latitudes, take the height of the pole above the plane, 2 8° i', and fet it from P to b and d -, and take the whole length of the line of the inclination of meridians, and fet from b to cut PC in O, and draw bO and dO. Then from the fame line of inclination, take thefe horary diftances (or 90 DIALLING. Fig. (or hour arches jl as you have them in the table, 30. viz. 11" 45', 26° 45', 41 45, 56-45* &c- and fet them feverally from O to a, a, a, &c. till you come at b. Alfo take the horary diftances 3 15, * 8 15, 33 15, &c. and fet from O towards d, as at r, r, r, &c. till you come at d. Then from P, thro' all the points a, a, a, and r, r, r, x&c. draw, lines quite thro' the dial, which will be the hour lines re- ^ quired. If any point falls above the horizontal line, draw a line from it, thro' the center, to the other fide. You may obferve, that any diftance Oa is equal to its correfpondent dr ; and any Or is equal to ba. Make the angle CPT zz 28 1 for the ftile. And the forenoon hours muft be markt towards the left hand, and the afternoon hours towards the right hand, of the meridian. If you would have the half hours, they muft be taken off the line of inclination, as before, and fet on the lines Ob and Od. As this is a very common and ufeful cafe, I fhall give another example of it. Ex. 2. Suppoje a S.plane declines eaft 49% in lat. 514. i. Geometrically. 3 1 . Take C for the foot of the ftile, thro' which draw the horizontal line AB ; and CF perpendicular to it, and equal to the height of the -ftile. Make the angle ,CFZ equal to the declination (490), to the right hand, becaufe the plane declines eaft ; and it will cut the horizontal line in Z. Thro' Z, draw PZE perpendicular to AB, for the meridian. N Set ZF from Z to X, in the horizontal line AB, 1 either way. Make the angle ZXP upwards, equal to the la titude (5 if) ; then P is the center of the dial. Thro' ~SS.'S\..B(U.igo. »¦ Sea. II. D I A L L I N G. 9* , Thro' C and P draw PCD for the fubftile. Fig Thro' C draw the contingent line QQ perpendi- 31 cular to the fubftile PC, to cut the meridian in N. Make CT equal to CF, and draw PT. Take the neareft diftance from C to PT, and fet from C to V in the fubftile PD. From ; the center V with any radius, defcribe the femicircle SCW. Divide the femicircle into parts of 15 degrees for hours, beginning at the line DN. Thro' all thefe points of divifion, draw lines from the center V, to cut the contingent QQ. in the hour points, 5, 6, 7, 8, 9, &c. From the center P of the dial,, draw lines thro' all the hour points, as P5, P6, P7, P8, &c. for the hour lines. Then the triangle PCT will be the ftile, which muft ftand perpendicular to the fubftile PC. 2. By Calculation. As radius — - 10. S. declination (490) 9-^7777 Cotan. lat. (gif> ¦ v 9,90060 Tan. fubft, dift. (30 50,) 9.77831 Radius — 10. Cof. -declination (49°) 9.81694"'; Cof. latitude {5 if) • 9.79414 S. ftile's height (24 6) 9.61 108 S. latitude (5 if) 9.89354 ¦ Radius- — 10. Tdn. declination (49) 10.06083 Tan. dif. longi t. (55 47) 10.16739 Then 92 DIALLING. Fig. Then having the plane's dif. longitude 55 47 ; 31. by adding and fubtraaing 15 degrees continually to and from it ; the hour arches, reckoned from the fubftile, will be had, as in the following table. hour hour arches. angles. 79" 13' 65 0 64 13 40 13 49 *3 25 20 34 13 15 3l 19 l3 8 6 4 13 1 44 Subftile .10 47 4 27 25 47 11 10 40 47 19 24 ¦ 55^ 47 3° 59 70 47 49 3l 85 47 79 46 1 Then for the hour angles, Radius — 10. S. ftile's height (24-6) 9.61 108 T. hour arch (4 13) 8.86763 T. hour angle (1 44) 8.47871 and fo of the reft, as in the table. ConjirutHon. Take fome point P near the top for the center of the dial, thro' which draw the meridian PE, for the 12 o'clock line. And make the an gle EPC zz 30 59 to the left hand, as the declina tion is eaft, and draw the fubftile PC. Then fet off the feveral hour angles from the fubftile, as you have Sea. II. DIALLING. 93 have them' in the table : CP8 zz 1 44 ; CP7 zz 8 6; Fig. CP6 zz 15 31,- &c. and you have the hour lines. 31. Make CPT zz 24 6, and CPT will be the ftile. 3. By the Scale. Having found the requifites by Gunter's fcale, or by calculation, take fome point P for the cen ter, from which draw the meridian or 12 o'clock line PE ; make the angle EPC zz 30 5g, to the left hand, becaufe the declination is eaft, and PC will be the fubftile. Thro' P draw a perpendicular to PC, on which fet the ftile's height 24 6, taken from the line of latitudes, from P to b and d. Then fet the whole length of the line of hours or inclination of meridians, from b Or d to O in the .line PC, and draw bO and dO. Then take the fe veral hour arches, which you have in the table, off the line of inclination, and iet them from O towards b, and from O towards d. Thofe on the left of the fubftile (4 13, 19 13, 34 13, &c.) muft be fet on Ob from O to a, a, a, Sec. And thofe on the right hand (10 47, 25 47, 40 47, &c.) fet from O to r, r, r, &c. Then from P, thro' all the points a, a, a, draw the hour lines P8, P7, P6, &c ; and thro' all the points r, r, r, &c. draw the hour lines P9, Pio, Pn, &c. Then make the angle CPT zz 24 6 for the ftile. PROB. XIII. To draw a north eretl declining Dial. To do this make a fouth declining dial, whofe declination is the fame, and lies the fame way, then turn it upfide down, and it will be . a north de clining dial. But the hours muft be numbered the contrary way. Thus, 94 DIALLING. Fig. Thus, if you want a north-eaft decliner, make' 31. a fouth-eaft decliner, if a north-weft decliner, make a fouth-weft decliner. Likewife, by extending the ftile and the hour lines quite thro' the center ; fo as the hours may be feen on the other fide of the plane ; the fouth- eaft decliner will then produce a north- weft decliner » and a fouth-weft decliner, a north-eaft decliner. But the hours muft be numbered 1, 2, 3, &c. from the meridian on the right hand; and 11, 10, 9, &c. on the left. But the midnight hours, or thofe before fun rife, and after fun fet, muft be left out ; as the fun, being then below the horizon, does not fhine on them. And in drawing the dial, the center H muft be taken about the middle of the plane, becaufe fome hours more are required, than in a fouth decliner ; and thefe are had by producing the other hour lines thro' the center. I judge it needlefs to draw a figure' for it ; but the calculation of fuch a dial will be as follows. Suppoje a north plane declining eaft 2 1 degrees, in lat. 544. Then we fhall find, by calculation, Or Gunter's fcale, The jubftile's diftance 140 20' Stile's height 32 50 Plane's dij. longitude, 25 151 And the hour arches, and hour angles will be. as in the following table. Hours h'allinfl P1T1L, ;i . qj. Sea. II. DIALLING. 95 Hours. hour arches. hour angles. 8 85 15 81 19 9 70 15 5^ 30 io 55 15 n 40 15 12 25 15 I 10 15 Subftile 1 2 4 45 3 *9 45 11 06 [ 4 34 45 20 37 5 49 45 32 38 6 64 45 4§ 59 7 79 45 7i 33 PROB. XIV. To draw an upright dial without a center ; or a far declining dial. Examp. Suppofe a plane declines 70 degrees eaft, in the lat. 54f- 1. Geometrically. In fuch dial planes, on which the pole has but fmall elevation, if the center of the dial be taken within the plane ; the hour lines, efpecially near the fubftile, will be fo clofe together, as not to be readily diftinguifhed ; and therefore fuch a dial would be ufelefs. But by making the ftile higher at pleafure, the hours will become wider ; but then the center will run out of the plane ; and therefore we muft ufe the following method to draw the dial without a center. Suppofe 96 DIAL LT N G. Fig. Suppofe the prefent center of the dial to be H, 32. thro' H draw the meridian HE, in which 'take fome point Z, thro' which draw the horizontal line XZB. Make the angle ZHX equal to' the com plement of the latitude, cutting the horizontal line in X. Make the angle BZF equal to the com plement of the declination, towards the right hand, when the declination is weft ; but to the left, when it is eaft. Make ZF equal' to ZX ; and from F draw FC parallel to HZ, to cut the horizontal line in C. Then CF is the prefent height of the ftile, and C the foot of it. Thro' H and C draw the fubftile HCD. Thro' C draw 1CI perpendicular' to the fubftile HD ; and thro' fome other point S, another perpendicular KSK, to cut HE in A. In CI take CN equal to CF, and draw HNO for the ftile. From S take the neareft diftance to HO, and fet from S to G in the fubftile, and draw GA. Then fince the ftile HO has but fmall elevation, draw Tt parallel to HO, at a convenient diftance, for the new ftile. Then take the neareft diftance from S to Tt, and fet from S to D in the fubftile, and draw the line DL parallel to GA. From the center D, defcribe a femicircle, cutting DL in L. Divide this femicircle into hours beginning at L, and draw lines from the center D, thro' thefe points* , to cut the contingent line KK in the hour points, ' that of L being 1 2 o'clock. Again, take the neareft diftance from C to Tt, and fet from C to d in the fubftile. From d de fcribe a femicircle, and make the angle Cdl zz 1 SDL. Then divide this femicircle into hours be ginning at/; and thro' all the points draw lines from the center d, to cut the contingent line II in the hour points, / being 12 o'clock. Laftly, thro' the correfpondent hour points, in both contingent lines, draw -right lines, and thefe will be the hour lines. And to number the hours, obferve Sea. II. DIALLING. 97 obferve the line DL runs to 12 o'clock, (which isFio-. generally out of the dial) and this direas all the 3 °.' reft; for the next hour point is 11, the next 10, &c. 2. By Calculation. The requifites being found as in the laft Problem, will be as follows. Subftile' s diftance 330 50' Stile's height 11 28 Plane's dij. longitude 73 31 Then fuppofe HC to be 2.7 inches ; HS, 7.3 ; then by plane trigonometry to find the leaft diftan ces from C and S to HO, As radius 10. S. i>HO (11 28) 9.29841 So HC (2.7; 0.43136 diftance jrom C (.537) — 1.72977 So HS {7.3) 0.86332 'iftance from S (1.45) o. 161 73 Thefe diftances being too little, increafe them with any quantity you think proper, fuppofe 1.1 ; then the neareft diftances from C and S to Tt, will be 1.64 and 2.55 ; that is, the radius SD and Cd will be 2.55 and 1.64. Then to find the hour points. After the hour arches are had by help of the dif. longitude 73 31, and put into a table; the correfponding diftances, along the lines KK and II, from the fubftile, are had by this proportion. As radius 10. Tan. hour arch {61 29) 10.26493 So SD (2.55) 0.40654 diftance Jrom S (4.70) 0.67147 So Cd (1.64) 0.21484 diftance Jrom C (3.02) H 0.47977 and 98 DIALLING. Fig. and fo on thro' the Whole, which muft be put 32. down in the table, againft their correfponding hour arches, as follows. Hours. hour diftance . diftance arches. from S. from C. 3 61 29 4.70 3.02 4 46 29 2.69 »-73 5 31 29 1.56 1. 01 6 16 29 0.75 0.48 7 1 29 0.06 0.04 Subftile 8 13 3l 0.61 o-39 1 9 28 31 1.38 0.89 10 43 31 2.42 1.56 11 58 31 4.16 2.68 Conftrutlion. Draw an obfcure line HE perpendicular to the horizon, and make the angle EHD = 33 50, and HD is the fubftile. Make the angle DHO zz 1 1 28, the height of the ftile, and draw HO, then draw Tt barallel to HO, at a diftance 1.1 from it; then HST7 Will be the ftile, to be fet perpendicu lar upoh the fubftile HD. Make HC == 2.7 ; and HS zz 7.3. And thro' C and S, draw two perpen diculars to the fubftile HD, as II and KK. Then fet off upon II and KK, from the points C and S, the diftances you find in the table ; thofe above the fubftile, to the left ; and thofe below, to the right of it. Laftly, thro' the correfpondeht points in the two lines, draw the hour lines of the dial. If you look thro' the back of the paper, you will, fee a dial declining as far to the wfeft. Z-By Sea. II. DIALLING. 99 Fifi. 3. By the Scale. * Having found -the requifites by Gunter's fcale, or by calculation, draw HE perpendicular to the horizon ; and make the angle EHD the fubftile's diftance, and DHO the ftile's height, and draw HD and HO for the fubftile, and ftilar line. Chufe a point in the fubftile as S, thro' which draw a line KK perpendicular to the fubftile. Then take the diftance from the beginning to the hour III, on the greater polar line, and fet it from S to T, and draw Tt parallel to HO ; and T* is the new ftile. Then open the compaffes from the beginning to the hour 111, on the leffer polat line, and fetting one foot at S in the fubftile HD, move it along the fubftile (keeping the feet of the compaffes parallel to KK,) till the other foot touch the line Tt; as at r ; thro* that point r, draw a line 1CI parallel to KK. Then from the table of hour arches, take the feveral arches from the greater polar line, and fet them from S, upon the line KK, firft to the left hand, and then to the right, as they lie in the ta ble, in refpea of the fubftile ; then you have the hour points on the line KK. Again, from the fame table of hour arches take every arch from off the leffer polar line, and fet both Ways from C, on the line II, as before. Then lines drawn thro' the correfpondent points in both lines KK and II, will be the hour lines of the dial. The ftile CS 17 muft be placed perpendicular on the fubftile CS, and made thin at the top -, unlefs you have a mind to leave a fpace there, the thick nefs of the ftile. H2 PROB. ioo DIALLING. Fig. PROB. XV. To draw a dial upon a diretl Jouth reclining plane, or upon a diretl north inclining one. \. If a fouth plane reclines, and the reclination is lefs than the complement of the latitude, add the reclination to the latitude, the fum is the lati tude, for which you muft make an erea fouth dial, by Prob. VIII. 2. If the reclination be equal to the complement of the latitude, then the pole falls in the plane of the dial ; and then you muft make an horizontal dial under the equinoaial, by Prob. VI. 3. If the reclination be greater than the comple ment of the latitude ; fubtraa the complement of the latitude from the reclination ; and. the remain der is the latitude, for which you muft make a ho rizontal dial, by Prob. VII. And thefe are all the varieties that can happen in a reclining fouth dial. In north inclining planes, if a dial be made on the fouth reclining fide, and the ftile and hour lines con tinued thro* the plane ; there will be made a north inclining or proclining dial, as required. Otherwife , thus, which comes to the fame thing ; make fuch a dial as is before direaed for a reclining plane, then turn it upfide down, and it will be a dial for its correfponding north inclining plane. But in all of them, the hours muft be numbered the con trary way, from the meridian or 12 o'clock line. Thefe inclining dials are the worft fort of dials, by reafon the fun does not fhine fo long on them, as upon other dials. PROB. Fig. 33 1 ///// / JJ 1 \ / '"" Z^> ^//j X ^^^ vi«F r=--\~"'S ^Mm^ A\y s I^P^^v /(,--' p / / » ' 'X 1 ^ Jr / r-7 / V x/ ^f^w/ ; / \ \ / V:X . / slf /// // /¦ / ~N\ '¦ / ^ > '" 3 /JaF / S{ / fl / ¦ \ / ^sA / \ it , /m\\// / M-7) 4 7~-7£% // y' "'¦/¦" \f y \_ / A \ / ¦cv\ 5 y^^A^-^j '"" / 6 / / / / / / / ' ' ?¦ / /¦'¦'" / i\ / E 7 8 9 IC > 11 Jfiia/Zuif TiyELfli ./-- Sea. II. DIALLING. 101 Fig. PROB. XVI. To make a direSl north reclining dial, or a diretl fouth inclining one. i. If a north plane reclines, and the reclination is lefs than the latitude ; fubtraa the reclination from the latitude, and the remainder will be the latitude, for which you muft make a direa fouth dial -, and turn it upfide down to face the north, and let the hours be numbered weftward, i, 2, 3, &c. andeaftward, 11, 10, 9, &c. but fome of thefe midnight hours muft be left out ; and fome taken in, after 6 in the morning, and before 6 at night. Otherwife thus, add the reclination to the com plement of the latitude ; and the fum is the lati tude for which a horizontal dial is to be made ; then the north end is to be turned upward, and the thing is done. But leave out the midnight hours. 2. If the reclination be equal to the latitude, then the plane coincides with the equinoaial ; and con- fequently a horizontal dial under the poles muft be made by Prob. V. 3. If the reclination be greater than the latitude, add the complement of the reclination to the lati tude ; and the fum is the latitude, for which a ho rizontal dial muft be made, by Prob. VII. which will fuit that plane. In fouth inclining planes, (by Prob. VIII.) make a direa fouth dial, for a latitude which is the dif ference between your latitude and proclination. Only, if that difference is o, it will be an upright dial under the equinoaial, and the ftile perpendi cular; and made by Prob. V. And if the procli nation be greater than the latitude, the dial will be long to fouth latitude ; and is made by Prob. VII. H 3 but 1Q2 DIALLING. Fig. but the angle of the ftile will lie the contrary way ,33. upon the meridian, as in fig. 33. A north recli- clining dial becomes a fouth inclining one, when turned upfide down. There need be no more than 12 hours in any of thefe incliners. v Example, To make a north dial in lat. 6o°, that reclines 5-f degrees. To the comp. lat. 30 add the reclin. 54. new latitude 354 for, a horizontal dial. 1. Geometrically. 34. Take the point P about the middle of the plane, for the center of the dial, and draw the meridian DPE, and a line perpendicular to it thro' P for the 6 o'clock line; Make the angle DPR zz 354. the new latitude. Take any point G in the meridian, thro' which draw the perpendicular AGB for the contingent line. Then fet one foot of the com paffes in G, and take the neareft diftance to the line PR, and fet it from G to D. With the cen ter D, and any radius, defcribe a femicircle LSM ; divide each quadrant LS, SM into 6 equal parts, thro' which and the center D draw obfeure lines to cut AB in the hour points. Thro" all thefe points or marks in A B, draw lines from the center P of the dial, and thefe will be the hour lines. But the midnight hours are left out, when the fun does not fhine. And the hours after 6 in the morning, and before 6 at night, are drawn thro' the center ; and on thefe the fun will fhine in fummer. DPR is the ftile, which will point to. the pole when the dial is rightly placed. 2. By Sea. II. DIALLING. 2. By Calculation. As radius : Sin. new latitude (35f) : : Tan. hour arch : Tan. hour angle. Hence the following table. 103 Fig. 34- Hours. hour arches. . hour angles. 3 9 4 8 5 7 6 45 0 60 0 75 0 90 0 3° 9 45 10 65 H 90 0 Conftrutlion. Draw the meridian DE, and take P for the cen ter ; at P make angles with PD and PE, as you find them in the tabfe, and draw the hour lines. JVlake DPR = 35f , and you have the ftile. 3. By the Scale. Let P be the center, about the middle of the plane -, and draw the meridian PE, and Pd perpen dicular to it, for the 6 o'clock line ; on which fet the extent to the new latitude (35T ), taken from the line of latitudes, from P to b and d. Then fet the whofe length of the hour line from b to a a.nd q, on the meridian DE ; and draw ba, da, bq* dq. Then on the line of hours, take the extent to t the firft hour, and fet from b to 5 and 7, and from d to 5 and 7. Then take the extent to the fecond hour IL, and fet from b to 4 and 8, and from d to 4 and 8. Alfo take the extent to III, and fet from i to 9, and from d to 3. Then from P draw lines H 4 thro' 104 DIALLIN G. Fig. thro' all thefe points, for the hour lines. Make 34. DPR zz 354- for the ftile. The ' mid-day or midnight hours need not be drawn, as the fun never fhines on them. PROB. XVII. To make an inclining or reclining eaft dial. Example. Let an eaft dial recline 2 1 deg. in lat. 5^. 1. Geometrically. 25. Take fome point C for the foot of the ftile, near the middle of the plane ; thro' which draw the per pendicular line DE. Draw CF perpendicular to DE, or parallel to the horizon, and equal to the height of the ftile. Make the angle CFH equal to the complement of the reclination or proclina tion ; downwards, if it recline ; or upwards, if it incline, to cut the vertical line DE in H. Thro' H draw the meridian or 1 2 o'clock line PHI per pendicular to DE. Make HG zz HP", and make the angle HGP zz the complement of the latitude, to the left hand when the plane reclines, and to the right hand when it inclines ; and P is the center of the dial, which will be at the top, as well as the meridian PI, when the plane inclines. Thro' P and C draw PCM for the fubftile. Thro' C draw.Qp^perpendiculaf to the fubftile PM for the contingent, in which make CT zz CF, and draw PT for the ftile. Take the neareft dif tance from C to the ftile PT, and fet it from C to V. Draw the line VI, from V to the interfeaion I of the contingent and meridian. From the center V, with any radius, defcribe a femicircle, which divide into hours, beginning at VI. Draw lines from the center V, thro' thefe divi- J?llK.va'.jc>4. Sea. II. D I A L L I N G; 105 divifions, to cut the contingent QQJn the feveral Fig. hour points. Then lines drawn from the center of 35. the dial P, thro' thefe points, will be the hour lines. And the triangle CPT, ereaed on the fubftile, PV will be the ftile. For the hours, PI being 12 o'clock, the reft pro ceed towards the left hand, being 11, 10, 9, 8, &c. in order. 2. By Calculation. 1. As radius 10. Coj. reclination (21) 9-97OI5 ' Tan. latitude (547) 10.14673 Tan. Jubftile's dift. 52 37, 10.11688 2. Radius * S. latitude (54f ) S. reclination (21) S. ftile's height, 16 58, 10. 9.91068 9-55432 9.46500 3. Radius —— • Cof latitude (54f) Tan. reclination (21) 10. 9-763959.58417 Cotan. dif longitude, 77 26; 9.34812 Then make a table for the feveral hour arches, by continually adding and fubftraaing 15 degrees, to and from the difference of longitude 77 26. And then to find the hour angles, Radius — — — 1.0. S. ftile's height (16 58) 9.46500 Tan. hour arch from ) 1010602 thejubftile (57 34) J Tan, hour angle 24 40, 9.66192 And fo of the reft, as in the following table. Hours io£ 35- DIALLING. Fig Hours. hour arches. hour angles. 3 57 34 2.4 4<> 4 42 34 15 0 5 27 34 8 40 6 J2 34 Subftile 3 43 7 2 26 0 43 8 17 26 5 H 9 32 26 10 30 IO n 47 26 62 26 *7 37 29 12 12 77 26 52 37 Conftrutlion- Draw PI parallel to the horizon at the bottom of the plane, for the 1 % o'clock line ; or at top of the plane if it inclines ; in which take any point P for the center of the dial, toward the left hand. Make the angle IPV on the right hand, equal to the fubftile's diftance from the meridian 52 37, and draw the fttbftile PV. Alfo make the angle VPT equal to the ftile's height 16 5%, and draw the ftile PT. Then make the hour angles on each fid.e of the fubftik> as you have them in the table. as VP7 = o 43, VP8 zz 5 14, &c ; and V£6 '=? 3 4^3, VP5 ==; § 40, &c. And draw the hour lines, 3. By the Scale. Having found the requifites as above, by, Gun ter's fcale, or by calculation ; and made a table of the horary diftances (or hour arches,) from the fub ftile ; take a point P near the bottom for the cen ter, thro' which draw the 1 2 o'clock line PI pa rallel to the horizon. Make the angle IPa zz fub ftile's Sea. II. DIALLING. ioy- ftile's diftance 52 37, and draw the fubftil® Pa. Fig. Thro' P draw bd perpendicular to Pa, in which 35. make Pb and Pd 16 58, tl]e pole height, taken from the line of latitudes. Then take the whole line of hours, or the line of inclination, and fet from b to a in the fubftile, and draw ba and da ; then from the fame line of inclination, take the hour arches as you have them in the table, and fet thofe above the fubftile from a towards b ; and thofe below,, from a towards d, on the lines ab' and ad. Thus, make ab zz 1a 34, 05 zz 27 34, a\ zz 42 34, 03 zz 57 34; alfo ay zz 2 26, aS =-17 26, ag zz 32 26, &c. And draw the hour lines a 3 04, a$, a6, 07, &c- And make the angle aPT zz 16 58, for the ftile. PROB. XVIII. To make an inclining or reclining weft. dial. This dial }s made by the fame rules as the eaft incliner or recliner ; only the angle PGH, which is made equal to the complement of the latitude, muft be taken to the right hand, for a weft recli ner ; but to the left hand for a weft incliner. And the meridian PI will he at the bottom of the plane, if it reclines ; but at top, if it inclines, The hours muft be reckoned the contrary way in the weft dial ; but the meridian or twelve o'clock line will gujde all the reft. If you turn the back of the paper towards you, and look thro' it, the eaft recliner will be changed into a weft recliner, for you will then fee the figure of a weft recliner thro' the paper. And fo the back fide of an eaft incliner, becomes a weft incliner. Excmp. io8 Fig. 35- DIALLING. Examp. Suppoje a weft dial reclines 50°, in lat. $4\. As all .the work is conftruaed as before, I fhall only give the calculation of it, as follows. Subfile's diftance 42 ° i' Stile's height 38 35 Plane's dif. longitude. 55 19 Then the table of hour angles and hour arches, will be as follows. Hours. hour arches. hour angles. 10 85 19 82 41 11 70 19 60 10 12 55 *9 42 1 1 40 19 27 53 2 25 19 16 26 3 10 19 6 29 Subftile 4 4 4i 2 56 5 19 41 12 34 6 34 4i 23 21 7 49 4i 36 19 8 64 4* 52 51 9 79 41 73 44 And if an eaft dial inclines 50% In lat. 54^ ; the hour angles will be the fame as above. PROB. XIX. To draw a fouth declining, reclining or inclining dial. Example 1. Suppofe a S.plane declines weftward 250, and reclines 15% in lat. 54f. 1. Geo- Fig 35 Ut umbra fi.c vita JDialti, nf nr IN'" V VI vir 47 Dl -H: e m IX ZH. 2. By CahulMion. As fin. reclination ( 1 5b) 9.41299 Cotan. declination (25) 1 0.33 132 Radius — 10. Tan. beig. meridian, 83 9, 10.9183; As Radius 10. Cof. declination (25) 9-957^7 Cotan. reclination (15) 10.57194 Tan. arch A, 73 32 iG.,52921 lat. 54 30 Arch B 19 2 ; here A beirtg greater i ' - l!Mi(t than the latitude ; the oppofite jtole is elevated. As Cotan. arch B (19 2) 10.46220 Cof. reclination (15) 9.98494 Sin. declination (25) 9.62594 . 19.61088 v Tan. fubftile's dift. 81 9.14868 ' At Cof. fubftile's diftance (8 1) 9-99573 Coj. arch B (19 2) — 9.97558 Radius io. Cof ftile's height, 17 19 9-97985 As Sea. II. DIALLING. As Sin. arch B. (19 2) — Radius -*-*. Sin. fubft. dift. (8 1) —1. Sin. plane's dif. longitude, 25 19 111 9-5»337Fig. 10. 36, 9- 1 4445 9.63108 Then make a table for the feveral hours, and place the plane's dif. longitude 25 19, againft 12. To or from which add and fubtraa 15, 30, 45, 60, &c. to get the other hour arches, which put againft their refpeaive hours. Then for finding the hour angles* Radius — — — JO; S. ftile's height (17 19J 9.47370 T^n. hour arch (79 41) 10.73985 r r - 1 - . Tan. hour angle, 58 33, 10,21355 for VII. And fo on for all the reft, which will be as in the following table. Hours. hour arches. hour angles. 7 79 4i 58 33 6 64 41 32 11 5 49 4i 19 lo 4 34 4i ii 38 3 19 41 6 5 2 4 4* Subftile 1 24 ; I 10 19 3 7 12 25 19 ¦ 8 1 I I 40 19 14 1 1 10 55 *9 23 16 9 70 19 39 46 8 85 19 74 37 Con- 112 DIALLING. Fig. 36, Conftrutlion. Draw the horizontal line hH near the bottom of the plane, in which take a point H towards the right hand, if the declination be weft ; but to the left if eaft. Draw the line ZH making the angle hHZ zz 83 9, the height of the meridian, for the 1 2 o'clock line. Take any point P near the top, when A is greater than the latitude, otherwife near the bottom, and draw the fubftile PV to the right hand for weft declination, making the angle HPV zz 8° 1', the fubftile's diftance. Then from P draw the hour lines, making angles with the fubftile at P, as you find them in the table. Thus VPII zz i° 24'. VPIII zr 6° 5', &c. And draw PT ma king the angle VPT zz 17 19, the ftile's height, and PTC is the ftile. 3. By the Scale. Having found the requifites by Gunter's fcale, or otherwife, draw the meridian PH, making an angle with a horizontal line of 8 30 muft take the point G below C, and H above it j. fee fig. 37, and then the horizontal line GI will be belowy and the meridian IH will run upwards from I. And the angle IOP or lOp muft always tend to -that part of the meridian 1Z, above 1 ; and -made equal to the latitude ; and the interfeaion of Op or OP, will always find the center ; then all the„- reft is plain. 2. For the Calculation, the fame rules ferve as. for recliners ; only the arch A is always more than, 900. 3. The working by the Seale is, alfo. the fame,, when the requifites are found. We may here take notice, that a fouth weft re cliner turned upfidc down, becomes a north, weft, incliner ; and a fouth eaft recliner turned upfide down, becomes a north eaft incliner ; the quantity of the declination and inclination remaining the fame ; but the hours muft be numbered contrary. Alfo if a dial be turned upfide down, the back- fide (the hours appearing thro' the plane,) will be I 4 it» 120 DIALLING. Fig", its oppofite incliner. But the midnight hours muft 38. be left out, and fome others put in, by producing the reft thro', the center. PROB. XXL To draw a north declining, reclining or inclining dial. Examp. 1 . Suppoje a N. dial plane to decline eaftward 25°j and to incline 15°, in lat. 54f. 1. Geometrically. 39. Let C be the foot of the ftile, DCE perpendi cular to the horizon, CF the height of the ftile perpendicular to DE. Make the angle CFG zz the proclination ; down-i wards, becaufe it inclines, and CFH the comple ment of it, to cut DE in H the zenith. Thro' G draw the horizontal line AG perp. toDE. In GE take GR zz GF, and make the angle GRI equal to the declination of the plane, to the fame hand it declines, that is, eaft. Thro' I and H draw the meridian or 12 o'clock line HL Thro' C draw CMK perpendicular to HI. In the line CH fet CM from C to L, and ex-- tend from L to F, and fet it from M to O, in the line MK ; and O is the dividing center of HP. Draw 01, and make the angle I OP downwards,- equal to the latitude, and draw OP interfeaing Hi in P the pole for the center, Thro' P and C draw PCV for the fubftile. Thro' C draw the contingent QQ perpendicular to the fubftile PC, interfeaing the meridian PH inN. Take CT =z CF, and draw PT, and CPT is the ftije. Take Sea. II. DIALLING. 121 Take the neareft diftance from C to PT, and fet Fig. it from C to V, in the fubftile PCV, and draw VN. 39. From the center V defcribe a femicircle, which divide into hours, beginning at VN. Then draw lines from V thro' thefe points to cut QQj and then from P thro' the points in QQ, draw the hour lines. 2. By Calculation. Here the data being the fame as in ex. 1. Prob. XIX, all the requifites will be the fame, being found by the fame rules for an incliner, as for the oppofite recliner ; and are as follows. Height oj the meridian 83° 9' Arch A 73 32 Arch B — iq 2 Subftile' s diftance 8 1 Stile's height — 17 19 Plane's dij. longitude 25 19 Here A being greater than the latitude, your pole is elevated. Then fee the table belonging to that example. Conftrutlion. At the bottom of the plane, draw the horizon tal line PY, and make the angle YPH = 83 9, to the right hand, and PH is the meridian, or 1 2 o'clock. Make the angle HPV zz 8 1, to the right hand, becaufe the declination is to the left, and PV is the fubftile. Then from P draw the hour lines, making the angles with the fubftile P V, , as you find them in the table. Thus VPi zz 10 19, VP12 zz 25 19, and VP2 zz 4 41, VP3 zz 19 41, &c. Make the angle CPT = 17 19, and CPT is the ftile, 3. By the Scale. Find the requifites by Gunter. At the bottom pf the plane, make the angle YPH zz. 83 9, and angle 122 D I A L L I N G. Fig. angle HPV = 8 i, and PV is the fubftile. Draw 39. bPd perpendicular to PV. Set the pole's height (17 19), taken from the line of latitudes, from P to b and d, and fet the whole line of inclination from b to a, and draw ba, da. Then take the fe veral hour angles (as in the table), and fet frora a towards b and d. Thus take 4 4 1 and fet from a to 2, and from b to 8. Set 19 41 from a to 3, and from b to 9. Set 34 41 from a to 4, and from b to 10, &c. Then from P draw lines thro* all thefe points for the hour lines. Make the angle VPT zz 17 19, for the ftile. Example 2. Suppoje a north plane to decline weftward 6o°, and recline 52°, in lat. 54-. 1. Geometrically. 40. About the middle of the plane take C for the foot of the ftile •, thro' which draw DCE perpen dicular, and CF parallel to the horizon, and make CF zz height of the ftile. Make the angle CFG zz" reclination, upwards becaufe the plane reclines, and CFH the comple ment of it, cutting DE in G and H the nadir. Thro' G draw AG perpendicular to DE, for a horizontal line. In GD take GR zz GF, and make the angle GRI zz the declination {6o°) ; to the fame hand it declines, which is weft. Thro' I and H draw the 12 o'clock line IH. Thro' C draw CMK perpendicular to IH, cut* ting it in M. In the line CD fet CM from C to L, and extend from L to F, and fet it from M to O, in the line MK, for the dividing center' of PH. Draw 01, and make the angle IOP downwards (being a north plane.) equal to the' latitude (54f ), and , > ¦ 1 ^ / / ¦• "\. ^\ 7S4\ %$8- '"> /-. 7 "X'/\- i ^\-.;A > Js' s'^^ / 1 / r--* hi / HJ/ \ \^ -A/ /P 3> - \ ii " Y 'V7 / f /J2 // fe\ b U /^" 'v 39 \ * 1 .,-" Ml \ - - ( \/-^rl '- 'T7 'V •- gfa-^JT^ rt /.---, -P / -. \ A'i \ . • \ \\ • 1 //^B By / \ \ \ \ '-/ /2f / Jjjl F/ 7 \\ E ^^^^^ Y IP ur//t u-i, 9 FL:JSSr./oa-j2a. Sea. II. DIALLING. 123 and draw OP, interfeaing IH in the pole P, for Fig, the center of the dial. 40. Thro' P and C, draw PCV for the fubftile. Thro' C, perpendicular to the fubftile PC, draw QQ, for the contingent, interfeaing the 1 2 o'clock line PH in N. Take CT = CF, and draw PT for the edge of the ftile. Take the neareft diftance from C to PT, and fet it from C to V in the fubftile PCV, and draw VN. From the center V defcribe a femicircle, which divide into hours as ufual, beginning at VN. Then drawing lines, from V thro' the points in the circle to cut QQ ; and then from P thro' the points in QQ ; and thefe laft will be the hour lines. And the hours muft be numbered from PN, 1, 2, 3, 4, &c. to the left hand, and the midnight hours left out. 2. By Calculation. As Sin. reclination (520) 9.89653 Radius ¦ 10. Cotan. declination (60) 9.76143 Tan. height meridian, 36 14, 9.86490 As Radius — 10. Cof. declination (60) 9-69897 Cotan. reclination (52) 9.89280 Cotan. arch A, 68 40 Colat. 35 30 Arch B 104 10 Here the north pole is elevated, Then, Cotan. 124 Fig. 40. DIALLING. Cotan. arch B (104 ioj 9.40212 Cof. reclination (52) 9.78934 S. declination (60) 9-93753 19.72687 2a#. fubft. diftance, 64 40 10.32475 or rather 115 20. ¦ As Cof. fubftile's dift. (64 40) 9.63132 Cof arch B ("104 10) 9.38871 Radius — 10. Cof. ftile's height, 55 7, "9-75739 As Sin. archB (104 10) 9.98658 Radius — 10. Sin. Jubftile's dift. (64 40) 9.95608 $/». <$/". longitude, 68 47,' 9.96950 or rather 11 1 13. - » Then make a table of hour arches placing 68 47 againft 12, and adding or fubtraaing 15 degrees, find the reft -, and from them the hour angles, by the following analogy, which put into the table as ufual. But the midnight hours muft be left out, Radius : S. ftile's height : : Tan. any hour arch : Tan. hour angle jrom the. Jubftih. Hours. Sea. II. DIALLING. Hours. hour arches. hour angles. 1 1 83 47 82 26 12 68 47 64 41 I 53 47 48 15 2 38 47 33 24 3 23 47 »9 53 4 8 47 7 13 Subftile 5 6 13 5 6 6 21 13 17 40 7 36 13 31 0 8 5i »3 45 3^ 9 66 13 61 45 io 8r 13 [ 79 20 125 Fig. 40. Conftrutlion. From a convenient point P for the center, draw the horizontal line PY. Make the angle YPI zz 36 14, the height of the meridian to the right, and draw the 12 o'clock line IPN quite thro' the center. Alfo make the angle NPV zz 64 40, the fubftile's diftance, on the left hand, as the declination is weft, and draw the fubftile PV. Then from P draw the hour lines, to make angles with the fubftile as in the table ; and the hours that are wanting draw thro' the center, leaving out the midnight hours. Make the angle VPT zz 55 7 the ftile's height, and CPT is the ftile. 3. By the Scale. Find the requifites by Gunter, or otherwife ; and draw PY parallel to the horizon. And make the angle YPI zz 36 14 the height of the meridian, and draw IPN; and make NPV zz 64 40, the fubftile's diftance ; and draw the fubftile PV -, and draw b?d perpendicular to PV ; and from the line i26 DIALLING. Fig. line of latitudes take 55 7 the pole's height, and 40. fet from P to b and d. Then fet the whole line of inclination from b to a, and draw' ba, da. Then having a table of hour arches, take them from the line of inclination, and fet them from a towards b and d, in the lines ab, ad; and draw lines from P through thefe points 5, 6, 7, 8, 4, 3, 2, &c. for the hour lines. Make the angle CPT zz 55 7, and CPT is the ftile. The other hour lines that are wanting, muft be drawn thro' the center. If A be zz the latitude, then B is 90% and the fubftile's diftance, and plane's1 dif. longitude are alfo 900, and the ftile's height is found by this pro portion ; rad : S. declination ; : coj. reclination : coj. ftile's height. If A be lefs than the latitude ; then B, the fub;- ftile's diftancej and plane's dif. longitude, will be lefs than 90. If A be greater than the latitude, then B and fhe fubftile^ cM'ance, are greater' than 900 ; as in- this example, where IPV is greater than a right' angle. For inclining dials. In thefe, the point G will be below Cr and H above it -, and the angle IOP muft be taken down wards from L which reprefents a point of the ho rizon. The calculation, and projeaion by tiW fcale, are the feme as in the oppofite recliners ; but the contrary pole will be elevated. Hence we may know, that if we want a north eaft incliner, we muft make a fouth. eaft recliner, and turn it upfide down. Or if we want a north weft incliner, we muft make a fouth weft recliner, and turn it up fide down : but the hours muft be numbered the contrary way from the meridian. Alfo any dial being made for a reclining plane ; the ftile and hours, Sea. IL DIALLING. uf hours produced thro' the plane, will make a dial, Fig. for the oppofite or inclining fide of the plane. I fhall here add two or three more examples, with the requifites, hour diftances, and hour angles •, leaving the conftruaion thereof, for the exercife of the young ftudent. Ex. i. Suppoje a S. E, plane defines 25% and reclines 19% in the lat. 54f . Height of the meridian 8i° 22' Arch A — 69 1 2 Subfile's diftance 5 59 Stile's height 13 27 Plane's dif. longitude 24 15 Ex. 2. ^1 N. W. plane declines 55° Hours hour arches. hour angles. 5 80 45 55 °° 6 65 45 27 18 7 50 45 L5 53 8 35 45 9 30 9 20 45 5 2 10 5 45 Subftile 1 20 11 9 J5 2 10 12 24 15 5 59 and reclines 20f ;' lat' 54° Height of the meridian Arch A — Subftile' s diftance Stile's height Plane's dif. longitude. 25' 6 6? 33 62 57 36 39 73 3 Hours 128 DIALLING. Hours. hour arches. hour I angles. 2 3 4 76 57 61 57 46 57 68 47 48 15 32 55 56 3r 57 16 57 20 25 10 i() ¦ 7 1 57 Subftile 1 10 8 13 3 7 53 9 28 3 J7 39 £x. 3. ^ AT. £. jj/flKf declines 50?, *z/zi retlines 70° 54f- Height of the meridian Arch A — — Subftile's diftance — Stile's height — . Plane's dif. longitude in lat. 41° 46' '3 10 147 29 «./p, Sea. II. DIALLING. Hours. hour arches. hour angles. 45 84 28 6g 28 83. 48 67 14 678 54. 28 39 28 24 28 51 20 36 19 22 6 9 9 28 Subftile 8 28 IO 5 32 4 19 n ¦ 20 32 18 29 12 35 32 32 31 I 2 3 50 32 65 32 80 32 47 *9 62 59 79 25 129 Scholium. In all thefe forts of dials, the horizontal line is of great ufe, and ought to be kept on, when the other lines are rubbed out ; for by the help of that, the dial is fet up in its true pofition. Alfo in all kinds of dials, that fhow the hour, &c. by a per pendicular pin, the horizontal line cuts oft all fu- perfluous lines in every dial, which can never be touched by the fhadow of the ftile. For what is above that line is of no ufe, becaufe the fun is be low the horizon, and cannot then fhine on the dial, or eaft any fhade above the horizontal line ; except the dial happen to be fhifted to a new place. K PROB. 130 D I A LL I N G. Fig. PROB. XXII. To draw' any hour line, when the line from the cen- -' ter of the dividing circle, does not cut the contingent' line, within the plane. Let C5 be the hour line to be found, where the I2' dividing line N» drawn from the center N of the circle, does not reach the contingent line AB. Try how near the dividing line Nrt, approaches the con tingent AB, about the fide of the plane, as within f, f, 4, &c. of NQ. Take fuch a part of NQ, at which diftance from Q, draw an obfcure paral lel to AB, to cut Nn in c ¦, draw chr parallel to the fubftile NQ. Then take the fame part of QC (as f, ~, 8cc), and fet it from h to a, in the parallel cr, and thro' a draw the hour line C5. Or thus. When the dividing line N» approaches the contingent AB very (lowly; draw Cr, N/ pa rallel to AB. And fee what part of NQ^ (as f , f, f , &c.) the line Nn is diftant from Nf, near the edge of the plane. Set that part from N to wards Q> from which point draw an obfcure pa rallel to AB, to cut Nn in c -, then thro' c draw fr parallel to NQj and take the fame part of CQ, and fet from r to a ; and thro' a3 draw the hour line C5. ¦ When the end n of the line N» comes, nearer AB than Nf, the former way is to be ufed ; but when it comes nearer Nf, the latter method. The reafon of this operation is, that CQj QN : : ha : he : : ra : fc, by fimilar triangles. PROB. XXIII. To draw a dial upon the deling of a room, which will '-Jhew the hours by reflexion. Place a fmall piece of a looking glafs exaaiy horizontal in a window, where the fun fhines, ; mea fure. Sea. II. DIALLING. 13^ fure the perpendicular diftance from the glafs to Fig., the cieling, for the height of the ftile ; and where ,12. ' the perpendicular cuts the cieling, is the foot of the ftile. Then having the height of the ftile, and its foot ; make a horizontal dial thereto upon the cieling (by Prob. VII.) ; and that will be your dial. The glafs is beft placed pretty high, as upon the tranfom. This dial is no more than a horizontal dial, 41." turned upfide down. For let CB be the cieling, G the glafs, GC perpendicular to the cieling, and equal to CF. Then if AG be a ray of the fun, reffeaed from G to the cieling, and FB a direa ray. Then the direa rays AG, FB will be parallel, and therefore the angle CFB zz angle CGA, and CGA zz CGB, becaufe the angle of incidence is ^equal to the angle of reflexion. Therefore CFB zz CGB, and GC being equal to CF, the direa ray FB and the reffeaed ray GB will fall upon the fame point B, of the cieling. Therefore if a dial be made on the plane CB, for the ftile CF, to fhew the hour by the direa rays FB, it will ferve equally for the reffeaed rays GB. For the drawing of this dial, a meridian line muft be had ; therefore at 1 2 o'clock, when the fun is in the meridian, hang up a line and plummet clofe by the glafs, mark its fhadow in the floor with chalk ; transfer this to the cieling, by help of a plumb line, and it will be the 1 2 o'clock line. If your window does not face the fouth, you may draw an eaft and weft line, thus. Calculate the fun's altitude when eaft or weft ; then obferve with a quadrant when he has that altitude ; and at that moment hold up a line and plummet, clofe to the glafs, thick enough to eaft a fhade to the oppofite wall, this carried up to the cieling gives the point . of eaft or weft. K 2 It i'34. DIALLING. Fig. It will be beft to draw your dial firft uport a 4*1. large paper, and when- that is done, draw a line perpendicular to the meridian to cut all the hour lines ; and then meafure the feveral diftances of the hour lines from 12, upon this line, in inches; and alfo the diftance of this line from the foot of the ftile. Then draw a line on your cieling as ma ny feet from the foot of the ftile, on which fet off the feveral hour points in feet, which were mea fured before in inches ; thro' which draw the hour lines to the center of the dial. But becaufe the foot of the ftile is commonly within the wall, and the center without doors ; it will be neceffary to draw two perpendiculars to the meridian, and get the hour points i» both, thro' which the hour lines muft be drawn. ' If you would continue the hour lines along the fides of the room ; draw them firft quite thro' the cieling till they cut the walls. Then a plane pair ing through any hour line and the glafs, will cut arty wall in the fame hour line. Therefore extend a thread from the glafs to the extremity of any hour line upon the cieling, which keep fixt there. Then extend another thread acrofs it, from the o-^ ther end of the fame hour line, juft to touch the former thread, and to reach to the oppofite wall ; it will touch that wall in a point, thro' which the fame hour line is to pafs. Note, inftead of a glafs, you may ufe a little water, which of itfelf, will always have its furface horizontal ; for if the glafs be not horizontal, the error by the reflexion will be doubled. And the water being always in motion, by the agitation of the air ; makes the point of reflexion on the cieling more eafily diftinguifhed. Cor. Ajter the Jame manner, a dial may he made en the floor oj a room; to Jhew the hour, by the jha dow of a black Jpot in the window. For SqOl. II. DIALLING. 133 For let CB be the floor, and F a place in theFig. window upon which you muftftick a patch fo large 41. as to give a fhadow in the floor, Then if FC be taken for the height of the ftile, and C for it's foot. And a horizontal dial be made on the floor, to the ftile CF, whofe height you muft meafure ; it is evident, the fhadow of the point F upon the floor will fhew the hour of the day. PROB. XXIV. To find what latitude a dial is made for. Meafure the angle that the ftile makes with the plane of the dial, and that will be the latitude, if it was a horizontal dial ; or the complement of the latitude, if it was an erea fouth dial. But if the dial has no ftile, as it may happen to be broken off by fome accident. Then meafure the angle between the hour lines of 12 and 3, or 12 and 9. Then find the tangent of this angle; feek that tangent among the fines, and the arch be longing to it, is the latitude, if it is a horizontal dial ; or the complement of the latitude, if it is a full fouth dial, or a full north one. Or in general, meafure the angle between 12 and any hour line ; then take the hour arch belongings and fay, As Tang, hour arch : Tan. hour angle : : So Radius : S. latitude for a horizontal dial, or cof. lati* tude for a direa fouth or north dial. PROB. XXV. How to place a dial in a true fituation. "Before any dial can be truly drawn upon a plane* it is neceffary to have the fituation of that plane, K x fuch 134 DIALLING. Fig. fuch as the inclination and declination, and the la titude of the place, which are the neceffary data for conftruaing it. And when the dial is drawn, it is equally neceffary to place it in the fame fitua tion, or elfe it can never fhew the time truly. If the dial was drawn upon a fixt plane, there is no more to be done with it ; but if it is upon a loofe moveable plane, it will require the fame operati ons to fet it up, as were ufed in determining its fituation. Tn the placing any dial, the three following di- reaions muft be obferved. i. That the horizon tal line, which is drawn for this purpofe, be placed parallel to the horizon. 2. That it be. placed fo as to have its proper declination. 3. That it may have its proper degree of reclination or proclina tion. .Thefe rules ferve generally for all dials, and muft be exaaiy performed, if you would expea the dial to go truly. In a horizontal dial, the meridian muft be fet horfh and fouth, which maybe done by Prob. I. pr by help of a line and plummet held up at i 2 o'clock.;* Then the meridian or 12 o'clock line muft be placed perfeaiy level ; and fo" muft the 6 o'clock hour, lines. And this may be done by help of a level1, or with a quadrant. ] - In a full fouth dial, the plane of it muft face the fouth direaiy, which may be done by a compafs, or fetting it to twelve o'clock, exaaiy at noon. And the 12 o'clock line muft be perpendicular to the horizon, which may. be fet by a plumb line. Alfo the plane of the dial muft be placed perpen dicular to the horizon, to be done alfo by a plumb line. In upright declining dials, the 12 o'clock line muft be fet perpendicular by a line and plummet ; as alfo the plane of the dial. And it muft have its proper declinationj and this may be done by help ot Sea. II. DIALLING. 135 of a compafs, or more exaclly by the fun's akitudcFig.-- and azimuth, as deferibed in Prob. III. And laftly in declining reclining dials, the hori zontal line muft be fet parallel to the horizon. And the declination, fet out by the compafs, or by the fun's altitude and azimuth. And the fame recli nation muft be given to the dial as it was cal culated for ; the method whereof is deferibed in Prob. II. ' The greateft difficulty is to fet a dial to the true declination. Therefore if you have a watch truly fet, and the dial have its proper pofition in other refpeas. Then if the fun fhines, you'll fee whether - . the watch and the dial agree or not ; if not, alter the declination till they do agree. And if the ho rizontal line be now parallel to the horizon, and the dial have its proper inclination, and the time agrees with the watch, you may conclude your dial truly fet j and there it muft be fixt. PROB. XXVI. Any dial being made for one place, how to fet it up in any other place, to Jhew true time. If you would fet it by the rules of this art, you muft know the latitude it was made for, and the , declination and inclination, if it has any. Thefe things being had, its fituation, in its original place is known. Therefore to fet it in any other latitude, you muft place it in a fituation quite parallel to its firft fituation, and the thing is done. For a horizontal dial, note the difference of la-* • titude between its original place and your place. Then raife the north fide of the dial, if it be car ried northward, or the fouth fide, if it is carried fouthward ; fo many degrees, as is the difference of latitude ; and there fix it. K 4 -F« 136 DIALLING. Fig. For an upright fouth or north dial, fet it to lean towards the fouth if it is carried northward ; or towards the north, if carried fouthward, as many degrees as is equal to the difference of latitude. And if it reclined or proclined before, north or fouth ; it muft be made to recline or procline, fo much more, or fo much lefs, as the cafe requires. A full eaft or weft upright dial muft be fet to face the eaft or weft ; but the horizontal line muft be elevated at fhe north end, if it is carried north ; or the fouth end if carried fouth, as many de grees as the difference of latitude comes to. '42. For all forts of declining dials, new requifites muft be found to fet them up by. Let AB be the meridian, B the place it was made for, A the place it is to fet up in. Make the angle GBF zz decli nation of the plane, BD the reclination of the plane, and BF its complement. Let F be the pole of the great circle DE, and thro' A draw the parallel circle CA, alfo draw FAE. Then in the fpherical triangle FBA, there are given FB the comp. reclination, the angle FBA the fupplement of the declination, and BA the difference of lati tude. To find FA the comp. reclination at A ; the angle FAB the declination at A ; and angle AFB the elevation of the horizontal line of the dial, above the horizon at the place A. And ac cording to thefe new requifites, when found, the dial muft be fet up ; and not by the declination and reclination at B, where it was made. And this holds for the oppofite inclining plane. For fince BID is the reclination, if the dial is re moved to D, it will be an upright dial, and fo it will be at E ; and when brought to A, and fet pa rallel, AE will be its reclination, the fame as it would have at C, or in any place of the parallel circle AC, for it will always be in the plane of it. Therefore FAB will be the declination at A. And fince the horizontal line of the dial is parallel to the Sea. II. DIALLING. 137 the horizon, when the dial is at C, or any where Fig. in the circle FD ; therefore when it is removed from 42. C to A (in a quite parallel pofition) being ftill in the plane of the circle CA, that end of the hori zontal line next A, muft needs be elevated above the horizon at A, to an angle equal to the arch CA or DE, or the angle BFA ; or the end next C, depreffed fo much. It may be noted, that in an eaft and weft re cliner, FBA will be a right angle. And in an up right decliner, FB will be a quadrant, or BA will coincide with DE, and FA will be greater or lefs than a quadrant, according as the angle FBA is obtufe or acute. Otherwije. This Prob. may alfo be done mechanically, af ter this manner. Stretch a thread, or rather a wire, between two fixt points, fo that it may be parallel to the earth's axis, or point direaiy to the pole. Then take your dial, and fet the edge of the ftile clofe to the wire, or at leaft parallel to it, and con tinuing it fo, turn the dial gently about this axis, till the fun, fhining on it, fhews the true hour of the day ; which may be known by a clock, or by another dial. There fix your dial ; or at leaft keep k in that pofition, till you take the declination, and reclination of it, as has been fhewn before ; and then the dial may be fet, from thefe requifites obtained. Cor. Hence you may place a dial, Jo as tojhew the hour oj the day in any given place of the world. For if the ftile be held clofe to the wire, as be fore direaed, and the dial be turned gently about the wire and ftile, as an axis ; till the fun fhews the time fo much more than the true, as the lon gitude of the faid place is more eaft than your place ; or elfe that it fhew the time fo much lefs than 138 DIALLING. Fig. than the true* as the place differs in longitude weft ward. There fix the dial, and it will always fhew the true time at that place. Here 15 degrees of longitude is equal to an hour, 30 to two hours, 45 to three hours, &c. Therefore the dial being fet fo much fafter for a place of eaft longitude; or fo much flower for a place in weft longitude, will eonftantly fhew the hour of the day at the other place. PROB. XXVII. To find the hour of the night by the moon's Jhining upon a Jun dial. Get the moon's age, or the number of days from either change or full, to the prefent time ; and take 4ts- thereof for the number of hours that the moon is behind the fun. Add thefe hours to the time fhewn by the moon, on the dial ; throwing out 12 if it exceed, and you have the hour of the night. Or thus. If you know the time of the moon's fouthing, count how many hours and minutes the fhadow on the dial wants of 12 o'clock; fubtraa them from the time of her fouthing, for the hour of the night. But if the fhadow be after 12, add thefe hours and minutes (on the dial) to the time of her fouthing, rejeaing 12, if it exceed; and you haVe the hour of the night. ' Scholium. Before I leave this feaion, I fhall put together a few general obfervations, which a diallift ought always to remember. 1 . In drawing any dial, work with obfcure lines, or fuch as may be rubbed out again -, for we have no farther occafion for them after the hour lines, and. fubftile, are obtained. 2. It is beft to fix the ftile upon the fubftile, before Sea. II. DIALLING. 139 before the dial is drawn-, for for it will be diffi-Fig. cult to fix it truly after. And it muft always ftand perpendicular to the plane of the dial. 3 In all dials, efpecially fuch as are troublefome to draw ; to have it exaaiy done, firft make a draught of it upon paper -, and. then transfer all the lines from thence to the dial plane. 4. In every dial, the thicknefs of the ftile is. to be confidered ; and a fpace left at the fubftile, e- qual to its thicknefs ; juft as if the dial was to be cut thro' at the fubftile, and the two parts fepa- rated, to a diftance equal to the thicknefs of the ftile. And therefore it fhould be made fo at firft. 5. If the fun is to fhine more than 12 hours on the dial ; you may get the remaining hours, which you want, by producing the hour lines already drawn, thro' the center. 6. When the center of the circle is found, that is to divide the contingent line, which may be ta ken on either fide of it ; you may defcribe a cir cle with any radius, and it is beft done by a line of chords; and the hour arches fet off" thereby. For in all circles, that are alike divided, the radii will cut the contingent in the fame points. 7. If you would have half hours or quarters upon your dial, you muft divide each hour of your circle in halves or quarters, and draw lines thro' the points to cut the contingent line, as in whole hoursi And in the calculation, inftead of 15 degrees ; add continually 7° 3d for half hours, or 3" 45' for quarters, and calculate accordingly. 8. In any dial, leave out the midnight hours ; and all hours, at which time the fun can never fhine on the dial. 9. In any dial, place the numbers for the hours, fo as to be moft legible -, therefore it would be ab- furd to place them upfide down, in regard to the fpeaator, 10. Every 140 DIALLING. Fig. 10. Every dial .muft be fo placed, that the up* per edge of the ftile may point direaiy to the pole ; and that the horizontal line be perfectly level; and that it have its proper declination and inclination. And in an upright dial, that the 12 o'clock line be perpendicular to the horizon. 11. When any dial plane panes thro* the pole, it requires an equinoaial horizontal dial to be made. And the ftile will be parallel to the plane. 12. Every dial is too faft in the morning, and too flow in the evening; owing to the refraaionof the fun-beams, by which the fun is raifed higher, and the fhadow brought nearer the fubftile. I2. 13. To draw any hour line C5 (fig. 12.) when it does not cut the contingent within ihe plane. From the centers C, N, draw Cr, Nf, parallel to the contingent AB, Try how much the projeaing line N« advances towards the contingent A% near the edge of the plane ath, as f, f or ^, &c. of NQ1. Take fuch a part of NQ^ in your eompafles, and running one foot along Nf (or QB), mind where the other cuts Nn, as at c (the feet of the eompafles being parallel to NQ). Thro' e draw fr parallel to the fubftile NC. Then take ra, (or ha) the fame part of rh (as fc (or he) is of fh, as f, f, &c). Then thro' a, draw the hour line C«5. See alfo fig. 2 8. Or thus. To draw C7 ; draw gp parallel to C2, which is 6 hours from C8, to cut the hour lines C8, C9, in d and b. Set db from dtof; thro' / draw the hour lines C7. SECT. [ 141 ] SECT. III. The ConflruBion of fome other forts oj Dials ; and drawing the Parallels if Declination^ Parallels of Altitude, and other fuch Furniture upon Dials. A 'fable of Latitude and Longitude of Places. PROB. I. To make a Dial upon the furface of a fphere. THIS is done without any ftile, only by the 44. line bounding light and darknefs. There fore get a fphere or globe, AEBQ, and mark two points of it, diametrically oppofite, for the two poles, of which P is one. In the middle between thefe poles defcribe the equinoaial EQ, which may be done thus ; open the compaffes to a quadrant's diftance, and fetting one foot in P, with the other defcribe the equinoaial EQ, which divide into 24 equal parts, for hours. Then mark thefe divifi ons thus ; put 6 to the top, and 7, 8, 9, 10, 11, 12, fucceffively to the other divifions on the left hand, and 5, 4, on the right ; thefe are to be placed above the equinoaial for the forenoon hours. Again to the fame point 6, put 6 under the equi noaial, and 5, 4, 3, 2, 1, 12 on the right hand, and 7, 8, on the left; thefe are for the afternoon hours. Then the globe is to be fixed fo in the fun, that the pole P may be elevated above the horizon, as much as is the latitude, or fo that its axis may point 142 DIALLING. Fig. point direaiy to the pole ; and the point 6 muft be 44. on the top, fo that the circle P6B may be in the meridian. Then as the fun eonftantly illuminates half the globe ; the circle terminating the enlight ened part, will always fhew the hour of the dayy ; where it cuts the equinoaial EQ. And 1 2 o'clock is, fhewn at two places, E and Q; at E at the end of the forenoon hours'; and at Q, at the begin ning of the afternoon hours. % P R O B. II. To make a common ring dial for any latitude. 'a 5. This dial muft be made of a ftreight plate of brafs, before it is turned into a ring ; for it would be troublefome to engrave the numbers on the in- fide after it is turned. Let ABGD be the ring, per feaiy circular, whofe breadth-is li. This ring is to be fufpended at A, and is to fhew the hour of the day by the fun's fhining thro' a hole on the fide AE in the fummermonths, and falling among the figures on the oppofite fide 8, -7, 6, 5, &c. placed on the infide of the rim. And in the winter months the fun is to fhine thro' a hole in the fide AD, among the figures on the infide of AE. Now fince the fun's height is different at different times of the year, the hole it fhines thro', muft be moved highr er and lower according to the fun's declination, or which is the fame thing, according to the month and day. Therefore we muft (hew how the months and days are to be placed on the outfide of the rim, to fet the hole by ; and likewife how the hours are to be placed oppofite thereto. Let the line z\a be equal to the circumference of the rim AEIGK, and the correfpondent parts equal,, AF zz zf, FE =fe, EH zz el, HI zz M, JG zz 1g, GK — gk, and AK zz ak, &c. and lat. ff, gg, osr D Sea. III. DIALLING. 143 or kk, be, the breadth of the rim. Then firft for Fig. the fummer months. Take any place G near the 45. bottom for the 1 2 o'clock line, let gg be the line. Draw GH parallel to the horizon. And make the angle HGE zz complement of the latitude, or height of the equinoaial ; and EGF zz fun's great eft declination, 234°. Or which is the fame thing, make HCE (or HE) zz twice the comp. latitude, and ECF (or EF) zz twice the greateft declination. That is, if za be divided into 360 parts or degrees, then he zz twice the comp. latitude, and ef zz 470. Let FK be parallel to the horizon. Make the an gles KF8, KF7, KF6, &c. to KFi, equal to the fun's altitude at 8, 7, 6, &c. of the clock, when he is in the tropic of cancer. Or make K8, Ky, K6, &c. equal to twice thofe altitudes ; which let be equal to £4, £5, k6, &c. in the line yx. Again draw EL.parallel to the horizon, and make the angles LE5, LE4, LE3, &c. equal to the fun's altitude at 5, 4, 3, &c. o'clock, when he is at the equator. Or making k6 or kl (in the line az) zz KL ; and then 65, 64, 6^, &c. twice the fun's altitude at 5, 4, 3, &c. And if his depreffion at 7, and 8, be placed at 7 and 8. Then drawing lines thro' the correfpondent hours (for forenoon and afternoon), you'll have for the hour lines 48, 57, 66, ys, 84, 03, 10 2, 11 1, and 12 12. And thefe altitudes of the fun muft firft be computed by fpherical trigonometry. Then for the days of the month, which are to be put upon the outfide of the rim at EF, whofe place on the linejjw is ej; and let ersjbe that fpace. Then take the double of the fun's declination for the beginning of every month, and fet from e to ward x, thro' which points draw lines perpendicu lar to ej, and divide the fpace in each month into 3 parts, each reprefenting 10 days. And put let ters for the names of the months as in the figure ; M for 144 DIALLING. Fig. M for March, A for April, M for May, I for June, 45. I for July, A for Auguft, S for September. In the middle of this fpace is a channel for the Aider, with the hole, to move in. The next thing is for the winter months. Take the point M for the 1 2 o'clock line further from the bottom, becaufe the fun goes low in winter ; draw MN parallel to the horizon, make NO zz twice the comp. latitude, and draw OP parallel to the horizon ; and make the angles PO5, PO4, PO3, &c. equal to the fun's altitude at 5, 4, 3+ &o. in the afternoon, when the fun is in the equi noaial. Or if p or 6 be the place of P, make p7, pS, pg, &c. twice the height at 7, 8, 9, &c. o'clock, in the morning, in the line IZ. In like manner making OQ^zz 47% and drawing QR. pa rallel to the horizon, make the angles RQ3, RQ2, RQi, the fun's altitude when in the tropic of Ca pricorn, and his depreffion at 4, 5 and 6 ; let their places be at 6, 5, 4, &c. in the line ix. Then draw the hour lines 12 12, 1 11, 2 10, 3^ 48, 57, and 66. Laftly, the months muft be fet off on the part OQ, as was done upon EF. Let oq be that fpace in the line iy. Then twice the fun's declina tion, at the beginning of every month, muft be fet off from 0 towards q; and-alf completed as in the fummer months. Thefe two parts jr and qo muft be engraven on the outfide of the rim, at FE andQO. All the lines being drawn, and numbered, the plate azxy muft be bent into the form of a ring, truly circular, and the lines ay, zx, foldered toge ther at A, where a loop of wire muft be fixt, at which it is to hang by a thread. There is to be made a channel quite round the middle of the plate ; in this a thin flip of brafs is to be fitted, to move back and forward, having two holes in it, againft Sea. III. DIALLING. i45 againft the parts EF, and OQ. And in thefe places Fig. - EF and OQ, the channel is cut quite thro' the aT plate. *a' The ufe of the dial is this ; fet the proper hole againft the day of the month whether fummer or winter, which is eafily done by moving the flip of brafs back and forward. And turning the fide AE towards the fun in fummer, or AD in winter ; let the dial hang by the thread at A, then the fun's ray paffing thro' the hole, will projea a fpot of light among the hour lines, which fhows the hour of the day. But note, the hour lines not being parallel ; you muft make the fpot of light fall at or near the fide xy, when the fun is at or near either tropic ; and near the fide AZ, when near the equinoaial; whe ther it be beforenoon or afternoon. And in ge neral, keep further from az, as the fun is further from the equinoaial. PROB. III. To make a univerjal ring dial. Such a dial as this will fhew the hour in any .go latitude, and is thus made. AECF and ABCD are two brafs circles, the outermoft AC reprefents the meridian, and the inner one BD the equator. ABCD is divided into 24 equal parts or hours be ginning at the edge of the meridian AE ; this cirr cle turns upon an axis AC, whofe ends are fixt in the meridian at A and C ; and may be fet fquare or perpendicular to AECF, putting it againft the flops at A and C. The points E and F, which are at a quadrant's diftance from A and C, reprefent the poles -, here a flat piece of brafs FG is placed, which turns round upon its axis, going thro' two pieces of brafs fixt at E and F. ' Along- the middle of L FG, ,46 D I ALL IN G. Fio-. FG, there b a bng flit, in whiph a piece of brafs P, with a fmall hole in it, is moveable up or down, to be fet to the day of the month. The quadrant AE is divided into 90 degrees, fo that the nut N may be fet to the latitude of the place, by. Aiding it along. The circle AC is hollowed out between A and E, and a piece of brafs wire fits into it, arid is faftened beyond A and E ; this wire goes thro* a hole in the bottom of the nut N. The piece FG is graduated with the months after this manner ; the tangent of the fun's declination (to the radius of the inner circle), is fet from the middle towards F, for north declination ; and towards G for fouth. And each month divided into three parts of 10 days each. The ufe is this ; open the inftrument till the horary circle BD reft againft the flops at C and A. Move the nut N till the black line in it falls upon the latitude of the place. Move the Aider P till the hole be againft the day of the month. Then fufpend the dial by the thread at N, and turn the piece FG round its axis till it face the fun ; and move the dial, till the fun fhining thro' the hole, cafts the fpot of light upon the. black line,, which runs along the middle of the infide of the hour circle BD ; and the numbers on the upper fide, fhews the hour. At this time the circle AECF is in the plane of the meridian ; and the. circle BD parallel' to the equinoaial. PROB. IV. To make a dial upon a quadrant. „ Let CAB be the quadrant divided as ufual into 90 degrees, CP a line and plummet. Defcribe the arch HI at a convenient diftance from AB, for infcribing the months, which is done thus. Find the £$La/lin ti-.xv: OCLJ4? Sea. ill. DIALLING. 147 the fun's meridian altitude when he is in the two Fig* tropics ; lay the line CP over thefe altitudes, by 47* which draw the lines mn, op. Then all the months are contained in the fpace mopn. About the middle of the fpace CH or CI de fcribe the arch FG. Then lay the line CP to the height of the equinoaial among the degrees, and Where it cuts FG^ make a mark, as 12. Draw the right lines 12 m, and 12 0, for the 12 o'clock lines in fummer and winter, refpeaively. For infcribing the months ; find the fun's decli nation, and from thence his meridian altitude, at the beginning of every month ; and laying the line CP fucceffively over thefe degrees, draw fines by it, within the fpace mopn, as 'you. fee in the figure ; and thefe lines divide the months from one another^ which may be divided into days as you will, and the names of the months written within' them, or at leaft the initial letters. To draw the reft of the hour lines ; and firft" for the fummer months. To do this, we muft firft find all the hour points upon the line HL There fore when the fun is in the tropic of cancer, by fpherical trigonometry, find the fun's altitude for every hour ; which mark upon the arch HI at 1, 2, 3, 4, &c. by laying the line upon the feveral degrees of the quadrant. Again, find the fun's altitude for every hour to 6, when he is in the equi- nocriah Make marks for thefe in the arch FG» at 11, 10, 9, 8, &c. Then draw the hour lines I 11^ 2 10, 39, 48, &c. which will be right lines very nearly. Then for the winter months, you muft in like manner find the fun's altitude for every hour, when he is in the tropic of Capricorn, and make marks along the arch oL From which draw lines to the marks in FG, which will be 1 n, 1 10, 3 9, &c. and thefe will be the hour lines, for the winter.. " L 2 • But i4« D 1 A L L I N G; Fig> But as fome of the fummer hour lines do not 47. cut the arch GF; and fome of the winter hour lines, the arch IH ; we muft find the points where they cut the line GI, at fun-rife or furt-fet. There fore for any fuch hour you have the afcenfional dif ference, 1 50 for 1 hour from 6, 300 for 2 hours, &c. Then from the afcenfional difference, find the . . declination ; and by that the meridian altitude.- Then laying the thread over thefe degrees of alti tude, obferve where it cuts the line 12 12, take. the diftance from that point to the arch HI, and fet it from I towards G, and you'll have the point in IG, through which that hour line is to be drawn. If you would have the hour lines drawn very ex aa, as they deviate a little from right lines ; , pro ceed thus. About the middle of the fpace GI, defcribe the arch ED. Lay the thread over the point D or d, according as it is fummer or winter,; where that arch interfeas one of the 12 o'clock lines ; and note the degrees cut, which take for the fun's meridian altitude. From thence find his de clination. Then by oblique fpherical triangles; (Cafe 8), find his altitude for any hour, except 12. (which by conftruaion is a right line) ; lay the thread over thefe degrees of altitude, and where it cuts the arch ED, is a third point, to draw the hour. line thro'. If you have a table of the fun's altitude for every hour, it will fave you the labour of calculation. To ufe the quadrant in finding the hour of the day. There is a fmall bead which flips up and • down the thread CP. Lay the thread over the day of the month, and flip the bead up or down, till it falls on the 1 2 o'clock line, Then holding the quadrant upright, that the line and plummet may hang at liberty, and play freely, then holding it fo in the fun, that the fhadow of the perpendicular, .,- ' * - pin Sea. III. DIALLING. "149 pin at C, may fall on the line CF ; then the bead Fig. refting among the hour lines, will fhew the hour 47. of the day. And note the hour lines in fummer run, upwards towards the left hand ; and in winter, up wards towards the right. PROB. V. To draw the parallels of declination upon any dial. Any circles of the fphere, whether great or fmall, are eafily deferibed by the rules of the gno monic projeaion of the fphere before delivered ; which, if the reader underftands, he needs no other direaions. In thefe forts of Problems, I fhall therefore be as fhort as poffible, and deliver what I have to fay in a general way, which may eafily be applied to particular cafes. Here all things be longing to the dial muft be fuppofed known, as the height of the ftile, the place of the fubftile, &c. Then the parallels of the fun's declination will be found, by finding the dividing center, of every hour circle ; by which the declination is to be fet off from the equino&ial, upon thefe feveral hour circles, or the complement of the declination, from the pole, which is the center of the dial. Let PC be the fubftile, CF the ftile, and C the 4g. foot of it. Draw FO perpendicular to PF, and O is a point of the equinoaial. Thro' O draw the equinoaial QQ perpendicular to PO. Make the : angles OFB and OFD equal to the declination of the parallel, north and fouth, as fuppofe 2'3-f. Then B and D will be the points in the fubftile, thro' which the north and fouth tropics pafs ; or the vertices of the two hyperbola's reprefenting the tropics. Then to find the points upon the other hour lines, as fuppofe on Pn. From C the foot of the ftile, draw CGL perpendicular to, the hour L 3 line, j5o DIALLING. Fig. line, and CH parallel to it, and equal to CF. 48. Then fet GH from G to L, on either fide the hour line Pit, and Lis the dividing center of Pu, Draw LE to the point where the hour line interfeas the equinoaial ; then make the angles ELM and ELN equal to the declination of the parallel (23f); atid M and N will be two points where the fame two parallels pafs thro ; and by the like procefs, the points in every hour line are found. ' Then two curves drawn thro' all thefe points, as RDS and VBX, will be the parallels required* one fouth, the other north. And thus all the parallels may be drawn, or thofe at the beginning of every fign, having their declinations given ; and being marked with the figns of Aries, Taurus, &c. the fhadow of the top of the ftile F, will fhew when the fun enters any qf thefe figns. In a horizontal, or a direa north qr foiith dial, having found the points on one fide the meridian ; they may be fet off on the hour lines on the other < fide. And even in any declining reclining dial, where the fubftile falls upon fome hour line, the points on the hour lines, on One fide, may be fet off upop the hour lines equidiftanti oh the other , fide. The procefs is juft the fame when the dial is a horizontal equinoaial dial, where the hour lines are parallel ; and the fame, in eaft and weft dials. in horizontals under' the poles, thefe parallels will be circles, whofe center is the foot of the ftile, and radius of any one equal to the co-tangent of the declination ; the height of the ftile being radius. If you make ufe. of a triangular ftile POT, you muft cut a notch out of the edge at F, for the fun to fhine thro' ; which will do the fame as the fha dow of the point F, And the point F, for the notch* Sea. III. DIALLING. 151 notch, will be found by ereaing CF perpendicular Fig. to the fubftile PO. 48. Cor. 1. Hence you may mark upon the dial, the time of fun rifing or fun fetting, or the length of, the day, when the fun enters into any of thefignS. For if the horizontal line be drawn on your dial, the interfeaion thereof with any parallel of the figns, will fhew among the hours, the time of fun rifing or fetting, which may be marked upon thefe figns in the meridian, or on the fide of the dial, or elfe the length of the day or night ; that is on one fide of the equinoaial. And if a parallel be, drawn to the horizontal line on the other fide of O, and at the fame diftance from O, its interfeaion with the parallels of the figns will fhew the hours on the other fide of the equinoctial. But if the horizontal line does not cut fome parallel circle, then the time of fun rife muft be calculated by fpherical trigonometry, having the declination of that parallel given. Cor. 2. Hence the dividing center L of any hour line PE, is diftant jrom the center of the. dial P, the length of the axis or ftile PF. And this gives ati eafier method of finding the dividing center of an hour line. For all ihefe centers will be in the circum ference of a circle deferibed from P with the radius PF. For PE is 90 degrees zz angle PLE ; and PL* zz PG2 4- GL> zz PG4 + GH1 zz PG1 + GO + CH1 = PC* + CFP zz PC2 + CF* = PF\ L4 PROB. 152 DIALLING. Fig. 48. PRO B. VI. To defcribe fuch lines upon a dial, as will fhew the rifing and fetting oj the Jun ; or the length of the days. To do this, you muft find what declination the , fun has when he rifes at 4, 5, 7, 8, &c. of the clock ; which is eafily done by right angled fphe rical triangles, having the afcenfional difference given ; or even by projeaion. Then having the ' declination of each hour, you muft defcribe fo many parallels for thefe feveral declinations, by the laft Prob. fuch as RDS, VBX, &c. which done, write upon each of them the hour of fun rife, correfponding to that declination. Or you may write the hour of fun rife on one fide, and that of fun fet on the other fide of the dial ; and the length Of the day, where each parallel cuts the meridian ; or after any other manner, which you think proper. Where thofe above the equinoctial ferve for the winter ; and thofe below for the fum mer. Then at any time, when the fhadow of the top of the gnomon F, traces out any of thefe pa rallels, you'll have by infpeaion the time of fun rife, or the length of the day. Cor. By a like method you may infer t in a dial any holy days, or remarkable days in the year. For you have no more to do but defcribe the fun's parallels for thefe days, and write upon them refpeaively, the names of the days ; and on thefe days the extremity of the fhadow will fall upon the refpeaive parallels, PROB. ..'J .-n/Atn? TLrXMl.pa./jS Sea. III. DIALLING. 153 Fig. PROB. VII. To defcribe the azimuths and circles of altitude upon any dial. The azimuths being great circles, are projeaed 40. into right lines ; and the parallels of altitude are projected into conic feaion, by the gnomonic pro- jeaicn of the fphere. In order to defcribe thefe, we muft have given us the place and height of the ftile, the horizontal line, the zenith, the dividing center of the horizon : but we have all thefe from the conftruaion of the dial. And we muft have befides the dividing center of every azimuth, by help of which we muft fet off the diftance of each parallel upon it. Let DE be a vertical line, AGB the horizontal line, CF the ftile, H the nadir, or rather the ze nith, becaufe the fhadow is always oppofite to the fun ; R the dividing center of the horizontal line ; HI the meridian or 12 o'clock line. From the center R, divide the horizontal line in parts of 10 or 20 degrees, beginning at the meridian, that is, at RI ; and through the points of divifion draw the azimuths Hk, HI, Hm, Hn, &c. from the zenith H. Then for drawing the parallels of altitude, we have F the dividing center- of the azimuth DH. Therefore at F make the angle GFa equal to the diftance of the parallel above the horizon, fuppofe 10 degrees, or HFa its zenith diftance ; and a will be the point thro' which the parallel is to pafs ; and thefe parallels muft be drawn below the horit zon AB, as the fhadow is oppofite to the fun. Then to find the point it muft pafs thro', in any other azimuth Hq ; firft find the dividing center, thus ; draw Chr perpendicular to Hq. Set CF _ . from 154 DIALLING. Fig. from h to t, and extend from / to C, arid fet that 49- extent from h to r, and r is the dividing center of Hq. Therefore at r make the angle qrd, equal to the parallel's diftance from the horizon, or Hrd its zenith diftance, and d is the point for the parallel to pafs thro'. Alfo O was the dividing center of the meridian HI. Therefore make the angle 10b feqUal the parallel's diftance (io°), and b is its point - in the meridian. And the points being thus found in all the azimuths, draw the curve xdaby, which will be the parallel of altitude of 1 o degrees; And Whenever the fhadow of the point F of the ftile* touches that parallel xay, the fun is 10 degrees high. And thus all the parallels may be drawnj at eVery 10 degrees diftance; up to the zenith. If the azimuths and parallels of altitude, were to be deferibed on a horizontal dial ; then the azi muths are right lines drawn from the foot of the ftile C, making equal angles with one another; And the parallels of altitude are circles, and the radius of any one is the tangent of the zenith dif tance, the ftile CF being radius, and C the center. For here the zenith falls in C the foot of the ftile. If the dial be ah erea one, then the zenith H is at an infinite diftance ; and therefore all the azi muths are parallel to one another* and perpendi cular to the horizon. And the horizontal line goes thro' C the foot of the ftile -, afld the dividing cen ters of the azimuths are all in the horizontal line; and the parallels of altitude will be hyperbola's. The azimuths are to be numbered from 12 O'clock; the weft azimuths towards the eaft; and fhe eaft towards the \veft. And if the tropics are deferibed in your dial, you need only draw fuch parts of the azimuths as are contained between ihe tropics ; or between the lower tropic and the horizontal line ; for the horizon cuts off all the fu-» perfluous parts.. Cor. Sea. III. DIALLING. 155 Cor. Here we may uje ajhorter way jor finding theFig. dividing center r, oj any azimuth ; like that jor divi- 49. fding the meridians ; which holds Jor all great circles meeting in a point, as H. About Hq defcribe a femicircle, or only a fitiall part thereof befide r ; with, the extent HF, and one foot in H, crofs it at r, for the center. For Hrq being a right angle, it is inferibed in a femicircle, whofe diameter is Hq ; and Hr is equal to HF, which is proved as before, in Cor. 2. Prob. V. PROB. VIII. To draw the Babylonian hours in a dial. The Babylonian hours are reckoned from fun 50." rifing to fun rifing, and are 24 equal hours, nearly of the fame length as the common hours ; only they are differently numbered. The way of defcribing thefe hours is the fame for all forts of dials ; fo that being fhewn for one, . it will ferve for all. Therefore in a foUth dial, let FC be the perpendicular ftile, P the center. Draw FO perpendicular to FP, to cut the meridian Pi 2 in O ; thro' Q draw the equinoaial AB perpendi cular to Pi 2. Then find what precife hour the fun rifes at, when neareft either tropic ; as fuppofe at 4 in fummer, and at 8 in winter; and in the equator he rifes, in all places, at 6: then, find his declination, when he rifes at thefe times ; and by Prob. V. draw two parallels for that declination, mm, and nn, on either fide the equinoaial. Draw alfo the two tropics cc, and dd, which are only of ufe to terminate the hour lines, when drawn. Obferve all the points where the hour lines of the dial cut the fouth parallel mm ; then fince the Babylonian hours proceed from 1 to 24; and }n this parallel he rifes at 8; therefore write 24 156 DIALLING. Fig. 24 at that point of the parallel, where the eight ¦50. o'clock line paffes ; and write 1 at 9, 2 at 10, 3 at 1 1, &c. Alfo note where the hour lines cut the equinoc tial AB ; and fince the fun then rifes at 6, call that 24; then 7 is 1, and at 8 write 2, at 9 write 3, at 10 write 4, &c. Again obferve all the points where the hour lines of the dial cut the northern parallel nn. And fince here the fun rifes at 4, call that 24, and,5 call 1, 6 call 2, 7 call 3, 8 call 4, 9 call 5, or at 9 write 5, at 10 write 6, at 11 write 7, at 12 write 8, &c. for the Babylonian hours. This done, you have nothing to do but draw lines thro' thefe points where the figures are the fame, as 22, 33, 44, 555, 666, 777, 888, 999, jo 10, &c. and thefe are the Babylonifh hours; which are right lines, becaufe they reprefent great circles of the fphere ; as will be evident at fight, by looking on a globe. We may obferve, if the horizontal line was drawn thro' the foot of the ftile C, it would pafs thro' the interfeaion of the parallel mm, with the 8 o'clock line ; becaufe the fun rifes at 8 ; and alfo thro' its interfeaion with the 4 o'clock line, •becaufe it fets at 4. All this being done, then at any time when the fun fhines, the top Fof the ftile, will fhew the Ba bylonian hour of the day, among the Babylonian hours ; as well as the true hour, among the hour lines of the dial. PROB. TL'XSHI.pa- :j Sea. III. DIALLING. 157 Fig. PROB. IX. To dejcribe the Italian hours upon a dial. The Italian hours begin at fun fet, and are num- 51. bered to 24 at fun fet next day, they are equal hours, nearly the fame as common hours. Thefe hours are drawn the fame way from fun fet, as the Babylonian hours were drawn from fun rife. Having deferibed upon your dial, the equinoc tial line AB, the two tropics, cc and dd, and the two parallels mm, nn, for the fun's fetting at 4 and 8 o'clock ; obferve where the fouthern parallel mm cuts the 4 o'clock hour line, and mark it 24. Alfo obferve where the other hour lines cut it, and write 23 at 3, 22 at 2, 21 at 1, &c. Alfo obferve where the hour lines cut the equi noaial, and reckon 6 (in the evening) 24, then 5 is 23, and at 4 write 22, at 3 write 21, at 2 write 20, &c. And obferve where the northern parallel nn is cut by the hour lines, and call 8 (at night) 24, and 7 call 23, 6 call 22, 5 call 21,4 call 20, 3 call 19, or at 3 write 19, at 2 write 18, at 1 write 17, &c. Then lines drawn thro' the points with. • the fame numbers, will be the. hour lines; as 14 14, 15 15 15, 16 16 16, &c. Then when the fun Alines, the end of the ftile F will fhew the Italian or common hours, among their refpeaive hour lines. Cor, Ij both the Babylonian and Italian hours were dejcribed upon one dial ; the hour lines oj both, will inter jetl in the parallel circles, where fhe Jun rijes at 4, 5, 6, 7, 8, o'clock, and their halj hours. Or wherein the day is 8, 9, 10, &c. to \6 hours long. For it is evident they interfea in the parallels mm- and nn ; and for the fame reafon the other in terfeaions *5* DIALLING. Fig.terfeaions would fall in fome other parallels. Sup- 51. pofe a Babylonian- hour d*awn from 16 to 20, it will crofs 3 Italian hour lines, in its way ; and fo [ there will be 3 parallels between 6 and 8 o'clock, or between 4 and 6 ; that is, one for every half hour. And they will anfwer refpedtively to thefe half hours, becaufe the parallels mm, nn, anfwer to the whole hours 4 and 8, Therefore they fhew fun rife or fun fet, every half hour ; or if you will, the length of the day, in whole hours, by doubling the time of fun fet. PROB. X, To draw the Jewifh hours in a dial, The Jewifh hours, otherwife called the old, un equal, planetary hours, are reckoned from fun rife ; and the day from fun rife to fun fet, is fuppofed to be divided into 12 equal parts or hours ; and therefore thefe hours in any one day will be equal to one another, but not equal to the hours of ano ther day. For when the days are long, as in fummer j the hours will be long ; but in winter, when the days are fhort, the hours are alfo fhort. The nights are alfo fuppofed to be divided into 12 hours ; but thefe hours" are longeft in winter," and fhorteft in fummer. But the hours of the day are never equal to. the hours of the night, but a£ the equinoxes ; and then the Jewifh hours are equal to the common hours. 52. Let P be, the center of the dial, CF the perpen dicular ftile ; EQ^ the equinoaial, which is fup pofed to be drawn ; if not, draw FA perpendicu lar to PF, and thro' A, draw the equinoctial EQ perpendicular to the fubftile PA. Set AF from A to D> fo is D the dividing center of the equi noaial. « Find Sea. III. DIALLING. 159 Find the declination of the fun, when he rifes Fio-. at fome even hour after 6 in fummer ; and as much rZ . before 6 in winter ; as fuppofe at 4 in fummer, and 8 in winter. Draw two parallels to the equi noaial, on each fide, for that declination, as mm and nn. Then when the fun is in the north pa rallel nn, the day is 1 6 hours long, which anfwers to 240 degrees, which divided into twelve parts, gives 20 degrees to a Jewifli hour at that time. Therefore with the center D, defcribe the circle SAW, which divide into parts of 20 degrees each, beginning at the line DN, where the 12 o'clock line cuts the equinoaial, for our 1 2 o'clock is the fame a,s the Jewifh 6th hour. Draw lines from D thro' all thefe points, to cut the equinoaial ; and thro' the points in the equinoaial draw lines from P, to cut the parallel nn in the points where the hour lines pafs. Then mark thefe points on the parallel thus, write 6 at the 1 2 o'clock line, and to the other points, towards the weft 5, 4, 3, 2, 1 ; and towards the eaft 7, 8, 9, 10, 11, 12. Again, when the fun is in the parallel -mm, the day is but 8 hours long, which anfwers to 120 de grees, and that divided by 12, gives 10 degrees to a Jewifh hour. Therefore from the fame center D defcribe another circle ; or rather, from a center as far on the other fide of the equinoaial EQ, de fcribe a circle, which divide into parts of 10 de grees each, beginning at the interfeaion of the 1 2 o'clock, and the equinoaial as before ; and draw lines from the center thro' thefe points to cut the equinoaial -, and from P, thro' thefe points in the equinoaial line, to- cut the parallel mm, in points for the hour lines. Or you may perform the fame thing, with the other femicircle SaW oppofite to SAW ; for drawing the line NDr, begin at r, to divide the femicircle into equal parts of 10 degrees ; from which draw lines thro' D, to cut the 'equi noctial -, i6o DIALLING. Fig. hoaial ; to which points in the equinoaial draw 52. lines from P, to cut the parallel mm in points for the hours. Then mark the point in the 12 o'clock ' line with 6, and the otherpoints towards the weft, 5> 4> 3> 2, 1 ; and towards the eaft, 7, 8, 9, io,- 11, 12; or as many of them as are wanted. Alfo upon the equinoaial at 12 write 6,- at 11 write 5, at 10 write 4, &c. and at 1 write 7, at 2 write 8, at 3 write 9, and 10 on. Then drawing. lines thro' all the three points that have the fame figures, thefe will be the hour lines for the Jevrifh hours ; which will be nearly ftreight lines.. But if you would be more exaa, take two more pa rallels between thefe and the equinoaial, and find the points for the hours as before. Note, the two tropics ought to be drawn for . terminating thefe hour lines ; but here is not room in this fmall fcheme to draw all the lines required. Otherwife thus. The Jewifh hours may .alfo be deferibed by help of the common hours, already drawn upon the dial. For fince in the parallel mm, a Jewifh hour is I- a common hour ; and in the parallel nn, 4 a common hour. Therefore dividing every hour into 3 parts, fet off from 1 2, in the parallel mm, firft 4, then 14, then 2, then 24, then 34, then 4 hours both ways ; and mark the points. Then on the parallel nn, fet off alfo from 12 both ways ; 14, 24, 4, 5f, 64, and 8 -hours, and mark the points. Then drawing lines thro' the correfpon- dent points, and thro' the points of the common hour lines, on the equinoaial ; and thefe are the hour lines required, and muft be numbered as be fore direaed. PROB. Sea. III. DIALLING. 161 Fig. PROB. XI. 5 To draw meridian lines in a dial, to fhew when it is noon at any particular places on the earth. The meridian of any place is eafily drawn in a dial, if you know the longitude of . that place, reckoning from yours, either eaft or weft. For allowing 15 degrees to an hour; by reducing the longitude to time ; you will find what hour artd minute it correfponds with. Therefore reckon fo many hours and minutes frqm 12, upon your dial ; towards the weft, if the place lies eaft -, or towards the eaft, if weft ; and thro' that point draw a line frqm the center. Then when the fhadow falls up on that line, the fun is in the meridian of the faid place. Or having the difference of longitude, or the hour arch, you'll find the hour angle, by the common proportion in dialling. Suppofe Quebec, whofe longitude from London 53; is 820 weft ; which is now 5f hours. Therefore thro' 5f hours, draw a line from P, and you have the meridian of Quebec. And the like for other places, as you fee in the figure. Many other things of like fort might be in- fcribed upon dials, as the circles of the 12 ccelef- tial houfes, and curves fhewing what iign afcends or defeends -, curves fhewing the latitude of parti cular places, or when the fun is vertical at fuch places, &c. But as thefe things are of little confe- quence here, 1 fhall fpend no more paper about them. Scholium. Some people may defire to know, how a dial plane is to be painted -, for if it be made of wood, it will not endure the weather without painting. The ground or firft painting, muft be with Spanifh M brown, i62 DIALLING. Fig. brown, and the laft with white lead; and are to be ground with linfeed oil boiled. To grind any colour, put only a little oil to it at firft, after it ' have been ground fometime without '; then grind the colour and oil together, adding oil by degrees, to make it like an ointment, and thinner ftill as you grind the longer. It will be apt to work off the ftone, if you do not fometimes fcrape it toge ther with a wooden knife or lat. Your plane muft be coloured feveral times over, to abide the wea ther, letting it dry between the times. 1 8. For glueing the joints together, boil your glue in blue or old milk ; boil the milk firft,- and take off all the fcum, before you put in your glue. This holds better than glue boiled in water. As fome people may want to know the latitude and longitude of fome remarkable places, I fhall therefore annex the following table. A Table Sea. III. DIALLING. 163 A Table of the Latitude and Longitude of Places ; the Longitude reckoned from London. Thofe places that are in South Latitude, are particularly marked with S ; and thofe in Weft Longitude, with W. Lat. Long. I Aberdeen Agra Aleppo Alexandria Algiers AmfterdamArda (Ardra) B. Bagdat Belfaft ^Belgrade Bergen , (Berlin " I Berwick Bofton(America) ' Breflaw Briftol C. Cadiz (Cales) Cairo (grand) Carabaia Cambridge Canterbury- Cardigan CarlifleChambalaCharles-town Chaxumo Chefter Chichefter Coca Conftantiople Copenhagen v Cork Coventry Cracow D. DahtzickDarlington 53 38 5+ 3» 18 35 1 19W Davids (Saint,) DrefdenDublin Dun cola DundalkDundee Dunkirk Durham E. Edenburgh Exeter F. Ferro (ifle) Fez Florence G. Geneva Genoa Glafcow Glocefter GoaGombroon , H. HagueHamburg Hartford Hanover HerefordHull I. Jerufale m. Ifpahan K. Kildare Kinfale I L" ILancafter Landaf |Lima 52 1 S3 1753 Lat. 1 Long. o IZ12 20 _ 5z 56 26 5I 2 54 54 55 56 S°4327 48 34 10 43 34 46 3i 44 25 55 56 5Z >5 36 27 49 5,2 12 53 4« 51 50 52 29 ;z 10 153 5° 31 46 32 34 5 4W 13 40 6 55W 33 4° 7 z 248W 2 27 1 40W oW 30W 17 40 W 5 20W 12 24 6 12 8 41 4 12W 2 10W 73 53 55 4^ 4 20 10 38 o 10W 10 o 2 36W o 23W 32 31 50 15 53 7 2; W 32 8 26W 54 10 5l 35 12 2S 2 40 W 75 52WJ Li m eric 164 D I A L L I N G. a , Lat. Long. R. Reading Lat. Long. Limeric 0 / 52 55 6 48W 0 / 51 28 0 , 0 53W Lincoln 53 H 0 34W Rhodes 36 24 20 p Lifbon 38 45 9 7W Rbchefter 51 30 0 32 Londoa M. 51 32 P 0 Rome s.. 41 47 15 46 Madrid 40 25 3 39w Marseilles 43 18 5 27 Salilbury 5i 4 I 52W Mexico 20 15 103 12W Sterling 56 30 3 45^ Iffongul 66 10 1 26 40 Stockholm 59 22 '9 3° Monorwotapa 25 oS 26 0 Sufa 45 5 7 ip Morocco 3i 56 9 12W T. Mofcow 55 47 38 '4 Tenerif 28 17 15 29W N. Tombute 14 20 11 12W Nanchang 1 (Nankin) J 32 10 11838 Troy ' Tunis 39 36 36 26 26 38 io 16 Naples 41 51 14 45 Turin 44 56 7 16 Neweaftle 5'5 0 1 4zW V. Nice 43 54 7 20 Venice 45 46 '3 12 Norwich 52 42 1 7 , Verfailles 48-42 2 ,20 Nottingham. 53 2 1 10W Vienna 48 29 16 24 Nuremburg O. Oxford 49 40 11 12 Villa rica * 20, 10 100 16W 51 45 1 10W W. ' P. Warfaw 52 21 21 IO Paris 48 50 2 25 Warwick 52 23 "l 33W Pegu 17 29 97 iz Waterford 52 7 7 SzW Pekin 40 15 in 10 Wexford 52 iS 6 56W Pembroke - 51 45 5 oW Winchefter 51 7 i 16W Peterborough - 52 33 0 20 W Worcefter 52 18 2 10W Petersburg 60 OC 36 6 ,*^ Portroyal 17 32 77 5W Y. Z. Frague 50 c 14 28 York 53 5« 0 ;6W « , 9=- Zuenziga 25 C ' 1 30W Que beck 47 35 74 10W %' F I N I S. ERRATUM, Dialling. Pa^e 9, line 15, read, tag, his rays. 52 53 11 1Z 1 « 3 South- , // Y ,J<^0^ du^f^ v/ • / \%\ \ 3 g l iz m. io Diallifty ILIM* md<.