A new method for discovering the longitude both at sea and land: humbly proposed to the consideration of the publick. By William Whiston, ... and Humphry Ditton, ...

Table of contents

- [titlepage]

- TO

- THE INTRODUCTION.

- LEMMATA, or Preparatory Propositions.

- 1

- 2

- 3

- 4

- 5

- 6

- 7

- 8

- 9

- 10

- 11

- 12

- 13

- 14

- 15

- 16

- 17

- OBSERVATIONS.

- APPENDIX.

PROBLEM. To find the LONGITUDE both at Sea and Land.

A New METHOD For Discovering the LONGITUDE BOTH AT SEA and LAND, Humbly Proposed to the Consideration of the PUBLICK.

BY

- William Whiston, M.A. sometime Professor of the Mathematicks in the University of Cambridg. and

- Humphry Ditton, Master of the New Mathematick School in Christ's Hospital, London.

LONDON: Printed for JOHN PHILLIPS, at the Black Bull in Cornhill. 1714.

TO

- The Right Honourable THOMAS Earl of Pembroke and Montgomery.

- The Right Reverend Father in God PHILIP Lord Bishop of Hereford.

- The Right Reverend Father in God GEORGE Lord Bishop of Bristol.

- The Right Honourable THOMAS Lord TREVOR, Lord Chief Justice of the Common Pleas.

- The Admirals of the Red, White and Blue Squadrons.

- The First Commissioner of the Admiralty.

- The First Commissioner of the Navy.

- The First Commissioner of Trade.

- The Master of Trinity-House.

- The Hon. Sir THOMAS HANMER, Bart. Speaker of the Honourable House of Commons.

- The Hon. General STANHOPE.

- The Hon. FRANCIS ROBERTS, Esq

- Sir ISAAC NEWTON, President of the Royal Society.

- WILLIAM LOWNDS, Esq

- WILLIAM CLAYTON, Esq

- Mr. JOHN FLAMSTEED, Astronomer Royal.

- Dr. EDMUND HALLEY, Savilian Professor of Geometry.

- Mr. JOHN KEILL, Savilian Professor of Astronomy.

- Mr. SANDERSON, Lucasian Professor of the Mathematicks.

- Mr. ROGER COTES, Plumian Professor of Astronomy.

Commissioners appointed by Act of Parliament for the Discovery of the LONGITUDE.

This New Method for that Discovery is with all due Submission humbly Dedicated by

The Authors.

THE INTRODUCTION.

BEfore we come to give an Account of this our New Method for the Discovery of the Longitude, both by Sea and Land, which we here take leave humbly to propose to the Consideration of the Publick, we think it reasonable to premise somewhat by way of Introduction: To give some Account of the Nature of the Problem before us; to speak a little of the Methods hitherto try'd, and the Reasons of their ill Success; and to add a brief Historical Narration, from what Occasions and by what Steps this our Method was first discover'd, and has arriv'd at its present Degree of Maturity.

As to the Problem it self, the Invention of the Longitude; it is plainly this: To discover in some measure a like sure way of frequently knowing, how far we are distant, on the Earth Spherical Surface, in Degrees, from any known Meridian, Eastward or Westward; as we can easily know, almost at any time, how far we are distant, in Degrees, on the same Surface, from the middle Circle or Equator, Northward or Southward. Now in this Case it must be noted, that as the Diurnal Motion does naturally imply fixed Poles, and a fixed Equator; which infer a different Meridian Altitude of those Poles, and of that Equator, and by consequence of all the heavenly Bodies, in different Latitudes; which different Altitude may in clear Weather be easily observ'd by proper Instruments, and thereby that Latitude may be readily discovered; so does not the same Diurnal Motion at all imply any Phaenomena, whence the Longitude may be discover'd to us: Because the same Parallel still bears, through its whole Circumference, the same Relation to those Poles and that Equator, without any Difference. The Diurnal Motion therefore, which affords an obvious Foundation for the Invention of the Latitude of every Place on the Earth, affords us no such Foundation for the Invention of the Longitude of the same Places. Nor is it therefore an easy Problem, either astronomical or practical to discover the same.

As to the Methods hitherto tryed, they are either Celestial, or Terrestrial; and may be reduc'd to these Seven, Four that are Celestial, and Three that are Terrestrial.

- (1) The Eclipses of the Moon.

- (2) The Eclipses of the Sun.

- (3) The Eclipses of Jupiters Planets.

- (4) The Motion of the Moon.

- (5) The Variation of the Needle.

- (6) Clocks, or Watches.

- (7) The Log Line or Dead Reckoning.

First, The Eclipses of the Moon are useful for the Longitude. For its Immersions into the Earths Shadow, its nearest Distances to that Center, and its Emersions therefrom, are all at distinct and known Points of absolute time. So that where and when they can be nicely observ'd; and the Difference of the apparent times at every Meridian noted; the respective Longitudes of those Places may be thereby found in time; and by allowing 15 Degrees of the Equator to an Hour, may be found in Degrees also.

Secondly, in the same manner may the Eclipses of the Sun be made use of; especially as now improv'd by our great Astronomer Mr. Flamsteeds Construction of them; and as they will, we hope, be farther improv'd by Mr. Whistons actual Exhibition of them, in his Instrument, just ready for Publication. Which Method, by the Difference of the apparent time of any Part of the Eclipse in different Places, gives the Difference of Meridians, or of Longitude in the like manner as before.

Thirdly, The Eclipses of Jupiters Satellits afford another like Method for the Discovery of the Longitude; and that on the same Foundation with those of the Moon.

Fourthly, the Motion of the Moon, with its Distance from the Sun, or rather its Appulse to and Occultation of those fixed Stars that ly along its Course, is another remarkable Method for this purpose; and is of the same Nature with the Eclipses of the Sun as to this matter.

These Four may justly be called Celestial, or Astronomical Methods of discovering the Longitude, because they make use of the Celestial Bodies, or of the Stars in order to that End. The Three Terrestrial Methods are as follows.

5ly, The Variation of the Needle from the North is now, especially since Dr. Halleys noble Observations and Map thereto relating, become one Method for the Discovery of the Longitude; particularly in those Parts, where that Variation is best known, and the North and South Position of its Lines are most remarkable. For by crossing the Meridians there, you also cross the Curves of equal Variation, and discover in some measure your Longitude thereby.

Sixthly, The Use of Clocks or Watches at Sea is another Method; and was attempted by the famous Hugenius. And indeed if they could be exactly kept to an even motion, and so shew the Hour at any one certain place at Land; the Comparison of the Time known by that Clock or Watch, with the apparent time at the Ship known by the Sun or Stars, or another Clock or Watch regulated by them, would discover the Longitude from the Place to which that first Clock or Watch was adjusted, in time, and so, as before, in Degrees also.

Seventhly, The Log-line and Dead Reckning, when all fails, is the last Remedy in this case; and from thence the Seamen guess, as well as they can, by the Angle and length of their Course, what Longitude and Latitude they are in: And when by Observation they find their Error in Latitude, they conclude upon a proportionable one in Longitude also. And so for want of a sure Guide, either Celestial or Terrestial, they are forc'd to depend on this; which yet is, as well as the rest, very uncertain and inaccurate.

For to come to the Reasons of the small Success of these several Methods.

As to the two first, the Eclipses of the Sun and Moon; to say nothing here of the slowness of the Moon's Motion, which renders any great degree of exactness impossible; or of the difficulty of Calculation and Construction, especially in the Sun's Eclipses; and of Observations in both: The single rareness of these Eclipses, which is not seldom made still rarer by cloudy Weather, renders them of very little use in Navigation.

As to the third Method, by the Eclipses of Jupiter's Planets; this must be own'd of much greater use: Since the quickness of their Motion, especially as to the innermost, makes the Moment of their Immersion into, or Emersion from Jupiter's Shadow very distinct and nice; and their frequency, which is almost one for every Day, renders them fit for the constant uses of Navigation. Nor have we hitherto had any other Method so useful at Land as this. Yet are there great Difficulties belonging to this Method; especially at Sea. The best Tables of their Motions are hitherto too imperfect to be at all times depended on, as to the exact absolute Time of their Celebration: And they require Telescopes of such a length as have not hitherto been manageable at Sea, in that state of Tossing and Agitation, which Ships there are subject to: Which difficulties, added to the impossibility of seeing these Eclipses for about three Months every Year, when Jupiter is near the Sun, renders this Method at present of small use in Navigation.

Nor can the fourth Method, or the distance of the Moon from the Sun, with its Appulse to, or Occultation of those fixed Stars, which lye along its course, give us the Longitude to sufficient exactness. For, to say nothing here of the slowness of the Moon's Motion, the want of the utmost accuracy of even the place of some of these fixed Stars themselves, and of the Sun it self; or of the necessity of the use of smaller Telescopes, even in this Case, as well as of the trouble of the Calculation and Construction, which are lesser Difficulties here also; 'Tis plain the Theory of the Moon, especially in some positions, is not exact enough hitherto for our purpose; as not serving for this Longitude nearer than to two or three Degrees: whereas the Seamen want it within one Degree, or less. Tho' indeed it must be allow'd, that if the Moon's Theory could be once so far perfected, that its place might be with certainty calculated nearer than to two Minutes of a Degree, this would be a very useful Method in order to the Discovery of the Longitude at Sea. Which Improvement therefore of it's Theory is a thing highly desireable in Astronomy.

We come now to the Terrestrial Methods, and to those difficulties which render them also incapable of discovering the Longitude, with that certainty, and to that degree of exactness, which the purposes of Navigation require. Thus the Curve lines of the Variation of the Needle, which is the first Terrestrial Method, are of small use, because the Laws of that Variation are not yet brought to a sufficient certainty, notwithstanding the most useful endeavours of Dr. Halley in that Matter: The Neighbourhood of Iron Mines, of Iron, or of Loadstones themselves, does sometimes disturb the general Rules, and deceive the Observers of that Variation: The Position of those Curves, too far Eastward and Westward, in a great part of the World, renders this Variation useless as to any general Discovery of the Longitude: and even there where the Position of these Curves is the most advantageous, as it is about the Cape of Good Hope, and a considerable way on both sides of it, yet is the distance of those Curves for the difference of one Degree of Variation, about 100 Geographical Miles, i. e. near two Degrees of a great Circle; and so this Method is incapable of shewing the Longitude very nicely in any Case whatsoever.

Thus the second Terrestrial Method, by Clocks or Watches, tho' the easiest to understand and practice of all others, has been so long in vain attempted at Sea, that we see little Hopes of its great usefulness there. Watches are so influenc'd by heat and cold, moisture and drought; and their small Springs, Wheels, and Pevets are so incapable of that degree of exactness, which is here requir'd, that we believe all wise Men give up their Hopes from them in this Matter. Clocks, govern'd by long Pendulum's, go much truer: But then the difference of Gravity in different Latitudes, the lengthening of the Pendulum-rod by heat, and shortening it by cold; together with the different moisture of the Air, and the tossings of the Ship, all put together, are circumstances so unpromising, that we believe Wise Men are almost out of hope of Success from this Method also.

And as for the Log-Line, and Dead-Reckoning, which is the third Terrestrial Method, they were the known deficiencies of this common way, as alter'd by Storms, and Currents, and the Inaccuracies of the way it self, and of even the Latitude, as commonly taken; together with the too frequent and enduring cloudy Weather, when they can take no Latitude at all; which have occasion'd the Seamen to desire some other Assistance for the Discovery before us.

We now come to our last Business, viz. to give the World a short History of our own Proposal; from what occasions, and by what steps this our Method was first discover'd, and has arriv'd at its present degree of Maturity. As to which matter, the Reader is to know, that somewhat above a Year ago, Mr. Whiston and Mr. Ditton, with some other common Friends, spent part of an Afternoon and the Evening together. Mr. Ditton took an occasion, among other common discourse, to observe to Mr. Whiston, that 'The nature of Sounds would afford a method, true at least in Theory, for the discovery of the Longitude'; since The difference between the apparent time, where the Sound is made, and where it is heard; abating only the time for its diffusion, which was now well known; is the difference of the Longitude of those two Places in time. Mr. Whiston immediately own'd the truth of the Proposition, and added, 'That as to the Propagation of Sounds, he remembred to have himself plainly heard the Explosion of great Guns about 90 or 100 Miles, viz. when the French Fleet was engag'd with Ours, off Beachy-head in Sussex; [which was A.D. 1690.] and himself was at Cambridge; and that he had been inform'd, that in one of the Dutch Wars, the sound of the like Explosions had been heard into the very middle of England, at a much greater distance. Upon this, Mr. Whiston, when they parted, told Mr. Ditton, that he took the thing to be so considerable, that tho it had been discoursed of in mix'd Company, after an unguarded manner, yet he look'd on it as fit to be conceal'd; since no body could tell, what Improvements might on farther Consideration be built upon such a Foundation.' Which Advice Mr. Ditton follow'd; and accordingly desir'd and obtain'd the Silence of those, that had then heard, what had pass'd. This Proposition about Sounds, and their distant Propagation, with respect to the Longitude, did upon this so fix it self in Mr. Whiston's Mind, and did occasion such Improvements there, that in less than two Days time he brought a small Paper to Mr. Ditton, containing a Scheme, how that Theory of Mr. Ditton's about Sounds might be reduc'd to Practice, and be actually apply'd to the discovery of the Longitude at Sea; which was then not much unlike the former branch of the following Essay, only more imperfect: Which Scheme Mr. Ditton approv'd of. Soon after this Mr. Ditton imparted this Discovery to a very good Friend, belonging to the Admiralty, in order to gain farther light as to its practicableness at Sea; and that proper Questions might by him be ask'd of Seafaring Men relating thereto, without any Suspicion; which could not well be avoided if we our selves had ask'd them; especially since the Notion was then got abroad, that we had a Project about the Longitude to propose to the World. The result of this Enquiry was, that those Sea-men our Friend enquir'd of, did not remember to have heard Sounds at Sea any whit near so far as the beforemention'd Examples shew'd they had been heard at Land; which difficulty put some stop to our Progress for a little while. However, at last, after farther enquiry, the final result was this, That tho' Sounds were not ordinarily either at Sea or Land heard very far, yet that was not at Sea, more than at Land, any certain Argument, that they could not spread so far; because Sounds had been heard a full Degree at least, or 60 Geographical Miles over Sea, even without any extraordinary Contrivance, either at the sounding Body, or the Ear; both which were yet, for certain, capable of great improvements, in order to the enlargement of that distance. So that the Objection started against the spreading of Sounds at Sea seem'd to be in a manner over, and we at liberty to prosecute our Design, as before, of discovering the Longitude by means of it. About this time Mr. Whiston discover'd and propos'd a great Improvement of his own to this Method; viz. That the Guns, which were to make the Explosions in the former Case, might also carry Shells, full of Powder, or such other combustible Matter, as would take fire at the utmost Altitude; and thereby certainly and exactly exhibit the point of the Azimuth, and the Distance of the sounding Body; and so join the use of the Eye and Ear together for the same purpose. Tho' at the first he must own he suspected, that the Apparent Diameter of that Light or Fire would in great distances be so small, as not to be there visible. In this very juncture a day of extraordinary Fireworks happen'd [it was the Thanksgiving day for the Peace, July 7th, 1713.] the Contemplation of which, did much revive and encourage this Notion: and the certain Account he soon had, that those Fire-works, nay, the small Stars, into which the Rockets commonly resolv'd themselves, were plainly visible no less than 20 Miles, put an end to his doubts immediately; and made him very secure, that such large Shells as might be fir'd at a vastly greater height, would for certain be visible for about 100 Miles; which he look'd on as nearly the limit of Sounds also, as to any purposes of Longitude. This Improvement of Mr. Whiston's, which was also then for the main the same with the second Branch now contain'd in this Paper, was also approv'd of by Mr. Ditton, and agreed to as fit to be a part of the former Design for the Discovery of the Longitude at Sea. Mr. Ditton did farther add, for Improvement, a sure Method of Trigonometrical Calculation, to ascertain from the Observations the horary Difference of Meridians (and by consequence the Difference of Longitude in Degrees) between the Ship's Place, and that of Explosion; without computing the Time of the Sound's propagation: but since this Method is somewhat more operose than that, which is propos'd hereafter, he chooses to omit it. He did also first observe that great Use of our Method at Land, in the Surveying of Countries, for the Perfection of Geography; which was also readily taken notice of by Sir Isaac Newton, and afterward by Dr. Halley, and that both of their own accord, upon our first communication of our Method to them. For when Matters were brought to so hopeful a Posture, and necessary Tables were preparing for the actual Practice of the whole Method, we began to think of intimating to the Publick, that we had a new Discovery, as to the Longitude, to propose to the World. Which we soon did, by our Letter inserted in the Paper call'd the Guardian, of July 14. and repeated by another in the same Author's Paper call'd the Englishman, of December 10. following. Having before communicated the matter to the illustrious Sir Isaac Newton, as we did afterward to those great Men, Dr. Clarke Rector of St. James's, Dr. Halley of Oxford, and Mr. Cotes of Cambridge. How far we profited by this Communication, and what their Opinions were concerning our Method, we need not say: because we do not give Account here of every occasional Improvement, either of our own or others; and because we now publish the intire Method, as it stands at present, to the whole World, for every one's open Judgment, and the farther Improvements of the skilful. Only so far their Opinions and Declarations appear to have been on our side, that upon hearing what they and we had to say, the Committee of the House of Commons, which was appointed to inquire into this matter, came unanimously to a Resolution in our Favour; and the Legislature have thereupon thought fit to pass an Act, appointing a noble Reward for such as shall discover a better Method than has been hitherto us'd for the finding the Longitude. Which Reward, whether we have any just Claim to, in whole or in part, we do hereby intirely submit to the Sagacity and Justice of those eminent Persons, whom the Legislature has been pleas'd to intrust with the Tryal, Experiment, Judgment, and Determination of all such Proposals.

We conclude all with our hearty Wishes as Men, that this our Design may tend to the common Benefit of Mankind: as Britains, that it may tend particularly to the Honour and Advantage of this our Native Country; and as Christians, that it may tend to the Propagation of our Holy Religion, in its original Purity, throughout the World.

London, July 7. 1714.
William Whiston, Humphry Ditton.

LEMMATA, or Preparatory Propositions.

I. ALL Sounds are propagated almost evenly; and are observ'd to move 14 Measur'd, or 12 Geographical Miles in one Minute of Time: i. e. one Geographical Mile or Minute of a Degree in five Seconds.

This is well known from the last and most accurate Observations[*] about the Velocity of Sounds, which are those of Mr. Dereham. Only a small Addition of Velocity is to be made, when a strong Wind carries the Sound with it, and Substraction when it opposes it.

II. The Sound of a great Gun may be heard by the Ear, duly assisted, if the Wind be favourable, or still, both by Sea and Land, at the least 100 measur'd, or 85 geographical Miles. In the open Sea also, the Point of the Compass may be nearly determin'd whence it comes.

This is very probable, as to the Distance, from many known Experiments[*]; wherein the Ear, even unassisted, has heard such Sounds much farther. And if the Sound were increas'd by a sounding Board, which might prevent its diffusion upwards, and so spread it farther Horizontally on all sides; and if the Ear were assisted by a hollow Tube of Metal, of the shape of a Bell or Tunnel, apply'd thereto, this Proposition would soon be more indisputable. Nor is there any great Difficulty, as to the Point of the Compass, whence the Sound comes at Sea, where nothing can reflect or echo the same in any other than the true Angle.

III. The Distance of the sounding Body, where the Sounds are of the same Strength, and Tenor, and Circumstances, may, within some Latitude, be determin'd by the Ear, duly assisted, and frequently exercis'd in such Observations; even at very considerable Distances from the sounding Body.

This appears from the obvious difference of the same Sounds at very different Distances at present; which is in a duplicate Proportion of those Distances: and from the great Improvements, Experiments made on purpose would probably afford us therein.

N.B. In order to determine accurately the Distance of a given Sound, there must be distinct Trials made, in an open Place, both by Sea and Land, in clear and in foggy Weather, with the Wind in all Positions, and of all Degrees of Strength; and this at several Distances of the Hearers: but till that is done, we must leave this matter to the Ear alone.

IV. A strong Wind carries Sound along with it in a Circle; where the Sounding Body is a Point in its Axis: and is more or less remote from its Center, according as the Wind is greater or less.

This appears by the Demonstration following. Let the Proportion of the Velocity of the Wind, to the Velocity of the Motion of the Air that causes the Sound, be as AB to AD.

Let the two equal Circles GDHE, GCHF, be described upon the Centers A and B; and let any Line, as KL, be drawn Parallel to DF. KI will therefore be always equal to ML. or to the Velocity of the Wind, and according to its direction: as AM = AK = BL = BI will be equal to the Velocity of the Motion of the Air that causes the sound, and according to its direction; or from the Center to the Circumference of that Circle which includes the sound. Whence the Diagonals BK, BM will be the distance or measure of the Equal sounds; and the points K. M will be in the Circumference of that Circle GDHE of which the sounding body B is a point in its Axis. Q.E.D.

Corollary (1.) Because the Lines AB and AK, and the Angle BAK are given; the distance of equal sounds BK is also given by plain Trigonometry. As the same line may be found Geometrically also, by applying its length to a scale of equal parts.

Coroll. (2.) Two equal Circles, sliding one upon the other, according the direction of the Axis FD is the readiest way of solving this Problem, for the use of Seamen; as being so very easy in Practice.

V. The Interval of apparent time, in two places, where a Sound is excited, and where it is received; besides that which is due to the real propagation of the Sound it self; is the Difference of their Meridians, or of their Longitude in Time.

Thus if a Sound, excited just at 12 a clock at one place, comes to another after the very same Time that is due to the Sounds propagation, as at the distance of 14 measured Miles, one minute after 12. At the distance of 28 such Miles, two minutes after 12. &c. 'tis evident the places are under the same Meridian, and have no difference of Longitude. But if it be heard sooner or later than those times, the Difference is what answers to the Temporary difference of their Meridians, or of Longitude, Westward or Eastward: and so is a sure indication of the same. As is very obvious on a little consideration; and as we shall shew presently by example.

VI. An Ordinary Great Gun is easily able to cast a Projectil about a Mile and a quarter, or 6440 English feet, in perpendicular height.

This appears by that known [*] Theorem in the Art of Gunnery, which demonstrates, that the utmost Altitude is always equal to half the utmost Random of the same Gun and Powder: which utmost Random, of ordinary Great Guns, with a very small charge of Powder, is known to be about two Miles and a half, or 12880 Feet.

N.B. That it appears by the same way that the largest Great Guns, with their largest charge of Powder, are able to cast a Projectil twice, nay thrice, and even four times the beforementioned height. But because the charges and trouble are in such cases much greater; and it is uncertain whether the advantages will be proportionably augmented, we choose to speak moderately, especially before tryal; and to propose nothing here but what is for certain cheap, practicable, and advantagious; and leave those more surprizing heights, to the consideration of the publick afterward: Only with this observation, that the Altitude will ever be as the Squares of the Velocity, with which the Projectil is thrown.

VII. The time of the Ascent or Descent of such a Projectil; without the consideration of the resistance of the Air; (which in the case of lead bullets, iron shels, or the like dense bodies is but very small, and in Wood not very great;) is 20″ or ⅓ of a Minute: and is always the same in the same height.

This appears from the known Velocity of descending or ascending bodies[*], which fall or rise 16▪1 English Feet in one second of time; and by consequence 6440 Feet in 20″. those lines of descent or ascent being known to be ever as the Squares of the Times.

VIII. Gunpowder may be discharged, or combustible matter set on Fire at that utmost height.

This all that deal in Rockets, Bombs, and Mortars do very well know. It being the great business of their art to proportion the Match or Fusee to any particular time when it shall give Fire; which may as well be always adjusted to 20″ as to any other number. Nor indeed is it impossible to contrive all so, that the very beginning of the descent shall be immediately instrumental in that matter, and thereby render the experiment more exact and infallible.

IX. Fire or Light 6440 Feet high, will be visible, in the night time, when the Air is tolerably clear about 100 measured, or 85 Geographical Miles: i. e. one whole degree, and 25 minutes of a great Circle, from the place where it is, even upon the surface of the Sea.

This is easily deduced from the Tables of Tangents and Secants, applyed to our Earth; as will appear presently. Only it may be noted that the Refraction of Light out of the somewhat thinner Air above, into the somewhat thicker Air beneath, increases this distance a little; as also that an Eye upon the Mast of a Ship will see such Fire or Light 10 Miles farther than one on a Level with the Surface of the Sea; as will appear presently also.

N.B. That the Distances this Fire or Light can be seen, abating the consideration of the Atmosphere, are nearly in a subduplicate proportion of the Altitudes; and so at four times the height here mentioned, to which yet we have observed Projectils may be thrown, this distance will be nearly twice as large; i. e. about 200 measured, or 170 Geographical Miles; even without the allowance for refraction, or for the elevation of Mountains, whereon such Guns may be plac'd: both which when allowed for will imply, that 'tis possible, if the light be strong enough, to extend this distance to between 200 and 300 Geographical Miles, or minutes of a great Circle. A vast extent this! and capable of affording proportionably vast advantages to Mankind, upon the present foundation!

X The Angle such fire or light is seen above the Horrizon will very exactly discover its distance; as will an easy observation its Azimuth.

The former branch is evident from the nature of a Sphere, with the usual Tables of Tangents and Secants: and may thus be computed by plain Trigonometry. Supposing the eye of the Spectator placed at the surface of the Sea; and not considering the very small difference by the refraction.

Let A represent the Earth's Center, BD the length of the Secant of 1°.

2′ 5. above the Radius; or 6440 feet. ED the Tangent of the same Angle. CB the length of the Secant above the Radius, at any lesser Angle, as BAF. and CF the Tangent of that last Angle. 'Tis evident that the Angle DFC is the elevation of the fire or light above the horizon at any given Point F. and that in the plain Triangle DCF the Angle DCF is given, equal to a right Angle, and to the Angle FAB. FCB is its complement; and equal to the sum of the remote Angles CFD, and CDF. The including sides also CF and CD are given; the former being the Tangent of the given Angle FAB, and the latter the difference of the Secant of the same Angle from the Secant of 1°. 25. So that by the known Rule of plain Trigonometry, as the sum of the sides, CF + CD, is to their difference, CF − CD, or CD − CF: So is the Tangent of the Semisum of the Angles, ½ CFD + ½ CDF = ½ FCB, to the Tangent of their Semidifference. Which Semidifference substracted at remoter and added at nearer distances to that Semisum; gives the Angle sought CFD. Q.E.I.

According to this Rule the following Table is made to every Minute, or Geographical Mile; for the ease of all that may use this Method, and may desire some exactness therein.

Miles distance.

Angle above the Horizon.

1

46—25

2

27—42

3

19—16

4

14—40

5

11—50

6

10—20

7

9—0

8

7—55

9

6—50

10

5—55

11

5—20

12

4—54

Miles. distance.

Angle above the Horizon.

13

4—30

14

4—8

15

3—52

16

3—37

17

3—23

18

3—11

19

3—0

20

2—50

21

2—41

22

2—33

23

2—25

24

2—18

25

2—12

26

2—6

27

2—1

28

1—55

29

1—50

30

1—46

31

1—41

32

1—37

33

1—33

34

1—28

35

1—24

36

1—20

37

1—17

38

1—14

39

1—12

40

1—10

41

1—7

42

1—4

43

1—1

44

0—59

45

0—57

46

0—55

47

0—53

48

0—51

49

0—49

50

0—47

51

0—45

52

0—43

53

0—41

54

0—39

55

0—38

56

0—36

57

0—34

58

0—32

59

0—31

60

0—30

61

0—28

62

0—26

63

0—25

64

0—24

65

0—23

66

0—21

67

0—20

68

0—19

69

0—18

70

0—17

71

0—15

72

0—14

Miles distance.

Angle above the Horizon.

73

0—13

74

0—12

75

0—11

76

0—9

77

0—8

78

0—7

79

0—6

80

0—5

81

0—4

82

0—3

83

0—2

84

0—1

85

0—0

N.B. It appears by this Table that the distance will never be less exact in this Method than is the Observation of the Altitude; since one Mile here never corresponds to less than one Minute; but that generally the distance is much more exact than the Observation: Since one Mile commonly corresponds to considerably more than one minute; nay at very near distances to more than one whole degree; as is evident by inspection. As for the observation of the Azimuth, 'tis too easie to need any demonstration.

N.B. If the Eye be elevated above the surface of the Sea, it will see the fire or light farther; according to the following Table.

Miles distant.

Elevation in feet.

1

1

2

4

3

8

4

15

5

23

6

34

7

45

8

57

9

71

10

88

11

107

12

128

XI. If the fire or light can be rendred compleatly visible during the intire time of the ascent or descent, as in the ordinary Sky-rockets, its Distance may be exactly determin'd also from the time it appears above the Horizon, by the use of the following Tables, even without the knowledge of the Angle of Elevation.

A Table of the number of feet that Bodies fall or rise, as far as 20″ of time.

″

feet.

1

16▪1

2

64▪4

3

145

4

259

5

402

6

580

7

789

8

1030

9

1294

10

1610

11

1948

12

2318

13

2721

14

3156

15

3622

16

4122

17

4653

11

5216

19

5812

20

6440

A Table of the Excess of the Secants in Feet, above the Earths Semidiameter, as far as 1°. 2′ 5.

′

feet.

1

1

2

4

3

8

4

15

5

23

6

34

7

45

8

57

9

71

10

88

11

107

12

128

13

151

14

174

15

199

16

227

17

256

18

288

19

321

20

357

21

393

22

430

23

470

24

512

25

556

26

601

27

647

28

695

′

feet.

29

745

30

798

31

853

32

909

33

968

34

1026

35

1088

36

1151

37

1216

38

1283

39

1352

40

1422

41

1493

42

1569

43

1642

44

1720

45

1800

46

1882

47

1963

48

2047

49

2134

50

2222

51

2312

52

2404

53

2497

54

2592

55

2688

56

2787

57

2887

58

2991

59

3093

60

3198

61

3305

62

3415

63

3526

64

3639

65

3755

66

3870

67

3988

68

4118

69

4229

70

4353

71

4479

72

4608

73

4735

74

4866

75

4998

76

5132

77

5269

78

5407

79

5546

80

5687

81

5830

82

5974

83

6121

84

6271

85

6422

N.B. The Rule for Practice is this: Observe the Number of the Seconds of Time that you see the Fire or Light, either ascending or descending, in the former Table; with its corresponding Number of Feet. Take this Number of Feet out of the entire Number 6440, and keep the Remainder. For where that Remainder is found in the latter Table, you will find the true Distance over against it, e. g. Suppose the Light or Fire is observ'd to take up 12″. or a fifth Part of a Minute, in its visible Ascent or Descent. The corresponding Number of Feet in the former Table is 2318, which deducted from 6440, leaves 4122, for the Difference: Which Number in the latter Table corresponds to somewhat above 68′. and shews that the real Distance sought, is somewhat above 68 Minutes, or geographical Miles. The Demonstration is easy from the former Scheme. For DB − DC = CB, and so DA − DC = CA. or the Difference of the Secant of 1°. 25′. and of any Part of it visible in another Horizon, as at F, is equal to the Secant of that Angle DAF, or of the Arch BF, which is the Distance required. Only if the Bottom of the Atmosphere be too thick to permit the Light or Fire to be seen to any certain Altitude, allowance must be made for the same, in the use of these Tables.

XII. When a Sound and a Light are made at the same Place, either at the same time, or at any given Interval; the Distance of such Sound and Light from the Auditor or Spectator may be exactly determin'd.

For if they are made at the very same time, the Difference of the Velocity of Light, which is, physically speaking, instantaneous, and of Sound, which goes 12 geographical Miles in a Minute, will, with great Exactness, determine that Distance. And if there be a given Interval between them, it is easily allow'd for.

XIII. If the Longitude or Latitude of one Place be known, and the Distance and Position of another be also known; the Longitude and Latitude of this other Place is known also.

This is too obvious to need a Demonstration; and may be easily shew'd on a Map, with a Pair of Compasses, apply'd to the Scale of that Map.

XIV. If the Longitude or Latitude of one Place be known, and its Distance from another be known also, and the Longitude of that other Place be otherwise known, its Latitude is thereby known. And if its Latitude be otherwise known, its Longitude is thereby known also.

This is also too obvious to need a Demonstration; and may be shewed on a Map, as well as the foregoing.

XV. Hulls of Ships, without Sails or Rigging, may be fixed at Sea in all ordinary Cases, by Anchors; and in extraordinary Cases, where the Ocean is vastly deep, by Weights let down from the Hulls quite through the upper Currents into the still Waters below, as near as possible to the Bottom.

This Matter belongs to Tryal and Experimenrs, and is not to be here particularly demonstrated. Only we may observe, that the lower Parts of the Waters in the Ocean are commonly found to be free from the Currents and Motions of the higher Parts; and that the Method by which those very Currents are discovered, is no other than by thus letting down the Lead far below them; which, tho' it touch not the Bottom, yet makes the Boat out of which the Lead is thrown, in the Words of an Eye-Witness, [*] ride as firmly as if it were fastened by the strongest Cable and Anchor to the Bottom.

N.B. If any Current or strong Wind does, in some measure, carry away such an Hull, with any ordinary Lead, or Weights, care is to be taken that the Cord or Chain be upward as small, and make as little Resistance to the flowing Water as possible; and that the same Cord or Chain with its Weights or Leads below, be as large and cumbersome, and make as great Resistance to the still Water below, as possible: that so the Motion of the Hull may be insensible. Note also that in case there appear still some Motion in the Hull, the Mariners are nicely to observe its Velocity and Direction; and at convenient Seasons to bring it back again, as near as possible, to its original Station.

N.B. These Hulls may be fixed in proper Places as to Latitude, by the known Methods of observing the Latitude; and as to Longitude, by Eclipses of the Sun, or Moon, or of Jupiter's Planets, or by the Moon's Appulse to fixed Stars; or rather by an actual Mensuration of Distances on the Surface of the Sea by Trigonometry, just as Monsieur Picard and Cassini measur'd the Length of a Degree of a great Circle on the Land; while the Light to be thrown up from the Ships will afford the same Advantage that any elevated Mark does at Land, and while the vastly greater Length of a Basis or measur'd Line on the Shore, the vastly greater Distances of the Ships; and the much greater Evenness of the Surface of the Sea than of the Land, do give us hopes of more Exactness in this Way of Mensuration than in any other.

N.B. By the same Method, if done with sufficient Accuracy, we may also hope to discover the Quantity of a Degree in all sorts of great Circles, and perhaps more exactly than even Monsieur Picard or Cassini have been able to do: because we may hereby actually measure a much larger Portion of such great Circles than they could. Which Advantage of this Method is in itself very considerable also.

XVI. If the Altitude of the Sun, at the best Advantage, can be taken within four Minutes of a Degree at Sea or Land; the time is thereby determined to about half a Minute: if to two Minutes of a Degree, the time is determin'd to about a quarter of a Minute, even in our Latitude; while nearer the Equator the like Limits determin the time still more exactly.

This the Astronomers well know: and any that observe in common Quadrants how an Hour, in the middle between Noon and either Morning or Evening, contains usually about 7 or 8 Degrees of Altitude; while no less than 15 Degrees makes an Hour upon the Equator, will easily agree to this Proposition.

XVII. The best time for the exact Discovery of the Hour at Sea, and of adjusting all Watches or Movements to shew the same afterwards, is that of the rising and setting of the Sun; that is, in case Allowance be made for the Horizontal Refraction of his Rays; but not otherwise.

For if the time while the whole Body of the Sun is rising or setting, which may be very nicely observ'd at Sea, be added at Night to, and substracted in the Morning from, the Time that any Table of its rising and setting, or a particular Trigonometrical Calculation, does determine; the Sum in the first, and Difference in the second Case will give the true Time when the Sun's Center will appear to be in the very Horizon. And this because the Sun's Horizontal Refraction is observ'd to be very nearly equal to his apparent Diameter.

N.B. The Exact time of the Sun's rising and setting, at all Declinations, and in all Latitudes, is found by the following Rule of Trigonometry.

Out of half the Sum of the Complement of the Sun's Declination, and of the Complement of the Latitude of the Place, and of an Arch of 90°. deduct severally the two former Sides, to gain two Differences. Then say,

As the Rectangle of the Sines of thofe two former Sides: to the square of the Radius:: so is the Rectangle of the Sines of those two Differences: to the Square of the Sine of half the Angle at the Pole, included between the same two Sides, which Angle is the Measure of the Time.

For Example. Suppose an Hull of a Ship was fix'd in the Latitude of London, and there were occasion to compute exactly the Time of the rising and setting of the Sun, for the longest Day of the Year. The Calculation is thus.

Compl. of Declin.

66°

31′

Compl. Lat.

38

30

Quadrant.

90

00

Sum

195

01.

Half

97°

30′

30″

Deduct.

66

31

00

First Differ.

30

59

30

Deduct from the former Half

38

30

00

Second Differ.

59

00

30

Log sin.

66°. 31′. 00″.

9.9624527.

A

Log. sin.

38. 30. 00

9.7941496.

B

A + B

19.7566023.

C

Log. rad. square

20.0000000.

D

Log. sin. First diff.

9.7117341.

E

Log. sin. Second diff.

9.9331794.

F

E + F

19.6449135.

G

D + G

39.6449135.

D + G − C

19.8883112.

½ = 9.9441556. or

(61°. 33′. 44″.

Its double

123°. 7′. 28″. or

(8h. 12′. 30″.

Note also that the Amplitude cannot be exactly taken even at Sea, without the like Allowance for Refraction. And the Difference of Amplitude, when the first Edge of the Sun touches, and the last leaves the Horizon, is to be added or substracted in this Case, to that when it appears to be half set; in order to obtain the Sun's true Amplitude: as well as we added or substracted the Difference of time before, for the exact Adjustment of the true Moment of its rising and setting.

The Solution of the Problem.

Let a great Gun, with a Shell that will take Fire at its utmost Altitude, be discharg'd perpendicularly 6440 Feet high above the Surface of the Sea, every Night exactly at 12 a-Clock, at all convenient Distances and Situations, and from known Places. This Discharge will, by the Distance and Point of the Compass of its Sound, nearly give the Longitude and Latitude to all Places or Ships within the hearing thereof: And it will, by the same Distance and Azimuth of its Light or Fire, exactly give the same Longitude and Latitude to all Places or Ships within the Sight thereof; according to the foregoing Lemmata. Q.EI.

For Example: Let us suppose an Hull fixed in a known place, 30 Degrees more Westward than the Meridian of London; and that every Midnight its Great Gun is discharged, as before; and that a Ship sailing by at 54′ 40″ after Eleven, sees the Fire or Light 30′ above the Horizon; i. e. by a foregoing Table, at 60 Minutes, or Geographical Miles distance. It was therefore 12 a Clock at the Hull, when it was only 11 h. 55′ at the Ship. So that the difference of time is 5′ and the difference of Longitude upon the Equator 1° 15′ and the Ships Longitude from the Meridian of London is hereby known to be 31° 15′ Westward.

Suppose also that the Weather be such that only the sound can be heard, and that it proves to be so weak as to be justly esteem'd 60 Geographical Miles, or one Degree of a great Circle, distant. This Distance answers to 5′ of time, for the interval of the Propagation of the Sound: which therefore, if it be heard just at 12 a Clock at the Ship, will imply that when the Explosion was made it was at the Ship only 55′ past eleven, the same moment that it was full 12 at the Hull; and that therefore the difference of Meridians is the same as before, viz. 5′ in time, or 1° 15′ upon the Equator, Westward.

Suppose farther, that the Light be seen at the same time that the Sound is heard; with no other than the small difference of the slowness of the Sound, in comparison of the instantaneous motion of the Light; and that the difference of time between the most elevated appearance of the Light and the hearing of the Sound; (which may be easily and exactly observ'd by any tolerable movement whatsoever, or by a Pendulum, that vibrates half Seconds:) be found to be 4′ 40″ or, which is all one, that the intire difference of the Explosion made, and of the Sound heard, be 5′ in time. This difference will imply the distance of the Ship from the Explosion to be 60 Geographical Miles. And if the sound is heard at the Ship 54′ 40″▪ after Eleven, the difference of Longitude upon the Equator will still be 1° 15′ and the real Longitude from London will be 31° 15′ Westward, as before.

If the Azimuth of the Fire or Light be also observ'd, take with your Compasses from your Scale the true distance, 60 Minutes, or Miles, and set it from the place of the Hull, on the true Angle, in any large Map or Sea-Chart. This will determine the very Point where the Ship is, both for Longitude and Latitude. The same thing may be done for the Sound, in case the Point of the Compass be observ'd also.

If the Latitude of the Ship be known, take the known distance, either by the means of the Light or Fire seen, or, if the Weather be too Foggy for that, of the Sound heard; and let it cross the known parallel of Latitude in the Chart; and this will determine the Longitude. The like is to be done for the Latitude, were the Longitude first known. But that not being the usual case, it needs not be farther insisted on. Nor need we shew how all this may be done by Calculation also: since those that understand any thing of Navigation cannot be to seek therein.

N.B. In case some parts of the Ocean prove so very deep and rough that no Hulls can be fixed in them, the way to recover the Longitude, which may be by this means interrupted, is rightly proposed by Sir Isaac Newton himself, in his Paper delivered in to the Committee of the House of Commons, viz. to sail obliquely from the last Hull into the Parallel of the next, and so along the same; till upon approaching to that next Hull the Longitude be anew recover'd, and the Voyage be continued as before. Nor is this Interruption of any consequence; because it cannot happen but in places where there is no Danger; and where Seamen are under no concern for the knowledge of the Longitude.

Notes

*
Philos. Transact. No. 247. Sir Isaac Newton's Princip. Edit. 2.
•
. 343, 344.

↵

*
Philos. Transact. p. 156, 201, 247.
↵

*
Halley ap Transact. Philosph. No 179. Mr. Anderson's Gunn. passim.
↵

*
Ubi Supra.
↵

*
Philos. Trans. No. 36. Abridg Vol. 3. p. 555, 556.
↵

OBSERVATIONS.

(1.) IF in all proper Roads of Ships such a great Gun be plac'd and discharg'd, exactly every Midnight, whether on Shoars, or Islands, or in Hulls, at the Distances of about 600 Geographical Miles or 10 Degrees, All Ships that sail within any tolerable Distance may commonly every Week or Ten Days thereby correct their Reckoning, and know their Longitude, as well as Latitude, even when the Heavens are not clear enough to make Celestial Observations for either.

(2.) The Ordinary Watches, Movements, or Log-Lines in Ships, when thus Corrected and Adjusted, once in a Week or Ten Days, will well enough shew the Longitude during every one of those short Distances between the Hulls; and so will render the knowledge thereof still more universal.

(3.) If one such Row of Hulls be any where found too defective, a double Row may there be laid, Pair by Pair, in the same [or equidistant] Meridians; with proper Distinctions in the Sounds, or the Light, to prevent mistaking one Row for the other. Nor will there, in this Case, be room for almost any Uncertainty, since even in Cloudy Weather, as much as the Wind carries away the Sound of any one, so much will it usually bring the Sound of another.

(4.) If it be any where necessary, Masts may be Erected upon Hollow Empty Vessels, with White Spheres at their Tops; and these Vessels may be fix'd in proper Places, at equal Distances between the Hulls, for the more sure guiding the Ships in Places of Danger. And since from the Top of any Mast that is 88 Feet High, the Top of another as high may be discovered at the Distance of 20 Miles, there will hardly be occasion for more of these Vessels any where than one every 120 Miles: Nor will these Masts and Vessels be any Annual Charges at all.

(5.) Besides the great Guns, and their solemn Explosions, In proper Places, at several Havens, where there is any Danger at the Entrance of Ships, as well as at other convenient Promontories jetting out into the Sea, a Rocket may be thrown up from the Top of a Neighbouring Steeple, or Hill, or the like most elevated Place every Midnight, for the Seamens better Direction and Security.

(6.) Signals of all Sorts may be given by this Method, by mutual Agreement. As suppose in Storms we would know which way, and how strong the Wind is at the nearest Explosion, &c. Ships may thus give Signals of Distress to the Hulls, or to one another. The News of great Events may be also this way carried very soon over the Sea; especially, if any Ships were plac'd within Sight and Hearing of each others Signals, as a Fleet may sail in Times of Peace, &c. In short, no one knows how far this Method of Communication by these Kinds of Signals may be improv'd; and how great a Convenience may hence arise to the several Parts of the Globe, especially in the Way of Trade and Commerce; and even for the Propagation of Knowledge both Divine and Human throughout the World.

(7.) If in any clear Night a sufficient Number of such Explosions were made at proper Distances in any Country, and convey'd in order from one to another; so that the Second Gun were fired when the Light of the First was seen, or otherwise; with the Observation of the exact apparent Times when they were made, when the Light was seen, what Angle, or how long that Light was above the Horizon, and what Azimuth it had; both the Longitude and Latitude, as well as the Distance and Position of all these Places, might by this means be readily determin'd at Land; especially if the Experiments were repeated several times, and were compar'd one with another. And by the same Observations every where, the Longitude and Latitude, Distance and Position, of all other Neighbouring Places from those, and so from another, might be readily determin'd also.

N.B. This Method of Survey is no hard Thing in Practice, even to those that know little of the Mathematicks: For any Right Angle, set by a Plummet or Level, with Two Pins or Points, for the Eye and for the Object, does by the Proportion of its Sides give the Angle above the Horizon; by the Angle its Horizontal Side makes with the Meridian, it gives the Azimuth; and by the Interval of Time between the Light and Sound, it gives the Distance of every Place from that of Explosion, according to the Figure following.

Where AB represents the Horizontal Side of the Norma or Square, and BC the Perpendicular Side; whose Proportion once given, as suppose 80 to 35, the Discovery of the Angle of Elevation CAB is most easy, as here 23° 38′ Where also NS. represents the Meridian Line lying from North to South; and the Angle SAB, suppose of 32 Degrees, 15 Minutes, represents the Azimuth, Eastward. We need not add that a Plummet of 9•8 Inches long will vibrate half Seconds, for the Interval of Time between the Light seen and the Sound heard. Nor is there, we think, any way yet discover'd of surveying Countries and Kingdoms that can compare with this, either for Expedition, Certainty or Cheapness. A Specimen of which we hope soon to give the World in an Actual Survey of Great-Britain and Ireland, and their Coasts hereby; if the Publick please to give us Encouragement and Assistance therein.

(8.) The way of casting a Shell 6440 Feet high is very easy: for the same Force or Charge of Gun-powder that will cast any Shell 12880 Feet for its utmost Random, at the Elevation of 45 Degrees, will certainly cast the same Shell perpendicularly upward 6440 Feet, as we have already observ'd. And since the time of this entire perpendicular Projection, or Ascent and Descent together, is 40″, or some very small Matter more; on Account of the Resistance of the Air; We have another sure Way of Adjusting the same Projection to that Height; viz. by observing what Quantity of Powder will cast the Shell high enough to stay very little above 40″ in the Air, before it falls to the Ground.

(9.) This Method of firing Powder, or other combustible Matter, at, or very near the utmost Height of 6440 Feet, may be well enough put in practice, even tho' some considerable Error should be committed in the adjusting the Fusee to give Fire in 20″. Since the Mistake of even a Fifth Part, or Four entire Seconds, in that Case, would produce but an Error of the 25th Part of the whole Altitude: and the Mistake of a Tenth Part, or Two Seconds, would occasion an Alteration of no more than the Hundredth Part thereof. This is evident, because this Time belongs to the highest Part of the Projectils Motion, which is the slowest: And because the Lines describ'd by all Ascending and Descending Bodies are still in a Duplicate Proportion of the Times of such their Ascent and Descent.

(10.) If one or more Rows of such Hulls were laid in the same or Equidistant Meridians, Southward or otherways, Ships might with greater Safety than formerly go to discover those Parts of the Globe which are hitherto undiscover'd. Nor can we at present guess what Advantages may thereby accrue to the Parts of the World already discovered.

(11.) Every one of these Hulls may be Places of Observation as to the Variation of the Needle, to the Currents, to the Soil, the Fowls, the Fishes, and other Phaenomena of the several Places where they are fixed; and an excellent Means of keeping up a mutual Correspondence between the several Parts of the Globe, for all useful Purposes whatsoever.

(12.) As this Method ought to be put in Practice by the Consent of all Trading Nations; so ought every one of the Hulls employ'd therein to have a legal Protection from them all; And it ought to be a great Crime with every one of them, if any other Ships either injure them, or endeavour to imitate their Explosions, for the Amusement and Deception of any.

(13.) Since the Charges of the Powder for each Gun will be very small; since the Shells and their Contents come to no great Price neither; since the Persons employ'd in the Hulls may be in part taken out of such Places where they are maintain'd at the Publick Charge already; and so will require only some Additional Rewards, or Future Privileges for such their Service; And since the Land-Explosions, which will be much the most numerous, will be withal much the cheapest; It will appear, upon the Whole, that the Annual or Constant Expences of this Method will be comparatively very small and inconsiderable: especially if they are, as they ought to be, equally distributed among the several Trading Nations of the World.

The Advantages of this Method.

- (1.) THIS Method requires no Depth of Astronomy, no Nicety in Instruments, and but seldom any Celestial Observations at all, either as to the Latitude, or the Hour at the Ship; and so is to even the common Sailors the most Practicable.

- (2.) It does generally determine the very Place of the Ship, both as to Longitude and Latitude at once, and so is the most Compendious.

- (3.) It does generally determine the very Place to a few Miles, at the farthest; and so is the most Accurate.

- (4.) It affords Help even in Cloudy and Foggy Weather, when no Celestial Observations can be made, and the Latitude it self cannot be otherwise found, and so is the most General.

- (5.) It will frequently afford a double Observation Two successive Nights, from the same Hull to the same Ship, and so is the most secure.

- (6.) It will commonly afford a double way of Observation at the same time, by the Eye and by the Ear, which will confirm or correct one another; and so is the most certain.

- (7.) The more inaccurate Branch, by the Sound, is not only more universal than the other; but is also much more exact than any Method formerly discover'd: So that in the very worst Circumstances this way is certainly the very best.

- (8.) It is the most undoubted and exact where there is the greatest Want and Danger: And if it should at all be deficient, it is there only where there is no Danger, and hardly any occasion for knowing the Longitude, as has been shew'd already. So that on all Accounts it is plainly the most Useful and Advantageous.

APPENDIX.

IT is farther humbly propos'd to the Learned, Whether it may not be proper for all Nations, upon this Occasion, to agree upon one first Meridian, or beginning of Longitude, for the common Benefit of Geography? And whether it may not be proper, in that Case, to fix it to the Pike of Tenariff, as the most noted Place already; and as the Place whence the Highest and most generally useful Explosion must in this Method be made every Midnight continually for the Discovery of the Longitude it self?

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A new method for discovering the longitude both at sea and land: humbly proposed to the consideration of the publick. By William Whiston, ... and Humphry Ditton, ...

Whiston, William, 1667-1752.

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1106010469

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9781106010469

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CW3306963341

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0161901100

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— A new method for discovering the longitude both at sea and land: humbly proposed to the consideration of the publick. By William Whiston, ... and Humphry Ditton, ..., Whiston, William, 1667-1752., Ditton, Humphry, 1675-1715.. 78,[2]p. : ill. ; 8⁰. London :: printed for John Phillips,. 1714.. [With an appendix.] [Reproduction of original from the British Library.] [English Short Title Catalog, ESTCT39882.] [Electronic data. Farmington Hills, Mich. : Thomson Gale, 2003. Page image (PNG). Digitized image of the microfilm version produced in Woodbridge, CT by Research Publications, 1982-2002 (later known as Primary Source Microfilm, an imprint of the Gale Group).]

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