LONGIMETRY

, the art of measuring lengths or distances, both accessible and inaccessible, forming a part of what is called Heights and Distances, being an application of geometry and trigonometry to such measurements.

As to accessible lengths, they are easily measured by the actual application of a rod, a chain, or wheel, or some other measure of length.

But inaccessible lengths require the practice and properties of geometry and trigonometry, either in the measurement and construction, or in the computation. For example, Suppose it were required to know the length or distance between the two places A and B, to which places there is free access, but not to the intermediate parts, on account of water or some other impediment; measure therefore, from A and B, the distances to any convenient place C, which suppose to be thus, viz, AC = 735, and BC = 840 links; and let the angle at C, taken with a theodolite or other instrument, be 55° 40′. From these measures the length or distance AB may be determined, either by geometrical measurement, or by trigonometrical computation. Thus, first, lay down an angle C = 55° 40′, and upon its legs set off, from any convenient scale of equal parts, CA = 735, and CB = 840; then measure the distance between the points A and B by the same scale of equal parts, which will be found to be 740 nearly. Or this by calculation,

 840, its half 62° 10′,
 735
Sum15751.1972806
Dif. 1050.0211893
Tang.  62° 10′10.2773793
Tang.   7 11 11/149.1012880
s. Sum or [angle] A = 99° 21′11/149.9711092
to s. [angle] C = 55 409.9168593
So BC = 8400 9242793
ToAB = 741.20.8699404

For a 2d Example—Suppose it were required to find the distance between two inaccessible objects, as between the house and mill, H and M; first measure any convenient line on the ground, as AB, 300 yards; then at the station A take the angles BAM = 58° 20′; and MAH = 37°; also at the station B take the angles ABH = 53° 30′, and HBM = 45° 15′; from hence the distance or length MH may be found, either by geometrical construction, or by trigonometrical calculation, thus:

First draw a line AB of the given length of 300, by a convenient scale of equal parts; then at the point A lay down the angles BAM and MAH of the mag-| <*>itudes above given; and also at the point B the given angles ABH and HBM: then by applying the length HM to the same scale of equal parts, it is found to be nearly 480 yards.

Otherwise, by calculation. First, by adding and fubtracting the angles, there is found as below:

37° 00′58° 20′       53° 30′
58  2053  30       45  15
53  3045  15sum 98  45 [angle] ABM
sums    148  50157  05
from    180  00180  00
[angle] AHB 31  1022  55[angle] AMB
Then,
as sin. AHB : sin. ABH :: AB : AH =465.9776,
and, as sin. AMB : sin. ABM :: AB : AM =761.4655;
their sum is  1227.4431
and their diff.  295.4879
Then as sum AM + AH : to dif. AM - AH ::
tang. 1/2 AHM + 1/2 AMH =71° 30′,
to tang. 1/2 AHM - 1/2 AMH =35  44
the dif. of which is AMH =35  46.
Lastly,

as s. , the distance sought.

LONGITUDE of the Earth, is sometimes used to denote its extent from west to east, according to the direction of the equator. By which it stands contradistinguished from the Latitude of the earth, which denotes its extent from one pole to the other.

Longitude of a Place, in Geography, is its longitudinal distance from some first meridian, or an arch of the equator intercepted between the meridian of that place and the first meridian.

Longitude in the Heavens, as of a star, &c, is an arch of the ecliptic, counted from the beginning of Aries, to the place where it is cut by a circle perpendicular to it, and passing through the place of the star.

Longitude of the Sun or Star from the next equinoctial point, is the degrees they are distant from the beginning of Aries or Libra, either before or aster them; which can never exceed 180 degrees.

Longitude

, Geocentric, Heliocentric, &c, the Longitude of a planet as seen from the earth, or from the sun. See the respective terms.

Longitude

, in Navigation, is the distance of a ship, or place, east or west, from some other place or meridian, counted in degrees of the equator. When this distance is counted in leagues, or miles, or in degrees of the meridian, and not in those proper to the parallel of Latitude, it is usually called Departure.

An easy practicable method of finding the Longitude at sea, is the only thing wanted to render the Art of Navigation perfect, and is a problem that has greatly perplexed mathematicians for the last two centuries: accordingly most of the commercial nations of Europe have offered great rewards for the discovery of it; and in consequence very considerable advances have been made towards a perfect solution of the problem, especially by the English.

In the year 1598, the government of Spain offered a reward of 1000 crowns for the solution of this problem; and soon after the States of Holland offered 10 thousand florins for the same. Encouraged by such offers, in 1635, M. John Morin, professor of mathematics at Paris, proposed to cardinal Richlieu, a method of resolving it; and though the commissioners, who were appointed to examine this method, on account of the imperfect state of the lunar tables, judged it insufficient, cardinal Mazarin, in 1645, procured for the author a pension of 2000 livres.

In 1714 an act was passed in the British parliament, allowing 2000l. towards making experiments; and also osfering a reward to the person who should discover the Longitude at sea, proportioned to the degree of accuracy that might be attained by such discovery; viz, a reward of 10,000l. if it determines the Longitude to one degree of a great circle, or 60 geographical miles; 15,000l. if it determines the same to two-thirds of that distance; and 20,000l. if it determines it to half that distance; with other regulations and encouragements. 12 Ann. cap. 15. See also stat. 14 Geo. II, cap. 39, and 26 Geo. II, cap. 25. But, by stat. Geo. III, all former acts concerning the Longitude at sea are repealed, except so much of them as relates to the appointment and authority of the commissioners, and such clauses as relate to the publishing of nautical almanacs, and other useful tables; and it enacts, that any person who shall discover a method for finding the Longitude by means os a time-keeper, the principles of which have not hitherto been made public, shall be entitled to the reward of 5000l. if it shall enable a ship to keep her Longitude, during a voyage of 6 months, within 60 geographical miles, or one degree of a great circle; to 7500l. if within 40 geographical miles, or two-thirds of a degree of a great circle; or to a reward of 10,000l. if within 30 geographical miles, or half a degree of a great circle. But if the method shall be by means of improved solar and lunar tables, the author of them shall be entitled to a reward of 5000l. if they shew the distance of the moon from the sun and stars within 15″ of a degree, answering to about 7′ of Longitude, after making an allowance of half a degree for the errors of observation, and after comparison with astronomical observations for a period of 18 1/2 years, or during the period of the irregularities of the lunar motions. Or that in case any other method shall be proposed for finding the Longitude at sea, besides those bef<*>re-mentioned, the author shall be entitled to 5000l. if it shall determine the Longitude within one degree of a great circle, or 60 geographical miles; to 7500l. if within two-thirds of that distance; and to 10,000l. if within half the said distance.

Accordingly, many attempts have been made for such discovery, and several ways proposed, with various degrees of success. These however have been chiefly directed to methods of determining the difference of time between any two points on the earth; for the Longitude of any place being an arch of the equator intercepted between two meridians, and this arc being proportional to the time required by the sun to move from the one meridian to the other, at the rate of 4 minutes of time to one degree of the arch, it follows that the difference of time being known, and turned| into degrees according to that proportion, it will give the Longitude.

This measurement of time has been attempted by some persons by means of clocks, watches, and other automata: for if a clock or watch were contrived to go uniformly at all seasons, and in all places and situations; such a machine being regulated, for instance, to London or Greenwich time, would always shew the time of the day at London or Greenwich, wherever it should be carried to; then the time of the day at this place being found by observations, the difference between these two times would give the difference of Longitude, according to the proportion of one degree to 4 minutes of time.

Gemma Frisius, in his tract De Principiis Astronomiæ et Geographiæ, printed at Antwerp in 1530, it seems first suggested the method of finding the Longitude at sea by means of watches, or time-keepers; which machines, he says, were then but lately invented. And soon after, the same was attempted by Metius, and some others; but the state of watch-making was then too imperfect for that purpose. Dr. Hooke and Mr. Huygens also, about the year 1664, applied the invention of the pendulum-spring to watches; and employed it for the purpose of discovering the Longitude at sea. Some disputes however between Dr. Hooke and the English Ministry prevented any experiments from being made with watches constructed by him; but many experiments were made with some constructed by Huygens; particularly Major Holmes, in a voyage from the coast of Guinea in 1665, by one of these watches predicted the Longitude of the island of Fuego to a great degree of accuracy. This success encouraged Huygens to improve the structure of his watches, (see Philos. Trans. for May 1669); but experience soon convinced him, that unless methods could be discovered for preserving the regular motion of such machines, and preventing the effects of heat and cold, and other disturbing causes, they could never answer the intention of discovering the Longitude, and on this account his attempts failed.

The first person who turned his thoughts this way, after the public encouragement held out by the act of 1714, was Henry Sully, an Englishman; who, in the same year, printed at Vienna, a small tract on the subject of watch-making; and afterwards removing to Paris, he employed himself there in improving timekeepers for the discovery of the Longitude. It is said he greatly diminished the friction in the machine, and rendered uniform that which remained: and to him is principally to be attributed what is yet known of watchmaking in France: for the celebrated Julien le Roy was his pupil, and to him owed most of his inventions, which he afterwards perfected and executed: and this gentleman, with his son, and M. Berthoud, are the principal persons in France who have turned their thoughts this way since the time of Sully. Several watches made by these last two artists, have been tried at sea, it is said with good success, and large accounts have been published of these trials.

In the year 1726 our countryman, Mr. John Harrison, produced a time-keeper of his own construction, which did not err above one second in a month, for 10 years together: and in the year 1736 he had a machine tried in a voyage to and from Lisbon; which was the means of correcting an error of almost a degree and a half in the computation of the ship's reckoning. In consequence of this success, Mr. Harrison received public encouragement to proceed, and he made three other time-keepers, each more accurate than the former, which were finished successively in the years 1739, 1758, and 1761; the last of which proved so much to his own satisfaction, that he applied to the commissioners of the Longitude to have this instrument tried in a voyage to some port in the West Indies, according to the directions of the statute of the 12th of Anne above cited. Accordingly, Mr. William Harrison, son of the inventor, embarked in November 1761, on a voyage for Jamaica, with this 4th timekeeper or watch; and on his arrival there, the Longitude, as shewn by the time-keeper, differed but one geographical mile and a quarter from the true Longitude, deduced from astronomical observations. The same gentleman returned to England, with the time-keeper, in March 1762; when he found that it had erred, in the 4 months, no more than 1′ 54″1/2 in time, or 28 5/8 minutes of Longitude; whereas the act requires no greater exactness than 30 geographical miles, or minutes of a great circle, in such a voyage. Mr. Harrison now claimed the whole reward of 20,0001, offered by the said act: but some doubts arising in the minds of the commissioners, concerning the true situation of the island of Jamaica, and the manner in which the time at that place had been found, as well as at Portsmouth; and it being farther suggested by some, that although the time-keeper happened to be right at Jamaica, and after its return to England, it was by no means a proof that it had been always so in the intermediate times; another trial was therefore proposed, in a voyage to the island of Barbadoes, in which precautions were taken to obviate as many of these objections as possible. Accordingly, the commissioners previously sent out proper persons to make astronomical observations at that island, which, when compared with other corresponding ones made in England, would determine, beyond a doubt, its true situation: and Mr. William Harrison again set out with his father's time keeper, in March 1764, the watch having been compared with equal altitudes at Portsmouth, before he set out, and he arrived at Barbadoes about the middle of May; where, on comparing it again by equal altitudes of the sun, it was found to shew the difference of Longitude, between Portsmouth and Barbadoes, to be 3h 55m 3s; the true difference of Longitude between these places, by astronomical observations, being 3h 54m 20s; so that the error of the watch was 43s, or 10′ 45″ of Longitude. In consequence of this, and the former trials, Mr. Harrison received one moiety of the reward offered by the 12th of Queen Anne, after explaining the principles on which his watch was constructed, and delivering this as well as the three former to the Commissioners of the Longitude, for the use of the public: and he was promised the other moiety of the reward, when other time-keepers should be made, on the same principles, either by himself or others, performing equally well with that which he had last made. In the mean time, this last timekeeper was sent down to the Royal Observatory at Greenwich, to be tried there under the direction of the Rev. Dr. Maskelyne, the Astronomer Royal. But it| did not appear, during this trial, that the watch went with the regularity that was expected; from which it was apprehended, that the performance even of the same watch, was not at all times equal; and consesequently that little certainty could be expected in the performance of different ones. Moreover, the watch was now found to go faster than during the voyage to and from Barbadoes, by 18 or 19 seconds in 24 hours: but this circumstance was accounted for by Mr. Harrison; who informs us that he had altered the rate of its going by trying some experiments, which he had not time to finish before he was ordered to deliver up the watch to the Board. Soon after this trial, the Commissioners of Longitude agreed with Mr. Kendal, one of the watch-makers appointed by them to receive Mr. Harrison's discoveries, to make another watch on the same construction with this, to determine whether such watches could be made from the account which Mr. Harrison had given, by other persons, as well as himself. The event proved the affirmative; for the watch produced by Mr. Kendal, in consequence of this agreement, went consrderably better than Mr. Harrison's did. Mr. Kendal's watch was sent out with Capt. Cook, in his 2d voyage towards the south pole and round the globe, in the year 1772, 1773, 1774, and 1775; when the only fault found in the watch was, that its rate of going was continually accelerated; though in this trial, of 3 years and a half, it never amounted to 14″1/2 a day. The consequence was, that the House of Commons in 1774, to whom an appeal had been made, were pleased to order the 2d moiety of the reward to be given to Mr. Harrison, and to pass the act above mentioned. Mr. Harrison had also at different times received some other sums of money, as encouragements to him to continue his endeavours, from the Board of Longitude, and from the India Company, as well as from many individuals. Mr. Arnold and some other persons have since also made several very good watches for the same purpose.

Others have proposed various astronomical methods for finding the Longitude These methods/chiefly depend on having an ephemeris or almanac suited to the meridian of some place, as Greenwich for instance, to which the Nautical Almanac is adapted, which shall contain for every day computations of the times of all remarkable celestial motions and appearances, as adapted to that meridian. So that, if the hour and minute be known when any of the same phenomena are observed in any other place, whose Longitude is desired, the difference between this time and that to which the time of the said phenomenon was calculated and set down in the almanac, will be known, and consequently the difference of Longitude also becomes known, between that place and Greenwich, allowing at the rate of 15 degrees to an hour.

Now it is easy to find the time at any place, by means of the altitude or azimuth of the sun or stars; which time it is necessary to find by such means, both in these astronomical modes of determining the Longitude, and in the former by a time-keeper; and it is the difference between that time, so determined, and the time at Greenwich, known either by the time-keeper or by the astronomical observations of celestial phenomena, which gives the difference of Longitude, at the rate above- mentioned. Now the difficulty in these methods lies in the fewness of proper phenomena, capable of being thus observed; for all slow motions, such as belong to the planet Saturn for instance, are quite excluded, as affording too small a difference, in a considerable space of time, to be properly observed; and it appears that there are no phenomena in the heavens proper for this purpose, except the eclipses or motions of Jupiter's satellites, and the eclipses or motions of the moon, viz, such as her distance from the sun or certain fixed stars lying near her path, or her Longitude or place in the zodiac, &c. Now of these methods,

1st, That by the eclipses of the moon is very easy, and sufficiently accurate, if they did but happen often, as every night. For at the moment when the beginning, or middle, or end of an eclipse is observed by a telescope, there is no more to be done but to determine the time by observing the altitude or azimuth of some known star; which time being compared with that in the tables, set down for the happening of the same phenomenon at Greenwich, gives the difference in time, and consequently of Longitude sought. But as the beginning or end of an eclipse of the moon cannot generally be observed nearer than one minute, and sometimes 2 or 3 minutes of time, the Longitude cannot certainly be determined by this method, from a single observation, nearer than one degree of Longitude. However, by two or more observations, as of the beginning and end &c, a much greater degree of exactness may be attained.

2d, The moon's place in the zodiac is a phenomenon more frequent than that of her eclipses; but then the observation of it is difficult, and the calculus perplexed and intricate, by reason of two parallaxes; so that it is hardly practicable, to any tolerable degree of accuracy.

3d, But the moon's distances from the sun, or certain fixed stars, are phenomena to be observed many times in almost every night, and afford a good practical method of determining the Longitude of a ship at almost any time; either by computing, from thence, the moon's true place, to compare with the same in the almanac, or by comparing her observed distance itself with the same as there set down.

It is said that the first person who recommended the finding the Longitude from this observed distance between the moon and some star, was John Werner, of Nuremberg, who printed his annotations on the first book of Ptolomy's Geography in 1514. And the same thing was recommended in 1524, by Peter Apian, professor of mathematics at Ingolstadt; also about 1530, by Oronce Finé, of Briançon; and the same year by the celebrated Kepler, and by Gemma Frisius, at Antwerp; and in 1560, by Nonius or Pedro Nunez.

Nor were the English mathematicians behind hand on this head. In 1665 Sir Jonas Moore prevailed on king Charles the 2d to erect the Royal Observatory at Greenwich, and to appoint Mr. Flamsteed his astronomical observer, with this express command, that he should apply himself with the utmost care and diligence to the rectifying the table of the motions of the heavens, and the places of the fixed stars, in order to find out the so much desired Longitude at sea, for perfecting the Art of Navigation. And to the fidelity and industry| with which Mr. Flamsteed executed his commission, it is that we are chiefly indebted for that curious theory of the moon, which was afterwards formed by the immortal Newton. This incomparable philosopher made the best possible use of the observations with which he was furnished; but as these were interrupted and imperfect, his theory would sometimes differ from the heavens by 5 minutes or more.

Dr. Halley bestowed much time on the same object; and a Starry Zodiac was published under his direction, containing all the stars to which the moon's appulse can be observed; but sor want of correct tables, and proper instruments, he could not proceed in making the necessary observations. In a paper on this subject, in the Philos. Trans. number 421, he expresses his hope, that the instrument just invented by Mr. Hadley might be applied to taking angles at sea with the desired accuracy. This great astronomer, and after him the Abbé de la Caille, and others, have reckoned the best astronomical method for finding the Longitude at sea, to be that in which the distance of the moon from the sun or from a star is used; for the moon's daily motion being about 13 degrees, her hourly mean motion is above half a degree, or one minute of a degree in two minutes of time; so that an error of one minute of a degree in position will produce an error of 2 minutes in time, or half a degree in Longitude. Now from the great improvements made by Newton in the theory of the moon, and more lately by Euler and others on his principles, professor Mayer, of Gottengen, was enabled to calculate lunar tables more correct than any former ones; having so far succeeded as to give the moon's place within one minute of the truth, as has been proved by a comparison of the tables with the observations made at the Greenwich observatory by the late Dr. Bradley, and by Dr. Maskeline, the present Astronomer Royal; and the same have been still farther improved under his direction, by the late Mr. Charles Mason, by several new equations, and the whole computed to tenths of a second. These new tables, when compared with the above-mentioned series of observations, a proper allowance being made for the unavoidable error of observation, seem to give always the moon's Longitude in the heavens correctly within 30 seconds of a degree; which greatest error, added to a possible error of one minute in taking the moon's distance from the sun or a star at sea, will at a medium only produce an error of 42 minutes of Longitude. To facilitate the use of the tables, Dr. Maskelyne proposed a nautical ephemeris, the scheme of which was adopted by the Commissioners of Longitude, and first executed in the year 1767, since which time it has been regularly continued, and published as far as for the year 1800. But as the rules that were given in the appendix to one of those publications, for correcting the effects of refraction and parallax, were thought too difficult for general use, they have been reduced to tables. So that, by the help of the ephemeris, these tables, and others that are also provided by the Board of Longitude, the calculations relating to the Longitude, which could not be performed by the most expert mathematician in less than four hours, may now be completed with great ease and accuracy in half an hour.

As this method of determining the Longitude depends on the use of the tables annually published for this purpose, those who wish for farther information are referred to the instructions that accompany them, and particularly to those that are annexed to the Tables requisite to be used with the Astronomical and Nautical Ephemeris, 2d edit. 1781.

4th. The phenomena of Jupiter's satellites have commonly been preferred to those of the moon, for finding the Longitude; because they are less liable to parallaxes than these are, and besides they afford a very commodious observation whenever the planet is above the horizon. Their motion is very swift, and must be calculated for every hour. These satellites of Jupiter were no sooner announced by Galileo, in his Syderius Nuncius, first printed at Venice in 1610, than the frequency of their eclipses recommended them for this purpose; and among those who treated on this subject, none was more successful than Cassini. This great astronomer published, at Bologna, in 1688, tables for calculating the appearances of their eclipses, with directions for finding the Longitudes of places by them; and being invited to France by Louis the 14th, he there, in the year 1693, published more correct tables of the same. But the mutual attractions of the satellites rendering their motions very irregular, those tables soon became useless for this purpose; insomuch that they require to be renewed from time to time; a service which has been performed by several ingenious astronomers, as Dr. Pound, Dr. Bradley, M. Cassini the son, and more especially by Mr. Wargentin, whose tables are much esteemed, which have been published in several places, as also in the Nautical Almanacs for 1771 and 1779.

Now, to find the Longitude by these satellites; with a good telescope observe some of their phenomena, as the conjunction of two of them, or of one of them with Jupiter, &c; and at the same time find the hour and minute, from the altitudes of the stars, or by means of a clock or watch, previously regulated for the place of observation; then, consulting tables of the satellites, observe the time when the same appearance happens in the meridian of the place for which the tables are calculated; and the difference of time, as before, will give the Longitude.

The eclipses of the first and second of Jupiter's satellites are the most proper for this purpose; and as they happen almost daily, they afford a ready means of determining the Longitude of places at land, having indeed contributed much to the modern improvements in geography; and if it were possible to observe them with proper telescopes, in a ship under sail, they would be of great service in ascertaining its Longitude from time to time. To obviate the inconvenience to which these observations are liable from the motions of the ship, a Mr. Irwin invented what he called a marine chair; this was tried by Dr. Maskelyne, in his voyage to Barbadoes, when it was not found that any benefit could be derived from the use of it. And indeed, considering the great power requisite in a telescope proper for these observations, and the violence, as well as irregularities in the motion of a ship, it is to be feared that the complete management of a telescope on ship-board, will always remain among the desiderata in this part of nautical science. And farther, since all methods that depend on the phenomena of the heavens have also this other defect, that they cannot be observed at all times, this| renders the improvement of time keepers an object of the greater importance.

Many other schemes and proposals have been made by different persons, but most of them of very little or no use; such as by the space between the flash and report of a great gun, proposed by Messrs Whiston and Ditton; and another proposed by Mr. Whiston, by means of the inclinatory or dipping needle; besides a method by the variation of the magnetic needle, &c, &c.

Longitude of Motion, is a term used by Dr. Wallis for the measure of motion, estimated according to its line of direction; or it is the distance or length gone through by the centre of any moving body, as it moves on in a right line.

The same author calls the measure of any motion, estimated according to the line of direction of the vis motrix, the Altitude of it.

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ABCDEFGHKLMNOPQRSTU and VWXYZABCEGLMN

Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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LOG
LOGARITHM
LOGARITHMIC
LOGISTICS
LONG (Roger)
* LONGIMETRY
LONGOMONTANUS (Christian)
LOXODROMIC Curve
LOXODROMICS
LOZENGE
LUBIENIETSKI (Stanislaus)