The LONGITUDE of a star (is an arc of the ecliptic from the first point of Aries , [1] to the place where the star’s circle of latitude cuts the ecliptic.

Thus, the longitude of a star such as S, (Astronomy plates, fig. 32) [2] is an arc of the ecliptic T L , included between the commencement of Aries, and the circle of latitude T M , which passes through the center S of the star, and through the poles of the ecliptic.

Longitude is with regard to the ecliptic what the right ascension is with regard to the equator. See Ascension.

In this sense the longitude of a star is nothing other than its place in the ecliptic, reckoned from the commencement of Aries.

To find the longitude of a star, as well as its latitude, the difficulty comes down to finding its inclination and its right ascension . See these two words , for knowing these last two, and also knowing the angle of the equator with the ecliptic, and the point at which the ecliptic cuts the equator, it is apparent that one will determine by the rules of spherical trigonometry alone the longitude and the latitude of the star. Therefore we have given and shown at the words Declination, Star, Ascension and Globe, the different ways to find the right ascension and the declination of stars or planets.

The longitude of the sun or of a star from the equinoxial point closest to the star, is the number of degrees and minutes there are from the commencement of Aries or of Libra , up to the sun or the star, either ahead of, or behind, and this distance can never be more than 180 degrees.

The longitude of a place, in geography , is the distance from that place to a meridian that is regarded as the prime, or an arc of the equator, included between the meridian of that place and the prime meridian. See Meridian.

The prime meridian was formerly set at the island of Hierro, the most westerly of the Canaries, and Louis XIII had so ordered it to make geography simpler; today, almost all geographers and astronomers reckon their longitudes from their meridian, that is to say, from the meridian of the place from which they are observing: this is of little consequence in itself; for it is the same to take as the prime meridian one meridian or another, and one will always have the longitude of one spot on the earth when they have the position of their meridian in relation to the meridian of some other place, such as Paris, London, Rome, etc. It is however true that if all astronomers were to agree on a common meridian, one would never be required to make the reductions that are necessary so as not to confuse modern geography. In general one may define longitude as the number of degrees of the equator contained between the meridian of a place and that of any other place that is proposed. You wish, for example, to know how far Peking, the capital of China, is distant from Paris in terms of longitude , so set Paris on the common meridian, and then extend this point to the east, while counting how many degrees are passed from the equator under this meridian, until you see Peking coming under the meridian; following the great globe of M. de Lille, [3] you will find 113 degrees from the equator, gathered between the meridian of Paris and that of Peking.

In the numeration of degrees, the Arctic pole always being towards the top, the distance that extends to the right up to 180 degrees, marks by how much a proposed place is more to the east than another. The distance that extends in the same way to the left up to 180 degrees, marks by how much more one place is to the west than another. It would be useful to name eastern longitude the degrees that are to the right of the meridian of a place, up to 180 degrees, and western longitude those that stretch to the left of the same meridian, up to the same number; but it is a universal practice to count only one progression of longitude up to 360 degrees. [4]

Longitude , in Navigation, is the distance of the vessel, or of the place where one is to another place, counted from east to west, in degrees from the equator. [5]

The longitude of two places on the sea can be estimated in four ways: either as the arc of the equator between the meridians of these two places; or by the arc of the parallel that passes by the first of these two places, and which is bounded by the two meridians; or by the arc of the parallel between these two meridians, and which passes through the second of these two places; or finally by the sum of the arcs of different parallels found within the different meridians that divide the space contained between the two meridians. Hence by whichever manner one chooses it is always necessary to estimate the distance of the meridians in degrees, and it appears more useful to mark this by the degrees at the equator than in any other way. But it must be noted that these degrees do not at all give the distance between the two points: for all these arcs, whether of the equator or of the parallels contained between the same two meridians, all have the same number of degrees, and all the places situated under these meridians have the same difference of longitude , but they are that much closer to one another, the closer they are to the pole; this is why it is necessary to remember this in calculating the distances of places whose longitudes and latitudes are the same, and seamen have tables specifically intended for that.

The search for an exact method to determine longitudes at sea, is a problem that has much occupied mathematicians of the past two centuries, and for the solution of which the English have publicly offered great rewards: [6] there have been unsuccessful efforts to bring this about, and different methods have been postulated, but without success; the projects have always been found to be bad, envisioning operations that are too impractical, or that are somehow flawed, so that no person has yet been able to claim the prize.

The approach that most have proposed, is to determine the difference in time between any two points on earth: for one hour is equal to 15 degrees at the equator; that is, 4 minutes in time for each degree at the equator, 4 seconds of time for each minute of a degree; and thus the time difference being known and converted into degrees, this will give the longitude, and the same in reverse.

To determine the difference in time, clocks, watches and other machines have been used, but always in vain, having only, among all the instruments meant to mark time, only the pendulum can be exact enough for this purpose, and the pendulum cannot be used at sea. Others with sounder, and more likely to be successful views, search in the skies for the means by which to determine longitude on earth. In effect, if one knows for two different places the exact time of the appearance of some celestial phenomenon, the difference in these two times will give the difference in longitude between these two places. For we have in ephemerides the movements of the planets, and the times of all celestial phenomena, such as the beginnings and ends of eclipses, and the conjunctions of the moon with the other planets in the ecliptic calculated for a certain place. If one could then observe exactly the hour and the minute at which these phenomena occur at any other place whatsoever, the difference in time between those moments and that listed in the tables being converted into degrees, that would give the difference in longitude between the place where one made the observation and that of the place upon which the tables were built.

The difficulty consists not in finding the exact time, that comes from observing the height of the sun; but what is missing is a sufficient number of appearances that can be observed, for all these slow movements, for example those of Saturn, are first of all left out, because a small difference in appearance can only be perceived over a great stretch of time, and as what happens here is that the phenomenon varies detectably by at least two minutes, an error of two minutes in time produces one of thirty miles in longitude . Hence among the phenomena found in this case, those that have appeared most suited to this objective, are the different phases of lunar eclipses, the longitude of this heavenly body or its place in the zodiac, its distance from fixed stars, or the movement where it joins to them, and the conjunction, the distance and the eclipses of the satellites of Jupiter: we will speak of each of these methods one after another.

1º. The method by lunar eclipses is very easy, and would be exact enough if there were eclipses of the moon every night. At the moment that we see the start or the middle of a lunar eclipse, we have only to take the height or the zenith of some fixed star, and deduce its time, this supposes that we already know the latitude, and then all that remains is to solve for a spherical triangle whose three sides are already known, the first is the distance from the zenith to the pole, complement of the latitude; the second is that of the star at its zenith, complement of the star’s height; the third, that of the star at the pole, complement of the star’s declination, for one will derive from there the value of the angle formed by the meridian and the circle of declination passing through the star, which when adjusted for the difference of the right ascension of the sun and the astral body for that day, will give the distance of the sun to the meridian, or the time that one is searching for, that is to say, the hour of the day at the moment and at the place of observation; one will have no need even to know the height of the star, if the star is on the meridian. In effect, the time at the moment of observation will then be given by only the difference between the right ascension of the eye and of the star for that day, converted into time; this moment that one will have found in this manner, on being compared to those that are listed in the tables for the same eclipse, will give us the longitude. See Eclipse.

2º. The place of the moon in the zodiac is not a phenomenon that has, like the former, the fault of being something that is observed only rarely, but on the other hand the observation of it is difficult, and the calculation is complicated and burdened because of the two parallaxes to which one must pay attention; to the point that one can scarcely make use of this phenomenon with the least assurance, to determine longitudes . It is true that if one waits for the moon to pass the meridian of the place, and one then takes the height of some notable star (one supposes that one already knows the latitude of the place) the latitude will deduce the time exactly enough, although it would be still better to use for this the observation of some stars situated on the meridian.

The time being determined, it will be easy to know which point of the ecliptic then passes through the meridian, and by that we will have the place of the moon in the zodiac corresponding to the time where we find ourselves; we then search in the ephemerides at what hour of the meridian of the ephemerides the moon should be found at the same point of the zodiac, and we will then have the times of the two places at the same instant, so that their difference converted into degrees of the great circle, will give us the longitude .

3º. As it often happens that the moon should be observed at the meridian, astronomers have for this reason turned their gaze towards another, more frequent phenomenon by which to deduce longitudes , which is the occultation of fixed stars by the moon; in effect, the entry of the stars into the disc of the moon, or their exit from behind this disc, could determine the true place of the moon in the sky at the given moment of observation; but the parallaxes to which it was necessary to pay attention, these spherical oblique angled triangles that had to be solved for, and the variety of cases that can present themselves, render this method so difficult and so complicated, that mariners have made only very little use of it to the present. Those who would wish to use it will find a very great aid in the zodiac of the stars, published through the efforts of doctor Halley, and which contains all the stars whose lunar occultation can be observed. [7]

But despite the little use made up to now of this method, the majority of this century’s most able astronomers believe that observation of the moon is perhaps the most exact way to determine longitudes . According to them, it is not necessary to observe the occultation of stars by the moon to mark a determined point in time; the movement of the moon is so rapid, that if one relates its position to two fixed stars, it forms with these stars a triangle that, constantly changing its shape, can be taken for an instantaneous phenomenon, and to determine which moment at which one makes the observation. There is no more hour of the night, there is no other hour in which the moon and the stars are visible, that offers our eyes a like phenomenon, and we can, by the choice of stars, by their position, and by their splendor take from among all the triangles that which will appear the most suitable for the observation.

Now, to come to the knowledge of longitudes , two things are required, the first is that at sea, one observes with sufficient exactitude the triangle formed by the moon and the stars, and the other is that one knows exactly enough the movement of the moon to know what hour will be marked by the pendulum regulated at the place from which one departed, when the moon forms with the two stars the triangle that one observes. One can make the observation pretty exactly, because at sea one knows exactly enough the time at the place where one is, and moreover one has had for some years an instrument with which one can, despite the vessel’s motion, take the angles between the moon and the stars with sufficient precision to determine the triangle of which we speak. The difficulty comes down to the theory of the moon, to knowing exactly enough its distances and its movements to be able to calculate at any moment its position in the sky, and to determine at what instant in this or that place, the triangle it forms with the two fixed stars will be this or that. We will not in any way conceal that in this lies the greatest difficulty. This heavenly body that has been given to the earth as a satellite, and which seems to promise it the greatest usefulness, gets away from the uses to which we wish to put it, by the irregularity of its path: however, if one thinks of the progress that has been made over time on the theory of the moon, one would not know how to keep oneself from believing that the time is near when this heavenly body that dominates the sea, and causes its ebb and flow, will teach mariners how to find their way across it. Préface du traité de la parallaxe de la lune  [8] by M. Maupertuis. One will see in the article Moon details of the work done by the ablest geometricians and astronomers on such an important matter.

It must be admitted that this method to discover longitudes will demand more learning and more care than should have been necessary for it, had one been able to find clocks that, at sea, would keep the regularity of their movements; but it will be for mathematicians to take upon themselves the trouble of the calculations; provided that one has the elements upon which the method is based, one will be able, with either tables or instruments, to reduce to something very easy the execution of a difficult theory.

However, prudence demands that at the beginning one make only the most circumspect use of these instruments or tables, and that in using them one does not neglect any of the other methods by which one estimates longitude at sea; long usage will lead one to know the certainty.

As the locations of the moon are different from different points on the earth’s surface, because of the parallax of this planet, it will be necessary in the observations that one will make of the moon’s locations, to be able to reduce these locations one to another, or to the place of the moon as viewed from the center of the earth. In his Discours sur la parallaxe de la lune , from which we have taken some of the preceding, M. de Maupertuis provides for this some methods that are very elegant, and more exact than any that have been published up to now. See Parallax.

4º. In general, when trying to determine longitudes on land, it is preferable to make observations of the moons of Jupiter than of the moon, because the former are less subject to parallax than the others, and furthermore they can always be made easily whatever is the position of Jupiter on the horizon. The movements of the satellites are quick and should be calculated for each hour; hence to discover the longitude by means of these satellites, you will observe with a good telescope the conjunction of two of them or of one of them with Jupiter, or some other suitable appearances, and at the same time you will note the hour and the minute for the observation of the meriodional height of some stars. Then, by consulting tables of satellites, you will observe the hour and the minute at which this appearance should occur at the meridian of the place for which the tables were calculated, and the difference in the time will give you, as above, the longitude . See Satellites.

This method of determining longitudes on land is as exact as one could wish, and since the discovery of the satellites of Jupiter, Geography has made very great progress for this reason; but it cannot be used at sea. The length of the lenses needed up to now to be able to observe the disappearance and reappearance of the satellites, and the smallness of their fields of vision, mean that at the smallest agitation of the vessel one loses sight of the satellite, assuming that one had been able to find it. The observation of lunar eclipses is more practicable at sea, but it is much less good for knowing longitudes , because of the uncertainty about the precise time at which the eclipse begins or finishes, or is at its mid-point; which necessarily produces uncertainty in the resulting calculation of longitude .

The methods that have for their foundations the observation of celestial phenomena all have this defect, that they cannot be used every day, because the observations cannot be made in all weathers, and apart from that they are difficult to perform at sea, because of the movements of the vessel; it is for this reason that there are mathematicians who have abandoned methods that the moon and satellites can provide; they have recourse to clocks and other instruments of this type, and it must be admitted that if they are able to make these accurate enough and perfect enough so that they can run precisely with the sun without advancing or falling back, and otherwise without either the heat or cold, the air, and the different climates bringing about any alteration, one could in that case have the longitude with all imaginable exactitude; for one would have only to set one’s pendulum or clock by the sun at the moment of departure, and when one wanted to have the longitude of a place, no more would be required than to examine in the sky what hour and minute it is; this is done at night by means of the stars, and during the day by means of the sun: the difference between the time being thus observed, and that of the machine, would clearly give the longitude . But such a machine has not to this day been invented; this is why one still has recourse to other methods.

M. Whiston [9] has conceived a method for finding longitudes by the flash and report of large cannon. Sound, as is known, moves fairly uniformly in all its waves, whatever the sounding body from which it emanates, and the medium through which it is transmitted. If one then fires a mortar or a large cannon at a place whose longitude is known, the time between when the flash, which moves as in an instant, will be seen and the sound, which moves at the pace of 173 toises per second, [10] is heard, will give the distance of two places from each other; thus, supposing one knows the latitude of the places, one could by this method arrive at an understanding of the longitude . See Sound, etc.

Furthermore if the hour and the minute at which one fires the cannon are known for the place at which one fires it, then observing, by the sun and the stars, the hour and the minute at the place for which one seeks the longitude , and where we suppose one can hear the cannon even without seeing it, the difference in these two times will be the difference in longitude .

Finally, if the mortar is loaded with a hollow ball or a type of bomb filled with combustible material, and if one aims it vertically, it would carry its charge to a mile in height, and one could then see the flash from nearly a hundred miles away. If one were then to find oneself in a place from which one could neither see the flash of the cannon, nor hear its report, one could nevertheless derive the distance from where one was, to that from which the mortar was pointed, by the height above the horizon to which the bomb would climb; hence, once the distance and the latitude were known, the longitude could easily be determined.

Following this idea, one could propose placing these mortars at set distances, and in known locations, along all coasts, islands, capes, etc., that are frequented, and to fire them at certain set times during the day for the use and advantage of mariners.

This method, which could be very satisfying in theory, is however entirely useless, because it is very clumsy and at the same time assumes too much. It supposes, for example, that the report could be heard from 40, 50 or 60 miles, and it is true that there are examples of this, but these cases are very rare, and normally the report of a cannon is not heard over, at most, half that distance, and sometimes much less. It also supposes that the sound always moves at the same speed, in place of the fact that its speed can rise or fall depending on whether it is moving in the same direction as the wind, or against it.

It is true that following some experiments the wind does not at all change the speed of sound; but these experiments need to be repeated very many times so that one could derive general rules; and there are also cases that appear contrary, as one often hears bells when the wind drives the sound to one’s ears, and one ceases to hear them when the wind is against them.

Finally, this method supposes that the propellant force of the gunpowder is uniform, and that the same measure will always lift the same ball to the same height; but there is not a single cannoneer who does not know the opposite. We say nothing of cloudy and dark nights when one cannot see the moon at all, or of stormy nights when one cannot hear the report at all, even at very short distances.

This is why sailors are reduced to very imperfect methods for finding longitude : what follows gives a general idea of the principal methods. They estimate the path the vessel has taken from the point from which they wish to reckon the longitude , which is something that up to now only be done with very imperfect instruments. They observe the latitude of the place to which the vessel has come, and they compare that to the latitude of the other place to know by how much they have moved in terms of latitude; and knowing more or less the rhumb line [11] of the wind before which they have run during that time, they determine through the combination of these different elements the difference in longitudes .

One sees well enough how many suspect elements come into this calculation, and how the research into longitude in this regard is still far from the perfection that one desires.

One can also use the declination of the compass to determine longitude at sea. See for this M. Bouguer’s Traité de Navigation,  [12] p. 313 , along with the methods most used by mariners to determine longitude .

Translator’s notes

1. The “first point of Aries” is the celestial location of the vernal equinox. When defined by Hipparchus in 130 BC it was in the constellation Aries. It is one of the two points at which the sun crosses the celestial equator during its year-long progression through the constellations and defines the geographical coordinate 0̊, 0̊, the opposite point being the First Point of Libra, exactly 180̊away.

2. Row 3, left, of the Encyclopédie ’s Planche III of Recueil de planches sur les sciences, les arts libéraux, et les arts méchaniques, avec leur explication. Astronomie .

3. Guillaume de l’Isle (1675-1726) was an extremely active French geographer and cartographer. A modern reproduction of his terrestrial globe from about 1700, Totius mundi / adornata juxta observationes dun Academiæ Regalis Scientiarum et nonnullorum alorum, secundum annotationes recertissimus edta per Guilielcum de L’Isle, celeberrinum regis gallia geomaphum is held by the Library of Congress, under call number G3170 1700.L51.

4. See d’Alembert’s article Degree.

5. In this instance the word “equator” must be understood as representing a great circle passing through the two poles (the technical definition of “meridian”) as the earth’s equator was understood even in d’Alembert’s day as being perpendicular to the axis of rotation and midway between the North and South Poles.

6. The British Parliament passed the Longitude Act in 1714; this act established the Board of Longitude, which encouraged the development of working devices to determine longitude at sea and offered substantial prizes ranging from £10,000 to £20,000 (depending upon its accuracy) to any person who could demonstrate the success of such a device (see http://cudl.lib.cam.ac.uk/view/MS-RGO-00014-00001/19). The most successful device was developed and refined by John Harrison (see http://cudl.lib.cam.ac.uk/view/MS-H-17809), whose series of chronometers had, by the early 1760s, essentially solved the problem of fixing time at sea and served as the model for all subsequent mechanical marine timekeeping devices. A prize of £20,000 in 1714 was an enormous sum of money, almost impossible to render in contemporary terms: the Bank of England inflation calculator suggests that £20,000 in 1714 would be worth £3.58 million today (see http://www.bankofengland.co.uk/education/Pages/resources/inflationtools/calculator/default.aspx), but this calculation takes no account of purchasing power in a period of sustained low inflation, no income taxes and a minimal state. Other calculations yield amounts ranging from £2 million to several hundred million pounds. As a comparison, in the mid-18th century Royal Navy, the captain of a first-rate warship (that is, the commanding officer of one of the largest and most complex sailing vessels afloat at the time) received pay for time at sea of 20 shillings (£1) a day, or £365 a year (N.A.M. Rodgers [1996] The Wooden World: An Anatomy of the Georgian Navy. New York & London: W.W. Norton, page 252).

7. Edmond Halley (1679) Catalogus stellarum Australium; sive, Supplementum catalogi Tychonici, exhibens logitudines & latitudines stellarum fixarum, quæ, prope Polum Antarcticum sitæ, in horizonte Uraniburgico Tychoni inconspicuæ fuere ... ad annum 1677 completum correctas; Cum ipsis observationibus in insula S. Helenæ ... depromptis ... London: Thomas James.

8. Pierre Louis Moreau de Maupertuis (1741) Discours sur la parallaxe de la lune, pour perfectionner la théorie de la lune et celle de la terre. Paris: Imprimerie royale.

9. William Whiston (1667-1752) was an English theologian and mathematician who succeeded Isaac Newton as Lucasian Professor of Mathematics at Cambridge University. He supported passage of the Longitude Act of 1714 (see Note 5, above) and in 1714, with Humphry Ditton, published A new method for discovering the longitude both at sea and land: humbly proposed to the consideration of the publick. London: Printed for John Phillips.

10. A toise was a measure equal to 6 French feet, or 6.39 English feet, the French foot being equal to 1.066 English feet. Therefore, the speed of sound as given by d’Alembert would be equivalent to 1,038 French feet/second, or 1,106 English feet/second. The modern standard accepted speed of sound in dry air at sea level at 20̊C is 343.2 meters/second, or 1,126 feet/second, so d’Alembert’s measurement was within 1.8% of the modern standard.

11. A “rhumb line” is a path across the earth’s surface (in practical terms for navigation, on the open sea) with a constant bearing relative to true or magnetic north. Before longitude could be precisely calculated, mariners would sail to the known latitude of their destination, and would then sail east or west along that latitude (that is, they would maintain the same bearing relative to true or magnetic north) until they reached their destination.

12. Pierre Bouguer (1753) Nouveau traité de navigation, contenant la théorie et la pratique du pilotage. Paris: H.L. Guerin.