Geography and Navigation Compleated; BEING A New Theory and Method WHEREBY THE True LONGITUDE Of any Place in the WORLD May be Found: Whether Differing in Longitude only, or both in Longitude and Latitude, from any Other Place in the Habitable World. ― By GEORGE KEITH, M.A. Rector of Edburton in Sussex. ― Most Humbly Proposed to the Consideration of the First Present Parliament of GREAT BRITAIN after the Happy UNION. ― LONDON: Printed for B. Aylmer, at the Three Pigeons in Cornhill. MDCCIX.
The PREFACE TO THE Impartial Readers.

&Tic;Here are no Judicious and Skilful Persons in Geography and Navigation, who will find fault with the Title of this Small Treatise, if once they be satisfied, that the Discovery I here present to their View, is true. For that a True Discovery of the Longitude, would compleat both Geography and Navigation, is so far from being any Hyperbole in Speech, that it is confessed by all Learned Men in those Sciences, that a True Discovery of the Longi&rehy;tude would compleat them.

If any skilful in Mathematical learning have any thing to offer against this New Theory and Method I have set forth to Publick View, how to find the Longitude, I shall take it kindly, if they please to write to me their Objections, they paying the Postage of the Letters they send to me, directing them to Mr. Brab. Aylmer at the Three Pigeons in Cornhill. But I pray them not to trouble them&rehy;selves nor me with unlearned Objections, that have no seeming Weight of Reason in them; that any ordinary Schoolboy or Novice, who has but begun in the Study of those Sciences, would not be at the Pains nor spend Time to reply unto. But I withal desire them to put their Names to what they write to me, (otherwise I am not like to take any notice of them;) that I may not be troubled with Heaps of Letters that signify nothing to the Purpose. The only thing that they should object, is, what they think is not Mathematical, or what is not built on Fundamental Principles of Geometry and Astronomy: And let them not bring such Objections against this Discovery, that is of E&rehy;qual Argument against the finding the Latitude, as the Longitude. Also I desire them to take notice, that tho' these Twelve Principles set down by me in the first Four or Five Pages of my Book, are suf&rehy;ficiently clear unto me; and I doubt not but they will be to many, as well as unto me; yet I have not built my Theory upon them all, but only some of them; divers of them being only for Illustration, and not for Argumentation; so as that if any of them might seem doubt&rehy;ful to some Readers, that will be no sufficient Reason to overthrow my Theory. And let them not say the Thing is not demonstrated, or they are not satisfied with my Proofs, and yet give no Reason for their so saying; but they ought to shew wherein the Defect lies, and by Mathematical Evidence to convince me that any of my Grounds and Fundamental Reasons are false. Neither let them take occasion against me for some Circumstantial Matters, if possibly there may be any that hurt not the Foundation: Nor for Faults of the Press; one of which I find in the beginning of Page 3. which correct thus: The Declination of the Fixed Stars (as about some Seconds per Ann.) is no more an Argument against finding the Longitude, than finding the Latitude.

And if any that shall happen to read this little Treatise, who are both Learned and Impartial; as I doubt not but it will find many such both at Home and Abroad; when they have duly examin'd it, and that in due Time all the Learned World will be satisfied with it, and the Demonstration that I bring for it, is approved by them; if they please to signify so much to me, they will not only do an Act of Justice towards me, but to Mankind in general.

George Keith.
 ― ― Geography and Navigation Compleated. BEING A New Theory and Method WHEREBY THE True LONGITUDE Of any Place in the World, may be Found.

&Sic;OME Indubitable Principles in Geo&rehy;metry and Astronomy presupposed, for the better Understanding the De&rehy;monstration of this New Method of Finding the Longitude of Places, from any First Meridian, and the Diffe&rehy;rence of Longitude between any Two Places.

Any Two Triangles having Two Sides cor&rehy;responding and equal to each other, and the An&rehy;gle either Equal or Common; where these Two Sides meet, the other Two Angles respectively are Equal, and the Basis is Equal to the Basis. elem. Euclid, Lib. I. Prop. 4. All the Fixed Stars keep the same Distance one from another, whereever they are, and whatsoever Azimuth they appear on; the Distance of any Two Fixed Stars is always the same, on all Azimuths and Meridians. The Case is the same with respect to the Lon&rehy;gitude, whether we follow the Hypothesis of Ptolo&rehy;my or Copernicus: But if the Starry Firmament be supposed to move, the Earth must be supposed to rest; or if the Earth be supposed to move, the Starry Fir&rehy;mament must be supposed to rest. The Arch AB of the Greater Circle repre&rehy;senting a Circle in the Heavens, is equal to the Arch AB in the Lesser Circle, in any Azimuth and in any Meridian Circle; whether there be Two Stars, the One at A and the other at B, or One Star only, and the other a Fixed Invisible Point. (Vide Scheme.) The Points A and B in the Lesser Circle of the Earth being supposed to be Immoveable, have Im&rehy;moveable Zeniths. The Declination of the Fixed Stars (as about some Seconds per Ann.) is no more an Argument against finding the Longitude, than finding the Latitude. Seeing in this New Method of Finding the Longitude, the Operation is only by the Fixed Stars, and neither by the Sun nor by any other of the Pla&rehy;nets, no regard is to be had precisely to the Diversity of Time, when a Star comes to your Meridian; as if the Knowledge of the precise Time of the Night were necessary: For all such Diversity of Time, why a Star comes not every Night to your Meridian at the same Time, proceeds from the Sun's continual Declination, and his going from the Stars, and not the going of the Stars from him. The Longitude of Places, and Diversity of Time, are different Things, and not the same: For the Longitude of Places are Real Things that actually exist together, and are Permanent Things; whereas Time is not such, but is in a Continual Flux. Yet the Longitude first being found, gives the Difference of Time. All Arches of Circles, Great or Small, drawn from one Center, so that they are Concentrick and Intercepted by the same Radii, are equal in Degrees and Minutes; as in the Scheme. Where the Arch AB in the Celestial Orb is equal to the like Arch in the Lesser, that represents the Terrestrial Orb. The which Principle is the Foundation of the Quadrant, Cross-Staff, and Astrolabe, and other Mathematical Instruments for measuring Arches and Angles. And seeing this Manner of Operation has no regard to any precise Time of Day or Night, it is preposterous to presuppose the Knowledge of the Exact Time when any Star is either in the Zenith or on the Meridian of your Place: And any such Ob&rehy;jection that argueth for the Knowledge of the precise and exact Time when you are to Observe your Star, is to overthrow all finding the Latitude by common Practice of Seamen, who are never curious to know the precise and exact Time when they should Observe the Sun or a Star's coming to the Meridian, but they begin their Observation of the Sun's still Rising high&rehy;er and higher, until he Rise to the Highest for that Time; and generally they begin early to make their Observations, as commonly about Eleven or a little after; and they wait to see how the Star ariseth by their Instrument, until he come to his Greatest Height; and thence they conclude he is come to his Meridian for that Day. An Error of a few Minutes, in find&rehy;ing the Longitude, is no other or greater Er&rehy;ror, than what is frequently and unavoida&rehy;bly committed by all Seamen, in finding their Latitude: And Seamen would think themselves well helped at Sea, if any should Teach them to find their Longitude within Nine or Ten Minutes, or more. The Praxis or Operation I undertake to Perform at present, is, To Find the True Longitude of any Place in the World, whether at Sea or Land, where you are, or whither you happen to come; and not where you are not, and whither you are never to come; even as you can only find the Latitude of the Place where you are, and not where you are not, and whither you are never to come. But by Seamens observing the like Method as I shall use in this, who may happen to be in those Remote Places, I will show them by their keeping a Correspondence by Letters betwixt themselves and others in Remote Pla&rehy;ces, who may be taught to find their Longitude in those Remote Places where they are; and consequent&rehy;ly, by the Rules here given, the True Longitude of all Places in the World where Navigation is used, may be found by Astronomical Observations, and Tri&rehy;gonometrical Calculations. And thus Tables of Lon&rehy;gitude may be composed for the General Benefit of Seamen to all Navigable Places, and of all Places passable by Land, where they have never been, and whither they are never to come.
The Explication of the SCHEME.

&Tic;HE greatest Circle, ABCGHD, represents sometimes the Equator, and other times some other great Circle in the Starry Firmament, accord&rehy;ing to the several Cases to be resolved, about the Lon&rehy;gitude and Latitude of Places.

The Lesser Circle represents the Concentrick Cir&rehy;cle on the Globe of the Earth.

P represents the North Pole.

AG represents the Equinoctial Colure, where I chuse to begin the First Meridian; it being allowed by all skill'd in Astronomy, that it is left to Mens Choice where to begin it; seeing it is easy by Addi&rehy;tion or Substraction, as the Case requires, to reduce the vulgar first Meridian, which is Ten Degrees East from the Equinoctial Colure; and e contra, the first Meridian at the Equinoctial Colure to the vulgar Me&rehy;ridian; differing only Ten Degrees the one from the other: Withal minding, that what is call'd in the Starry Firmament the Right Ascension of the Stars from Aries, the Point of Intersection of the Equator and Ecliptick, on the Equinoctial Colure upon the Earth, may be called Longitude of Places; begin&rehy;ning at Aries upon Earth, on the Parallel Equinoctial Colure, at the like Intersection of the Equator and Ecliptick upon Earth. For that there are such Parallel Circles and Lines on Earth, artificially descri&rehy;bed by Artists in Astronomy and Geography, as the Equator, the Ecliptick, the Equinoctial and Solsti&rehy;tial Colures, and Ecliptick, and Points of Intersecti&rehy;on of Aries and Libra, is evident from those Circles and Lines usually described in Globes and Maps visi&rehy;ble to our sight, corresponding to these in Heaven, Parallel and Concentrick unto them.

CD represents the Solstitial Colure.

The Arch AB, marked with Two Stars, under the Two Letters A and B, represents the Distance on the Arch of a Great Circle in the Starry Firmament, betwixt the Star A and the Star B; and if there be no visible Star at B, but only at A, as oft happens, there is a Point at B in the Starry Firmament, that keeps the same Distance always from A, as if there were a visible Star at B.

A in the Greater Circle, and B in the same Greater Circle, represent Two Zenith Points of Stars, to A and B on the lesser Circle of the Globe of the Earth: And though they be but once, in the Revolution of 360 Degrees in the Zeniths of A and B in the lesser Circle, yet they are still at the same Distance, and may well be called Zenith-Points or Stars over A and B in the lesser Circle. But to take an Observation with your Astrolabe at Sea, and with your Quadrant at Land, you must have always one Star (as at A) to look unto, and make your Observation by the same.

The Arch EF represents the Distance in the Arch of a great Circle on the Globe of the Earth, of Two Places differing more or less in Latitude, but having the same Longitude; to which is to be conceived (though not described in the Scheme) a Parallel Arch in the Starry Firmament, having two Stars, or one Star and a Point; but such a Parallel is not BA in the greater Circle; and though not visibly descri&rehy;bed to the Eye, yet is easily conceived by the Mind.

As the Declination of any Star is equal to the La&rehy;titude of the Place over which it is Vertical, once in the Revolution of 360 Degrees, which I call a Com&rehy;mon Zenith Star or Point, which all Stars of the same Declination have in common together: So where a Star is to be found, or a Point, in the Firma&rehy;ment, that has both the same Declination and Right Ascension from the Equinoctial Colure that the Place over which it is at some time Vertical, hath the same Latitude and Longi&rehy;tude; that Star or Point I call a Proper Zenith; for such a Property belongs to no other Star in the whole Firmament. For as no one Place in Earth has the same Latitude and Longi&rehy;tude with any other Place on Earth; so no Star or Point in Heaven has the same Declination and Right Ascension. And one only Star there is, or Point in Heaven, that has its Declination equal to the Latitude of the Place over which it is a Proper Zenith; and also its Right Ascension equal to the Longitude of that particular Place, reckoning its Longitude from the Equinoctial Colure: The which is easy to be found, both on Globes and Planispheres, if they were made in a true Conformity, the Heavenly Globes and Maps to the Heavens and Starry Firmament, and to a true Conformity to the Places on Earth, so as to be set down in their true and proper Situation: But no such Terrestrial Globes or Maps have yet ever been extant, but abound with great and foul Errors, especially with respect to their Longitudes. But by this New Theory and Method, New Terrestrial Globes in process of Time may be made, to answer so exactly to the Heavenly, as the Impression of the Seal or Stamp answers to the Seal it self, on Paper or Wax: And if there were such Globes so justly made, the Declinations and Right Ascension of every Star in Heaven would exactly correspond in equality to the Latitude and Longitude of the Place over which that Star or Point is a Proper Zenith or Point: And two Planispheres, the one Celestial, and the other Terrestrial, so rectified by this New Theory and Me&rehy;thod of finding True Longitudes to all Places on Earth, being so adjusted, that the Equinoctial Colure in the Heavenly Planisphere be set upon the Equi&rehy;noctial Colure in the Terrestrial Planisphere; and the Solstitial Colure in the Heavenly Planisphere to the Solstitial Colure in the Terrestrial, and both set upon one Center; the finding only the Latitude of your Place at Sea, by Observation of your proper Zenith Star in Heaven, and setting it under the proper Zenith Star or Point in Heaven, would so exactly answer the one to the other, that your Latitude would still give your Longitude in all Places, where you are, at Sea or Land: And a small Prick with the Point of a small Needle, through your proper Zenith Star of Point in the Celestial Planisphere, would make a particular Impression or Mark upon the Place or City in the Terrestrial Planisphere, as Ocular In&rehy;spection would demonstrate: But as yet no such Pla&rehy;nispheres have been extant: However, such as they are, with all their Errors, may be of some Service, by the juxta Position of the one upon the other, so far as to detect the Error, and find out the Truth, by making New Observations.

But it is a great Mistake, if any think that this would be a Petitio Principii, or running round in a Circle: For to run round in that called a vicious Cir&rehy;cle, nothing is really and truly effected; but here there is: For you would have this as a necessary Quesitum to be found, viz. every Day that the Sky is clear, to find your true Latitude; and then you would have Three Sides of a Spherical Triangle; viz. the Complement of the Latitude of the Place whence you came, for one Side, and the Complement of the Latitude of the Place where you now are, for the other Side, and the Distance sailed, on the Arch of a great Circle for the Third Side; and by a Canon in the Spherical Oblique Triangles, the Three Angles can be found; and the Angle at the Pole is the Longitude between the Two Places; which if there were True and Just Terrestrial Planispheres, only by means of finding your Latitude, without any Trigonometrical Calculation, by Ocular Inspection, you might find your Longitude, or Difference of Longitude. But because no such Planispheres are yet made, the Ship-Master who would learn this New Method must be well taught both to project the Triangle, and also to find the Angle at the Pole, which is the Longitude of the Place, betwixt where you now are, and the Place from whence you came, both at your first setting out, and of your Advance and Progress every Day by the differing Latitude that you come into from Day to Day, until you are come to run out the whole Difference of your Latitude from the beginning of your Voyage; and then your are at the Place whither you were bound: For by running out your Latitude upon the Arch of a great Circle, where you began, you do likewise run out your Lon&rehy;gitude, and are at your Port you designed for. And if you be driven out of the Arch of your Circle by Currents or Storms, it is but to begin a New Arch, which is as good as the former which your were driven from.

Some Postulata, in Order to Landmens Finding the Longitude of the Place on the Earth, where they are; and Seamens Finding the Longitude of the Place where they are at Sea; taking their Ob&rehy;servation of some known Star, East or West from the Equinoctial Colure; if there be none upon it, answer&rehy;ing to the Latitude of the Place where they are. &Tic;O be furnished with a good Qua&rehy;drant of Two or Three Foot Ra&rehy;dius or more, if it can be conveniently had, for Land&rehy;men to find their Longitude at Land; and large and good Astrolabes and Cross-Staffs for Seamen to find their Longitude at Sea, whereever they are. That the Persons who would learn this Practice be so far skilled in the common Principles of Astro&rehy;nomy and Navigation, as many Ship-Masters are; and also to have some Knowledge in Arithmetick and Geometry. That they be taught by some Mathematick&rehy;Masters also the Names and Situations of some few Stars, with their Declinations and Right Ascensions, as correctly given by Learned Authors. And such as are more learned, ought not to rely altogether upon any Tables of Declination and Right Ascension, but rather find them out by Art; of which, for so few as are necessary for the finding the Longitude, there is no Difficulty, nor great Labour to be used by them who are well learned. But for vulgar Use to Seamen, the ordinary Tables of Declinations and Right Ascen&rehy;sions of Stars, as correctly given by late Authors, may suffice; for even Learned Men differ in their Practice of finding them, in some few Minutes, which Seamen regard not. But lest Seamen think it too much to learn the knowledge of Five or Six Stars, so as by Sight to distinguish them from all others; to ease them of that Trouble, and to encourage them in so good a Work, I shall name at present but one only Star, which they must learn to know by Sight, and note down in Wri&rehy;ting, viz. the Star call'd Scheder, in the Breast of Cassiopeia; the Declination of it, which in correct Ta&rehy;bles is 54 Deg. 55 Min. and its Right Ascension 6 Deg. 4 Min. or thereabouts. But such as have more Ca&rehy;pacity than others, may learn to know as many Stars as they can, so as by Sight to distinguish them; and to note down in Writing, their Declinations and Right Ascensions, for their Use in working any Question at Sea, about finding the Longitude by not one Star on&rehy;ly, but by as many as they please; and yet all, (if the Operation be rightly performed, according to the Rules here to be deliver'd) to agree to Admira&rehy;tion, as much as when any one Question in Trigonometry may be wrought many several ways, and by several Datums, and yet all to agree, with so little Difference as is not to be notic'd. For what Diffe&rehy;rence happens in working Questions in Trigonometry is not in the Rules, which are most certainly true, but in taking an Angle or a Side too much or too lit&rehy;tle, as a Minute more or less will occasion some Di&rehy;versity in resolving the Question; as is well known to the Learned. And that one Star above-mention'd will suffice to any Seamen to find their Longitude as far East or West, Southerly, or Northerly, as that Star is any where visible. Not that the said Star has any peculiar Property in it, to discover the Longitude, more than any others; for the Longitude may be found by any Star as well as by that: But it, and many other Stars that lye near the Equinoctial Co&rehy;lure in our Hemisphere, East or West, are more com&rehy;modious for Beginners. And that the working to find the Longitude by several Stars, whether remote or near, doth agree in the giving the same Longitude of your Place, is a sufficient Demonstration by pro&rehy;per Effects to proper Causes, to prove the Theory and Method I am hereafter to deliver to be true: For no fallacious Cause can constantly and uniformly pro&rehy;duce the same Effects.
The PRAXIS or Operation; How a Man, whether at Sea or Land, may find the Longitude in any Part of the World where he is.

It being allowed by all Learned Men in Astrono&rehy;my and Geography, that a Man may begin his first Meridian any where, from whence to compute his Longitude from that beginning, as a sort of Epo&rehy;cha; I chuse to begin it at the Equinoctial Colure, for Reasons above given: And having found the Longitude of any Place, whether at Sea or Land, from this beginning, it is easy by Addition or Substraction to reduce it to any vulgar Meridian, or the vulgar Meridian to it; the Vulgar being Ten Degrees East from the said Equinoctial Colure.

After you have well known this Star called Sche&rehy;der, so as to distinguish it by Sight from all other Stars, wait upon the said Star until it come upon its South Meridian, as Seamen do to find the Sun's high&rehy;est Altitude; and take its Distance from the Zenith of your Place, by looking to it, and using an Astrolabe; which is a proper Instrument frequently used by Sea&rehy;men in other Countries, to observe the Height of a Star, or its Zenith Distance when it is at its greatest Height. Suppose the Zenith Distance be found by your Astrolabe to be 30 Degrees; Note that down for your Use, in order to the finding the Longitude.

This is one Way, that has no more Difficulty in it, nor less Exactness, than for Seamen to find their La&rehy;titude.

Objection. But it is impossible that you can know when that Star comes to the Meridian of your Place, unless you know the precise and exact Time when it should come upon your Meridian.

Answer. This Objection seeks to overthrow all finding the Latitude equally as the Longitude. For this see Page 4. Numb. X.

Objection 2. When AB in the greatest Circle (See the Scheme) are Vertical to AB in the lesser Circle, B in the greater Circle is not in its least Distance from the Vertex, unless the Parallel Arches AB in the lesser Cir&rehy;cle, and AB in the greater Circle, be in the Meridian.

Answer. The Words of the Objection are wrong&rehy;ly expressed, and carry in them a Contradiction to the Fundamentals of Astronomy: For it supposeth that two Zenith Fixed Stars or Points in the Starry Firmament, such as AB in the greater Circle, can be nearer one another at one Place than another; which is contrary to Principle II. in Page 2. of this Treatise: The which said Principle is universally re&rehy;ceived among all the Professors of Astronomy, viz. That all Fixed Stars and Points in the Starry Firmament, keep the same Distance where they are, and whatsoever Azimuth they appear on, as well as Me&rehy;ridian: For otherwise they would not be Fixed Stars, but Planets: They change indeed their Horizontal Distance, so as to be nearer the Horizon of the Place where you are, at one time, than another: And such Stars as never descend to the Horizon so as to touch it, namely the Star above mention'd, call'd Scheder, having Latitude 54 Degr. 55 Min. never comes near&rehy;er to the Horizon than at the Distance of 19 Degr. 50 Min. which is its nearest Distance to the Horizon: And when it is at the nearest Distance of 19 Degr. 50 Min. it is then upon the North Meridian; and when it is 70 Degr. 10 Min. from its nearest Distance to the Horizon upwards, then it is at its greatest Height upon the South Meridian: But still it keeps the same Vertical Distance from the other Star or Point A; both being Vertical Stars or Points to their respective Places on Earth, such as BA in the lesser Circle. The Words of the Objection being thus cor&rehy;rected, to render them consistent with the Fundamen&rehy;tal Principles of Astronomy, by saying, when the Star B comes to its nearest Distance to your Horizon, the said Star is upon the Meridian; and then by your Observation you can take its true Distance from your Zenith; and both the Stars or Points are Zenith Points. But this is so far from impugning my New Theory and Method of finding the Longitude, that it con&rehy;firms it, and perfectly agrees with the Practice. But I know a Way to take the true Distance of the Point A (when there is not a Star at A) from the Star B, at any Time of the Night, as well when it is not on the Meridian, as when it is on it: But for Brevity sake I shall not here mention it: For without that other Way, the Longitude can be found as well as the Latitude, by waiting upon the Star until it comes to the Meridian of your Place, as is above shewed.

Now for a Conclusion of the whole Matter with Relation to the Practice.

First, By Observation how soon the Star Scheder comes to the South Meridian of your Place, take his Distance from your Zenith by your Instrument above&rehy;mentioned.

Secondly, By this Distance form to your self in Mind a Spherical Oblique Triangle, such as that in the Scheme, viz. PEF, whose Side PE is the Comple&rehy;ment of the Latitude of one of your Places, and PF the Complement of the Latitude of your other Place, and EF the Distance found; which Three Sides being given, by a Canon in Spherical Trigonometry, you can find the Angle at the Pole, which is the true Lon&rehy;gitude required of B from the Star called Scheder, at the Equinoctial Colure, by which you observed. And because he is 6 Degrees East of the Equinoctial Colure, you must add to the Longitude found by the Star Scheder the said 6 Degr. and that gives your whole Longitude of your Place from the first Meridian at the Equinoctial Colure. But to reduce it to the vulgar Me&rehy;ridian, you must substract 10 Degr. because the Vulgar Meridian is 10 Degr. East of the other.

Next, to find the Longitude of any Two Places ha&rehy;ving the same Latitude, lying due East or West, the one from the other; First observe the Distance betwixt your Zenith where you are, and the Zenith of the Star Scheder, as before, and take that Distance for one Side of your Tri&rehy;angle, and take the Complement of your Latitude for another, and the Declination of the Star Scheder, answer&rehy;ing to the Latitude of the Place whose Zenith Star it is for the Third Side, and that will give you the Angle at the Pole; and adding to it 6 Degrees, as above, that will give the Longitude of your Two Places, from the first Meridian at the Equinoctial Colure. G.K.

astronomical scheme referred to in the text