NAVIGATION is the art of conducting a ship across the ocean. It is here treated to the exclusion of seamanship, which forms a distinct subject. The present article will give, first, a view of the history of the art from the time of the epoch-making voyages of Columbus and the Portuguese, with special reference to advances made in England, and then a sketch of practical navigation as the art now stands.

Up to the time of the Portuguese exploring expeditions, sent out by Prince Henry, which led to the discovery of the Azores in 1419, and the rediscovery of the Cape Verd Islands in 1447 and of Sierra Leone in 1460, navigation had been conducted in the most rude, uncertain, and dangerous manner it is possible to conceive. Thousands of years had passed without the least improvement being introduced, except the magnetic needle about the beginning of the 14th century (see COMPASS and MAGNETISM). Prince Henry did all in his power to bring together and systematize the knowledge then obtainable upon nautical affairs; he also established an observatory at Sagres (near Cape St Vincent) in order to obtain more accurate tables of the declination of the sun. John IL, who ascended the throne in 1481, followed up the good work of his grand-uncle. He employed Roderick and Joseph, his physicians, with

The backward state of navigation at this time is best understood from a sketch of the few rude appliances which the mariner had. He had a compass, a cross-staff or astrolabe, a moderately good table of the sun's declination, a correction for the altitude of the pole star, and occasionally a very incorrect chart. The first map or sea chart seen in England was brought by Bartholomew Columbus in 1489, and the. first map of England was made in 1520. Decimal arithmetic was invented by Simon Stevin about the end of the 16th century. Watches were unknown till 1530, and immediately Gemma Frizon or Frisius seized the idea for the purpose of ascertaining the difference of longitude between two places. They were too rough to be of use, and their advocate proposed to correct them by water-clocks or sand-clocks. Almanacs were first published in Poland in 1470, and in London three years later. These contained tables of the sun's declination and that of many of the stars, and tables for finding the latitude by the pole star and the "pointers." There was not till 1607 any means whatever of measuring a ship's progress through the water, and none in general use till twenty or thirty years later (see LoG).

The " cross-staff " appears to have been used by astronomers at a very early period for measuring heights and distances, more recently by seamen for measuring altitudes. It was one of the few instruments possessed by Columbus and Vasco da Gama. The old cross-staff, called by the Spaniards " ballestilla," consisted of two light battens. The part we may call the staff was about an inch and a half square and 36 inches long. The cross was made to fit closely and to slide upon the staff at right angles; its length was a little over 26 inches, so as to allow the "pinnies" or sights to be placed exactly 26 inches apart. A sight was also fixed on the end of the staff for the eye to peep through at the other two sights and objects to be measured. It was made by describing the angles on a table, and laying the staff upon it (fig. 1). The scale of degrees was marked on the upper face. Afterwards shorter crosses were introduced, so that smaller angles could be taken by the same instrument. These angles were marked on the sides of the staff.

Another primitive instrument in common use at the beginning of the 16th century was the astrolabe, which was more convenient than the cross-staff for taking altitudes, but was incapable of measuring distances. Fig. 2 represents an astrolabe as described by Martin Cortes. It was made of copper or tin, about one-fourth of an inch in thickness and 6 or 7 inches in diameter, and was perfectly circular except at one place, where a projection was provided for a hole by which it was suspended. Weight was considered desirable in order to keep it steady when in use. The face of the metal having been well polished, a plumb line from the point of suspension marked the vertical line, which when carefully subdivided gave the horizontal line and centre. The upper left quadrant was divided into degrees. The second part was a pointer pt of the same metal and same thickness as the circular plate, about 1/ inches wide, and in length equal to the diameter of the circle. The centre was bored, and a line was drawn across it the full length, which was called the line of confidence. On the ends of that line were fixed plates s, s, having each a larger and a smaller hole, both exactly over the line of confidence, as sights for the sun or stars. The pointer moved upon a centre the size of a goose quill. When the instrument was suspended the pointer was directed by hand to the object, and the angle read on the one quadrant only. Some years later the other quadrant was also graduated, to give the benefit of a second reading.

Among the earliest writers who touched upon navigation was John Werner of Nuremberg, who in 1514, in his notes upon Ptolemy's geography, describes the cross-staff as a very ancient instrument, but says that it was only then beginning to be introduced among seamen. He recommends measuring the distance between the moon and a star as a means of ascertaining the longitude.

Thirty-eight years after the discovery of America, when long voyages had become comparatively common, Gemma Frisius wrote upon astronomy and cosmogony, with the use of the globes. His book comprehended much valuable information to mariners of that day, and. was translated into French fifty years later (1582) by Claude de Bossiere. The system adopted is that of Ptolemy. The following are some of the points of interest for the subject before us. There is a good description of the sphere and its circles ; the obliquity of the ecliptic is given as 23° 30'. The distance between the meridians is to be measured on the equator, allowing 15° to an hour of time ; longitude is to be found by eclipses of the moon and conjunctions, and reckoned from the Fortunate Islands (Azores). Latitude should be measured from the equator, not from the ecliptic, "as Clarean says." The use of globes is very thoroughly and correctly explained. The scale for measuring distances was placed on the equator, and 15 German leagues, or 60 Italian leagues, were to be considered equal to one degree. The Italian league was 8 stadia, or 1000 paces, therefore the degree is taken much too small. We are told that, on plane charts, mariners drew lines from various centres (i.e., compass courses), which were very useful since the virtue of the loadstone .had become known ; it must be remembered that parallel rulers were unknown. Such a confusion of lines has been continued upon sea charts till very recently: Frisius gives rules for finding the course and distance correctly, except that he treats difference of longitude as departure. For instance, if the difference of latitude and difference of longitude are equal, the course prescribed is between the two principal winds, - that is, 45°. He points out that the courses thus followed are not straight lines, but curved, because they do not follow the great circle, and that distances could be more correctly measured on the globe. The tide is said to rise with the moon, high water being when it is on the meridian and nadir. From a table of latitudes and longitudes a few examples are here selected, by which it appears that even the latitude was much in error. The figures in brackets represent the positions according to modern tables, counting the longitude from the western extremity of St Michael. Flores is 5° 8' farther west.

Alexandria 31° 0' N. (31° 13') 60° 30' E. (55° 55') Athens 37 15 (37 58) 52 45 (49 46) Babylon 15 0 (32 32) 79 0 (70 25) Dantzic 54 30 (54 21) 44 15 (44 38) London 52 3 (51 31) 19 15 (25 54) Malta 34 0 (35 43) 38 45 (40 31) Rome 41 50 (41 54) 36 20 (38 30) In 1534 Gemma produced an "astronomical ring," which he dedicated to the secretary of the king of Hungary. He admitted that it was not entirely his own invention, but asserted that it could accomplish all that has been said of quadrants, cylinders, and astrolabes, - also that it was a pretty ornament, worthy of a prince. As it displayed great ingenuity, and was followed by many i similar contrivances during two centuries, a sketch is here given (fig. 3). The description must necessarily be brief.

The outer and principal sustaining circle EPQr represents the meridian, and is about 6 inches in diameter ; P, r are the poles. The upper quadrant is divided into degrees. It is suspended by a fine cord or wire placed at the supposed latitude. The second circle EQ is fixed at right angles to the first, and represents the equinoctial line. The upper side is divided into twenty-four parts, representing the hours from noon or midnight. On the inner side of that circle are marked the months and weeks. The third ring CC is attached to the first at the poles, and revolves freely within it. On the interior are marked the months, and on another side the corresponding signs of the zodiac ; another is graduated in degrees. It is fitted with a groove which carries two movable sights. On the fourth side are twenty-four unequal divisions (tangents) for measuring heights and distances. Its use is illustrated by twenty problems ; it can do roughly all that any instrument for taking angles can. Thus, to find the latitude, set the sights C, C to the place of the sun in the zodiac, and shut the circle till it corresponds with 12 o'clock. Peep through the sights and alter the point of suspension till the greatest elevation is attained ; that time will be noon, and the point of suspension will be the latitude. The figure is slung at lat. 40°, either north or south. To find the hour of the day, the latitude and declination being known : the sights C, C being set to the declination as before, and the suspension on the latitude, turn the ring CC freely till it points to the sun, when the index opposite the equinoctial circle will indicate the time, while the meridional circle will coincide with the meridian of the place. In this we may see the germ of the " equatorial " now used in the principal observatories.

There is at present in the museum attached to the Royal Naval College at Greenwich an instrument, ticketed as Sir Francis Drake's astrolabe, prior to 1570. It is not an astrolabe, but may be a combination of astronomical rings as invented by Gemma with other things, probably of a later date. It has the appearance of a large gold watch, about 2i inches in diameter, and contains several parts which fall back on hinges. One part is a sun-dial, the gnomon being in connexion with a graduated quadrant, by but too small for actual use ; it may be simply an ornament is 51° 34', about three miles too much.

In 1537 Pedro Nuiiez (Nonius), cosmographer to the king of Portugal, published a work on astronomy, charts, of reading the exact angle on a sextant, the scale of a one of them, suppose the fifteenth division on the sixth scale, then the angle was 1-1 of 90° 15° 52' 56". This was a laborious method ; Tycho Brahe tried it, but abandoned it in favour of the diagonal lines then in common use, and still found on all scales of equal parts.

In 1545 Dr Pedro de Medina published Tire Art of Navigation, at Valladolid, dedicated to Don Philippo, prince of Spain. This appears to be the first book ever published professedly on navigation. It was soon translated into French and Italian, and many years after into English by John Frampton. Though this pretentious work came out two years after the death of Copernicus, the astronomy is still that of Ptolemy. The general appearance of the chart of the Mediterranean, Atlantic, and part of the Pacific is in its favour, but examination shows it to be very incorrect. A scale of equal parts, near the centre of the chart, extends from the equator to what is intended to represent 75° of latitude ; by this scale London would be in 55° instead of 51i°, Lisbon in 371° instead of 38° 42'. The equator is made to pass along the coast of Guinea, instead of being over four degrees farther south. The Gulf of Guinea extends 14 degrees too far east, and Mexico is much too far west. Though there are many vertical lines on the chart at unequal distances they do not represent meridians; there is no indication of longitude. A scale of 600 leagues is given (German leagues, fifteen to a degree). By this scale the distance between Lisbon and the city of Mexico is 1740 leagues, or 6960 miles ; by the vertical scale of degrees it would be about the same ; whereas the actual distance is 4820 miles. Here two great wants become apparent, - a knowledge of the actual length of any arc, and the means of representing the surface of the globe on flat paper. There is a table of the sun's declination to minutes; on June 12th and December 11th (o.s.) it was 23° 33'. The directions for finding the latitude by the pole star and pointers appear good. For general astronomical information the book is inferior to that of Gemma.

In 1556 Martin Cortes published at Seville The Art of Navigation. He gives a good drawing of the cross-staff and astrolabe, also a table of the sun's declination for four years (the greatest being 23° 33'), and a calendar of saints' days. The motions of the heavens are described according to the notions then prevalent, the earth being viewed as fixed. He recommends the height of the pole being found frequently, as the estimated distance run was imperfect. He devised an instrument whereby to tell the hour, the direction of the ship's head, and where the sun would set. A very correct table is given of the distances between the meridians at every degree of latitude, whereby a seaman could easily reduce the difference of longitude to departure. In the rules for finding the latitude by the pole star, the star is supposed to be then 3° from the pole ; it is now (1883) 1° 18' 54". Martin Cortes attributes the tides entirely to the influence of the moon, and gives instructions for finding the time of high water at Cadiz, when by means of a card with the moon's age on it, revolving within a circle showing the hours and minutes, the time of high water at the place for which it was set would be indicated. He deplores the loss of the earl of Niebla and other valiant captains of Spain, before Gibraltar, in 1436, because the mariners kept no account of the tides. In this instance it was more probably the effect of current. There is a chapter upon the signs which prognosticate fair or foul weather, from Pliny and Aristotle ; another upon " shining exhalations," the " fire of St Elmo," and other old superstitions. Directions are given for making a compass similar to those then in common use, also for ascertaining and allowing for the variation. The east is here spoken of as the principal point, and marked by a cross, after that the true north.

There is a table of difference of latitude and departure in proportion to the tangent of the course.

The third part of Martin Cortes's work is upon charts ; he laments that wise men do not produce some that are correct, and that pilots and mariners will use plane charts which are not true. In the Mediterranean and Channel of Flanders the want of good charts is (he says) less inconvenient., as there they do not navigate by the altitude of the pole.

As some subsequent writers have attributed to Cortes the credit of first thinking of the enlargement of the degrees of latitude on Mercator's principle, his precise words may be cited. In making a chart, it is recommended to choose a well-known place near the centre of the intended chart, such as Cape St Vincent, which call 37°, "and from thence towards the Arctic pole the degrees increase ; and from thence to the equinoctial line they go on decreasing, and from the line to the Antarctic pole increasing."' It would appear at first sight that the degrees increased in size as well as being called by a higher number, but a specimen chart in the book does not justify that conclusion. It is from 34° to 40°, and the divisions are unequal, but evidently by accident, as the highest and lowest are the largest. He states that the Spanish scale was formed by counting the Great Berling as 3° from Cape St Vincent (it is under 21°). Twenty English leagues are equal to 17i Spanish or 25 French, and to 1° of latitude. Cortes was evidently at a loss to know the size of a degree, and consequently the circumference of the globe. The degrees of longitude are not laid down, but for a first meridian we are told to draw a vertical line " through the Azores, or nearer Spain, where the chart is less occupied." It is impossible under such circumstances to understand or check the longitudes assigned to places at that period. Martin Cortes's work was held in high estimation in England for many years, and appeared in several translations. One by Richard Eden in 1609 gives an improved table of the sun's declination from 1609 to 1625 - the greatest declination being 23° 30'. The declinations of the principal stars and the times of their passing the meridian, and other improved tables, are given, with a very.poor traverse table for eight points. The cross-staff, he said, was in most common use ; but he recommends -Wright's sea quadrant.

William Cuningham published in 1559 a book called his Astronomical Glass, in which he teaches the making of charts by a central meridional line of latitude in equal parts, with other meridians on each side, distant at top and bottom in proportion to the departure at the highest and lowest latitude, for which purpose a table of departures is given very correctly to the third place of sexagesimals. The chart would be excellent were it not that the parallels are drawn straight instead of being curved. In another example, which is one-fourth of the sphere, the meridians and parallels are all curved ; it would be good were it not that the former are too long. The hemisphere is also shown upon a projection approaching the stereographic ; but the eighteen meridians cut the equator at equal distances, instead of being smaller towards the primitive. He gives the drawing of an instrument like an astrolabe placed horizontally, divided into 32 points and 360 degrees, and carrying a small magnetic needle to be used as a prismatic compass, or even as a theodolite (fig. 4). A sketch is given of Ptolemy observing the sun with a primitive instrument, likely from its great size to give good results after being correctly fixed, except for the amount of error caused by the shrinkage or expansion of the parts.

Gerhard Mercator's great improvements in charts have been noticed in the article MAP, where a sketch is given of his map of the world, of 1569 (vol. xv. p. 521). From facsimiles of his early charts in Jomard, Les Monuments de la Geographie, the following measurements have been made. A general chart of 1569, of North America, 25° to 79°, is 2 feet long north and south, and 20 inches wide_

Norman's discovery of the dip (1576) has been spoken of at vol. xv. p. 221. He mentions and condemns the practice of each country having compass cards set to their variation, and sailors using them indiscriminately in any part of the world.

In 1581 Michael Coignet of Antwerp published sea charts, and also a small treatise in French, wherein he exposes the errors of Medina. He was probably the first who said that rhumb lines form spirals round the pole. He published also tables of declination, and observed the gradual decrease in the obliquity of the ecliptic. He described a cross-staff with three transverse pieces, which was then in common use at sea. Coignet died in 1623.

The Dutch published charts made up as atlases as early as 1584, with a treatise on navigation as an introduction.

In 1585 Roderico Zamorano, who was then the lecturer at the naval college at Seville, published a concise and clearly-written compendium of navigation ; he follows Cortes in the desire to obtain better charts. Andres Garcia de Cespedes, the successor of Zamorano at Seville, published a treatise on navigation at Madrid in 1606. In 1592 Petrus Plancius published his universal map, containing the discoveries in the East and West Indies and towards the north pole. It possessed no particular merit ; the degrees of latitude are equal, but the distances between the meridians are noted. He made London appear in 51° 32' N. (which is near enough) and long. 22°, by which his first meridian should have been more than 3° east of St Michael.

In 1594 Blundeville published a description of Mercator's charts and globes ; he confesses to not having known upon what rule the meridians were enlarged by Mercator, unless upon such a table as Wright had sent him (see below). Wright's table of meridional parts is here published, also an excellent table of sines, tangents, and secants, - the former to seven places of figures, the latter to eight. These are the tables made originally by Regiomontanus, and improved by the Jesuit Clavius.

In 1594 the celebrated navigator John Davis published a pamphlet of eighty pages, in black letter, entitled The Seaman's Secrets, in which he proposes to give all that is necessary for seamen - not for scholars on shore. He defines three kinds of sailing : - horizontal, paradoxical, and great circle. His horizontal sailing consists of short voyages which may be delineated upon a plain sheet of paper. The paradoxical or cosmographical embraces longitude, latitude, and distance, - the getting together many horizontal courses into one " infallible and true," i.e., what is now called traverse and Mercator's sailings. His " paradoxical course" he describes correctly as a rhumb line which is straight on the chart and a curve on the globe. He points out the errors of the common or plane chart, and promises if spared to publish a " paradoxall chart." It is not known whether such a chart appeared or not, but he assisted Wright in producing his chart a few years later. Great circle sailing is clearly described by Davis on a globe ; and to render it more practicable he divides a long distance into several short rhumb lines quite correctly. His list of instruments necessary to a skilful seaman comprises the sea compass, cross-staff, chart, quadrant, astrolabe, an " instrument magnetical " for finding the variation of the compass, a horizontal plane sphere, a globe, and a paradoxical compass. The first three are sufficient for use at sea, the astrolabe and quadrant being uncertain for sea observations. The importance of knowing the time of the tides when approaching tidal or barred harbours is clearly pointed out, also the mode of ascertaining it by the moon's age. A table of the sun's declination is given for noon each day during four years, 1593-97, from the ephemerides of Stadius. The greatest is 23° 28'. Several courses and distances, with the resulting difference of latitude and departure, are correctly worked out. A specimen log-book provides one line only for each day, but the columns are arranged similarly to those of a modern log. Under the head of remarks after leaving Brazil, we read, "the compass varied 9°, the south point westward." He states that the first meridian passed through St Michael, because there was no variation at that place ; the meridian passed through the magnetic pole as well as the pole of the earth. He makes no mention of Mercator's chart, nor of Cortes or other writers on navigation. Rules are given for finding the latitude by two altitudes of the sun and intermediate azimuth, also by two fixed stars, by the globe. There is a drawing of a quadrant, with a plumb line, for measuring the zenith distance, and one of a curious modification of a cross-staff with which the observer stands with his back to the sun, looking at the horizon through a sight on the end of the staff, while the shadow of the sun, from the top of a movable projection, falls on the sight box. This remained in common use till superseded by Hadley's quadrant. The eighth edition of Davis's work was printed in 1657.

Edward Wright, of Caius College, Cambridge, published in 1599 a valuable work entitled Certain Errors in Navigation Detected and Corrected. One part is a translation from Roderico Zamorano ; there is a chapter from Cortes, and one from Nunez. A year after appeared his chart of the world, upon which both capes and the recent discoveries in the East Indies and America are laid down truthfully and scientifically, as well as his knowledge of their latitude and longitude would admit. Just the northern extremity of Australia is shown. Wright said of himself that he had striven beyond his ability to mend the errors in chart, compass, cross-staff, and declination of sun and stars. He considered that the instruments which had then recently come in use "could hardly be amended," as they were growing to "perfection," - especially the sea chart and the compass, though he expresses a hope that the latter may be " freed from that rude and gross manner of handling in the making." He gives a table of magnetic declinations, and explains its geometrical construction. He states that Medina utterly denied the existence of variation, and attributed it to bad making and bad observations. Wright expresses a hope that a right understanding of the dip of the needle would lead to a knowledge of the latitude, "as the variation did of the longitude." He gives a table of declinations for every minute of the ecliptic, and another for the use of English mariners during four years - the greatest being 23° 31' 30". The latitude of London he made 51° 32'. For these determinations a quadrant over 6 feet in radius was used. He also treats of the " dip " of the sea horizon, refraction, parallax, and the sun's motions. With all this knowledge the earth is still considered as stationary, - although Wright alludes to Copernicus, and says that he omitted to allow for parallax. Wright ascertained the declination of thirty-two stars, and made many improvements or additions to the art of navigation, considering that all the problems could be performed arithmetically by the doctrine of triangles, without globe or chart. He devised sea rings for taking observations, and a sea quadrant to be used by two persons, which is in some respects similar to that by Davis. While deploring the neglected state which navigation had been in, he rejoices that the worshipful society at the Trinity House, under the favour of the king (James I.), had removed "many gross and dangerous enormities." He joins the brethren of the Trinity House in the desire that a lectureship should be established on navigation, as at Seville and Cadiz ; also that a grand pilot should be appointed, as Sebastian Cabot had been in Spain, who examined pilots (i.e., mates) and navigators. Wright's desire was partially fulfilled in 1845, when an Act of Parliament paved the way for the compulsory qualification of masters and mates of merchant ships ; but such was the opposition by shipowners that it was left voluntary for a few years. England was in this respect more than a century behind Holland. It has been said that Wright accompanied the earl of Cumberland to the Azores in 1589, and that he was allowed £50 a year by the East India Company as lecturer on navigation at Gresham College, Tower Street.

The great mark which Wright made in the world was the discovery of a correct and uniform method for dividing the meridional line and making charts which are still called after the name of Mercator. He considered his charts as true as the globe itself ; and so they were for all practical purposes. He commenced by constructing a meridional line, upon the proportion of the secants of the latitude, for every ten minutes of the arc, and in the edition of his work published in 1610 his calculations are for every minute. His calculations were based upon the discovery that the radius bears the same proportion to the secant of the latitude as the difference of longitude does to the meridional difference of latitude - a rule strictly correct for small arcs only. One minute is taken as the unit upon the arc and 10,000 as the corresponding secant, 2' becomes 20,000, 3' = 30,000, &c., increasing uniformly till 49', which is equal to 490,001; 1° is 600,012. The secant of 20° is 12,251,192, and for 20° 1' it will be 12,251,192 + 10,642, - practically the same as that used in modern tables. The principle is simply explained by fig. 5, where b is the pole and bf the meridian. At any point a a minute of longitude : a min. of lat. : : ea (the semi-diameter of the parallel) : kf (the radius). Again ea : kf : : kf : ki radius : sec. akf (sec. of lat.). To keep this proportion on the chart, the points of latitude must increase in the same proportion as the secants of the arc contained between those points and the equinoctial, which was then to be done by the "canon of triangles."1 Subsequent writers, including Gunter, Norwood, and Bond, give Wright the credit of having been the first to establish a correct proportion between the meridians and parallels for every part of the chart. This great improvement in the principle of constructing charts was adopted slowly by seamen, who, putting it as they supposed to a practical test, found good reason to be disappointed. The positions of most places had been laid down erroneously, by very rough courses and estimated distances upon an entirely false scale, viz., the plane chart ; from this they were transferred to the new projection.2 When Napier's Canon .11-firificus appeared in 1614, Wright at once recognized the value of logarithms as an aid to navigation, and undertook a translation of the book, which he did not live to publish (see NAPIER). E. Gunter's tables (1620) made the application of the new discovery to navigation possible, and this was done by T. Addison in his Arithmetical Navigation (1625), as well as by Gunter in his tables of 1624 and 1636, which gave artificial sines and tangents, to a radius of 1,000,000, with directions for their use and application to astronomy and navigation, and also logarithms of numbers from 1 to 10,000. Several editions followed, and the work retained its reputation over a century. Gunter invented the sector, and introduced the meridional line upon it, in the just proportion of Mercator's projection.

The third edition of Gunter's work was published in 1653, and the fifth edition in 1673, amended by Henry Bond, a practitioner in the mathematics, in the Bulwark near the Tower - a thick octavo. A table of meridional parts is given, with instructions to construct it by the addition of secants as Wright did. The table has been found upon examination to be very correct. The degree is divided into 1000 parts.

With the latitude left, course steered, and difference of longitude made good, Bond found the latitude of ship, by projection on the chart, by the sector, or by the following rule: - tan of course x proportional duff, of lat. .

=cliff. long.

And conversely, suppose latitude left 50°, course 33° 45', difference of longitude 51°-330' ; then cot 30° 45' x 330' - 493.5, prop. diff. lat., radius which, added to the meridional parts corresponding to 50°, will give the number opposite 55°, and 55° is the latitude. Various problems in sailing according to Mercator are solved arithmetically upon the tangents, without the table of meridional parts, which may also be done geometrically upon the tangent line of the eross-staff. The following important proposition is in Bond's own words : - “First we must know that the logarithmic tangents from 45° upwards do increase in the same manner as the secants added together do, if we account every half degree above 45° to be a whole degree of Mercator's meridional line; and so the table of logarithmic tangents is a table of meridional parts to every two minutes of the meridian line, leaving out the radius."

The way of using this proposition is as follows. The table begins at 45°, and every 30 minutes is reckoned a whole degree ; therefore, when both latitudes are given, take the half of each increased by 45°, subtract the tangent of the lesser sum from that of the greater, and divide the remainder by the tangent of 45° 30' (radius omitted); the quotient will be the equal, or equinoctial; degrees contained between the two latitudes. Or multiply the aforesaid remainder by ten and divide by half the tangent of 45° 30', and the quotient will be equal to the equinoctial leagues contained between the two latitudes. The logarithmic tangents are here treated as natural numbers, and the division done by logarithms. Bond lays no stress on the above solution as being new ; it is merely used in lieu of a table of meridional parts.

The subsequent history of the problem of meridional parts may most conveniently be added here rather than in its chronological place. An important letter from Dr Wallis, professor of geometry at Oxford, is given with the Phil. Trans. for 1685, No. 176. The writer says that, tho old inquiry about the sum or aggregate of secants having been of late renewed, he thought fit to trace it from its original, with such solutions as seemed proper to it. Archimedes and the ancients divided the curvilinear spaces as figs. 6 and 7. If they reckoned the first four it was too large; if the last four, too small. As the segments increased in number the error diminished. The degrees of longitude decrease as the cosine of the latitude (which is the semidiameter of such parallel) to the radius of the globe or equator. By the straight lines " each degree of longitude • is increased above its due proportion, at such rate as the equator (or its radius) is greater than such parallel (or the radius thereof)." The old sea charts represented the degrees of latitude and longitude all equal. "Hereby, -among other inconveniences (as Mr Edward Wright observed in 1599), the representation of places remote from the equator were distorted." Wright advised that the degrees of latitude should be protracted in like proportion with those of longitude, "that is, everywhere in such proportion as is the respective secant of such latitude to the radius" (see Wright's explanation of this part, and fig. 5). Fig. 8 represents one quarter of the globe, the surface of which is opened out till the parallel LA becomes a straight line as la, and each of the four meridians reaches P, P, P, P. The equator is represented by EE; so that the position of each parallel on the chart should b at such distance from the equator "as are all the secants (taken at equal distances in the arc) to so many times the radius, . . . which is equivalent to a projection of the spherical surface on the concave surface of a cylinder, erected at right angles to the plane of the equator," while each division of the meridian is equal to the secant of the latitude answering to such part, as fig. 9. This projection, if expanded into a plane, will be the same as a plane figure whose base is equal to a quadrantal arc extended (or a portion thereof), on which (as ordinates) are erected perpendiculars equal to the secants, answering to the respective points of the arc extended, as fig. 10. The first answers to the equator, the last to the pole infinite. "For finding this distance answering to each degree and minute of latitude, Mr Wright added all the secants from the beginning to the position required. The sum of all except the greatest (answering to the figure inscribed) is too little. The sum of all except the least (answering to the circumscribed) is too great - which latter Mr Wright followed. It will be nearer the truth than either if we take the intermediate spaces ; instead of minutes, take 1, 11, 21, &c., or the double of these, 1, 3, 5, 7, &c., which yet, because on the convex side of the curve, would be rather too little. Either of these ways will be exact enough for a chart. If we would be more exact, Mr Oughtred directs, as did Mr Wright before him, to divide the arc into parts yet smaller than minutes, and calculate secants thereto." Wallis continued the subject and the doctrine of infinite series ; but more than sufficient has been quoted for the purposes of navigation. At the end he adds that the same may be obtained in like manner by taking the versed sines in arithmetical proportion.

The next writer who made his mark upon this problem was Dr E. Halley (Phil. Trans., No. 219, 1695). He states that the tangential proportion between the latitude and the divisions of the meridional line was discovered by chance, and first published by H. Bond, in Norwood's Epitome of Navigation. James Gregory furnished a demonstration in 1668; but it was long and tedious. Halley claimed for himself the honour of being the first to give a rule whereby the meridional parts between any two latitudes may be calculated at once by the relation of the logarithmic tangents; but it is practically the Caine as that published by Bond. Halley said that Wright's table nowhere exceeded the truth by half a mile. Sir Jonas Moore's system, he said, was nearer the truth, but the difference is not appreciable till beyond navigable waters.

A rather curious paper was read before the Royal Society, June 4, 1666, by Nicholas Mercator upon the meridional line ; he proposes to divide it into the hundred-thousandth part of a minute. Roger Cotes wrote upon the same subject an exhaustive paper in Latin, called "Logometria," Phil. Trans., No. 338, 1714. He gives an illustrative figure in which the rhumb line crosses the meridians at an angle of 45°. His demonstrations by the ratios arrive at similar conclusions to those clearly expressed by Halley.

All these rules assume the earth to be truly spherical, instead of spheroidal. For the history of inquiry into the exact figure of the earth, see EARTH. It may be mentioned that a pamphlet on this subject by Murdoch, published in 1741, in which meridional parts are adapted to a (very exaggerated) spheroid, shows that plane charts and the roughly-divided Mercator's charts were still used at that date. Plane charts, indeed are explained even later, as in Robertson's Navigation, 1755.

The power of taking observations correctly, either at sea or on shore, was greatly assisted by the invention which bears the name of Pierre Vernier, which was published in Brussels in 1631 (see VERNIER). As Vernier's quadrant was divided into half degrees only, the sector, as he called it, spread over 14i degrees, and that space carried thirty equal divisions, numbered from 0 to 30. As each division of the sector contained 29 minutes of the arc, the vernier could be read to minutes. The verniers now commonly adapted to sextants can be read to 10 seconds. Shortly after the invention it was recommended by Bouguer and Jorge Juan, who describe it in a treatise entitled La Construction, (N., du quadrant nouveau. About this period Gascoigne applied the telescope to the quadrant (see MICROMETER) ; and Hevelius invented the tangent screw, to give slow and steady motion when near the desired position. In 1635 Henry Gellibrand published his discovery of the change in variation of the needle, which was effected by his comparing the results of his own observations with those of Burrough and Gunter. The latter was his predecessor at Gresham College.

In 1637 Richard Norwood, a sailor, and reader in mathematics, published an account of his most laudable exertions to remove one of the greatest stumbling-blocks in the way of correct navigation, that of not knowing the actual size of a degree or nautical mile, in a pamphlet styled The Seaman's Practices. Norwood ascertained the latitude of a position near the Tower of London in June 1633, and of a place in the centre of York in June 1635, with a sextant of more than 5 feet radius, and, having carefully corrected the declination, refraction, and parallax, made the difference 2° 28'. He then measured the distance with a chain, taking horizontal angles of all windings, and he made a special table for correcting elevations and depressions. A few places which he was unable to measure he paced. His conclusion was that a degree contained 367,176 English feet ; this gives 2040 yards to a nautical mile, - only about 12 yards too much. Norwood's works went through numerous editions, and retained their popularity over a hundred years ; the last which the writer has seen - a good book for the time, free from nonsense - is dated 1732. In it he says that, as there is no means of discovering the longitude, a seaman must trust to his reckoning. He recommends the knots on the log-line to be placed 51 feet apart, as the just proportion to a mile when used with the half-minute glass.

Dr Hooke read a paper at the Royal Society, in 1666, upon deep-sea sounding by means of a weight which became detached on striking the bottom, and allowed a float to ascend to the surface, while the time was carefully noted - basing his calculations upon performances in known depths. He was on the verge of a great success ; he required Sir W. Thomson's piano-wire instead of the float.

In the same year a paper was read by Dr Wallis (who had previously published a discourse on tides) showing that the modern theory was not then generally accepted. This was followed by a paper by Sir Robert Moray, who recommended frequent and extended observations, and proposed to form a table which embraced every circumstance that would appear to be desirable even at the present day. Sir Robert also spoke of the irregularities in the tides past the western islands of Scotland. In Phil. Trans., 1683, vol. xiii. No. 143, there is an account of Flamsteed's tide table for London Bridge, which gave each high tide every day in the year. He justly condemns the old almanacs for deriving the moon's age from the epact, and then allowing forty-eight minutes for every day. Brooker was the first to amend this reckoning, but in a rough manner. Henry Philips, well known by his works on navigation, was the first to bring the inequality to a rule, which was found more conformable to experience than was expected ; but Flamsteed made corrections on his rule.

The necessity for having correct charts was equalled by the pressing need of obtaining the longitude by some simple and correct means available to seamen ; and we have seen how many plans had already been thought of for this purpose. At one time it was hoped that the longitude might be directly discovered by the variation of the compass ; in 1674 Charles II. actually appointed a commission to investigate the pretensions of a scheme of this sort devised by Bond,' and the same idea appears as late as 1777 in S. Dunn's Epitome. But the only real way of ascertaining the longitude is by knowing the difference of time at two meridians ; and till the invention and perfecting of chronometers this could only be done by finding at two places the apparent time of the same celestial phenomenon. The most obvious phenomena to select were the motions of the moon among the sun and stars, which as we have seen were suggested as a means of finding the longitude by Werner in 1514, and continued to receive attention from later writers. But to make this idea practical it was necessary on the one hand to have better instruments for observation, and on the other to have such a theory of the moon's motions as should enable its place to be predicted with accuracy, and recorded beforehand in an almanac. The very principles of such a theory were unknown before Newton's great discovery, when the lunar problem begins to have a chief place in the history of navigation ; the places of stars were derived from various and widely discrepant sources ; and almanacs gave little useful information beyond the declination of the sun, the age of the moon, and the time of high water.2 Another class of phenomena whose comparative frequency recommended them for longitude observations, viz., the occultations of Jupiter's satellites, became known through Galileo's discovery of these bodies (1610). Tables for these were published by Dominic Cassini at Bologna in 1688, and were repeated in a more correct form at Paris in 1693 by his son, who was followed by Pound, Bradley, Wargentin, and many other astronomers. But this method, though useful on land, is not suited to mariners ; when Whiston, for example, in 1737 recommended that the satellites should be observed by a reflecting telescope, he did not sufficiently consider the difficulty of using a telescope at sea, or the infrequency of the occultations, and it is the lunar problem which will chiefly concern us in what follows.

The study of this problem was stimulated by the reward of 100Q crowns offered by Philip III. of Spain in 1598 ; the states-general followed with an offer of 10,000 florins. But for a long time nothing practical came of this ; a proposal by J. Morin, submitted to Richelieu in 1633, was pronounced by commissioners appointed to judge of it to be incomplete through the imperfection of the lunar tables, and in like manner when the question was raised in England in 1674 by a proposal of St Piere to find the longitude by using the altitudes of the moon and two stars to find the time each was from the meridian, and when the king was pressed by St Piere, Sir J. Moore, and Sir C. Wren to establish an observatory for the benefit of navigation, and especially that the moon's exact position might be calculated a year in advance, Flamsteed gave his judgment that the lunar tables then in use were quite useless, and the positions of the stars erroneous. The result was that the king decided upon establishing an observatory in Greenwich Park, and Flamsteed was appointed astronomical observer on March 4, 1675, upon a salary of £100 a year, for which also he was to instruct two boys from Christ's Hospital. While the small building in the Park was in course of erection he resided in the Queen's House (now the central part of the Greenwich Hospital school), and removed to the house on the hill, July 10, 1676, which came to be known as "Flamsteed House." The institution was placed under the surveyor-general of ordnance, - perhaps because that office was then held by Sir Jonas Moore, himself an eminent mathematician. Though this was not the first observatory in Europe, it was destined to become the most useful, and has fulfilled the important duties for which it was established. It was established to meet the exigencies of navigation, as was clearly stated on the appointment of Flamsteed, and on several subsequent occasions ; and we see now what an excellent foster-mother it has been to the higher branches of that science. This has been accomplished by much labour and patience ; for, though the most suitable man in the kingdom was placed in charge, it was so starved and neglected that it was almost useless during many years. The Government did not provide a, single instrument. Flamsteed entered upon his important duties with an iron sextant of 7 feet radius, a quadrant of 3 feet radius, two telescopes, and two clocks, the last given by Sir Jonas Moore. Tycho Brahe's catalogue of about a thousand stars was his only guide. In 1681 he fitted a mural arc which proved a failure. Seven years after another mural arc was erected at a cost of £120, with which he set to work in earnest to verify the latitude, and to determine the equinoctial point, the obliquity of the ecliptic, the right ascension and declination of the stars, till he numbered two or three thousand which appeared in the "British catalogue." See FLAMSTRED and ASTRONOMY.

Flamsteed died in 1720, and was succeeded by Halley, who paid particular attention to the motions of the moon with a view to the longitude problem. A paper which he published. in the Phil. Traits., 1731, No. 421, shows what had been accomplished up to that date, and proves that it was still impossible to find the longitude correctly by the moon.' He repeats what he had published twenty years before in an appendix to Street's Caroline tables, which contained observations made by him (Halley) in 1683-84 for ascertaining the moon's motion, which he thought to be the only practical method of " attaining " the longitude to of longitude). Sir Isaac Newton's tables, corrected with great exactness, almost fifteen hundred times, or as desired accuracy.2 The last remark calls us to consider this great improve-- directions at once.

Their imperfections are clearly pointed out in a paper by Pierre Bouguer (1729) which received the prize of the Paris Academy of Sciences for the best method of taking the altitude of stars at sea. Bouguer himself proposes a modification of what he calls the English quadrant, probably the one proposed by Wright and improved by Davis. Fig. 11 represents the instrument as proposed, capable of measuring fully 90° from E to N. A fixed pinule was recommended to be placed at E, through which a ray from the sun would pass to the sight C. The sight F would look through F and C at the horizon, shifting the former up or down till the ray from the sun coincided with the horizon. The space from E to F would represent the altitude, and the remaining part F to N the zenith distance. The English quadrant which this was to supersede differed in having about half the arc from E towards .N, and, instead of the pinule being Axed at E, it was on a smaller are represented by the dotted line eB, and movable. It was placed on an even number of degrees, considerably less than the altitude ; the remainder was measured on the larger arc, as described.1 Hadley's instrument, on the other hand, described to the Royal Society in May 1731 (Phil. Trans., Nos. 420 and 421), embodies Newton's idea of bringing the reflexion of one object to coincide with the other. He calls it an octant, as the arc is actually 45°, or the eighth part of a circle ; but, in consequence of the angles of incidence and reflexion both being changed by a movement of the index, it measures an angle of 90°, and is graduated accordingly; the same instrument has therefore been called a quadrant. It was very slowly adopted, and no doubt there were numerous Mechanical difficulties of centring, graduating, &c., to be overcome before it reached perfection.2 In August 1732, in pursuance of an order from the Admiralty, observations were made with Hadley's quadrant on board the " Chatham " yacht of 60 tons, below Sheerness, in rough weather, by persons - except the master attendant - unaccustomed to the motion ; still the results were very satisfactory. Two years later Hadley published (Phil. Trans., 1733) the descriptio4 of an instrument for taking altitudes when the horizon is not visible. The sketch represents a curved tube or spirit-level, attached to the radius of the quadrant.

From the year 1714 the history of navigation in England is closely associated with that of the " commissioners for the discovery of longitude at sea," a body constituted by Act 13 Anne c. 14 (commonly called 12 Anne c. 15), with power to grant sums not exceeding £2000 to assist experiments and reward minor discoveries, and also to judge on applications for much greater rewards which were offered to open competition. For a method of determining the longitude within 60 geographical miles, to be tested by a voyage to the West Indies and back, the sum of £10,000 was offered ; within 40 miles, £15,000; within 30, £20,000. £10,000 was also to be given for a method that came within 80 miles near the shores of greatest danger. No action seems to have been taken before 1737 ; the first grant made was in that year, and the last in 1815, but the board continued to exist till 1828, having disbursed in the course of its existence £101,000 in a11.2 In the interval a number of other Acts had been passed either dealing with the powers, constitution, and funds of the commissioners or encouraging nautical discovery ; thus the Act 18 George II. (1745) offered £20,000 for the discovery by a British ship of the North-West Passage, and the Act 16 George III. (1776) offered the same reward for a passage to the Pacific either north-west or north-east, and £5000 to any one who should approach by sea within one degree of the North Pole. All these Acts were swept away in 1828, when the longitude problem had ceased to attract competitors, and voyages of discovery were nearly over. The suggestions and applications sent in to the commissioners were naturally very numerous and often very trifling ; but they sometimes furnish useful illustrations of the state of navigation. Thus, in a memorial by Captain H. Lanoue (1736), which seems to be designed to commend a substitute for the log (a box with something, not fully explained, let into the sea), he records a number of recent casualties, which shows how carelessly the largest ships were then navigated. Several men-of-war off Plymouth in 1691 were wrecked through mistaking the Deadman for Berry Head. Admiral Wheeler's squadron in 1694, leaving the Mediterranean, ran on Gibraltar when they thought they had passed the Strait. Sir Cloudesley Shovel's squadron, in 1707, was lost on the rocks off Scilly, by erring in their latitude. Several transports, in 1711, were lost near the river St Lawrence, having erred 15 leagues in the reckoning during twenty-four hours. Lord Belhaven was lost on the Lizard in 1722, the same day on which he sailed from Plymouth.

One of the first points to which the attention of the commissioners was directed was the survey of the coasts of Great Britain, which was pressed on them by Whiston in 1737. He was appointed surveyor of coasts and headlands, and in 1741 received a grant for instruments. An Act passed in 1740 enabled the commissioners to spend money on the survey of the coasts of Great Britain and the " plantations." At a later date they bore part of the expenses of Cook's scientific voyages, and of the publication of their results. Indeed it is to them that we owe all that was done by England for surveys of coasts, both at home and abroad, prior to the establishment of the hydro-graphic department of the Admiralty in 1795. But their chief work lay in the encouragement they gave on the one hand to the improvement of timepieces, and on the other to the perfecting of astronomical tables and methods, the latter issuing in the publication of the Nautical Almanac. Before we pass on to these two important topics we may with advantage take a view of the state of practical navigation in the middle of last century as shown in two of the principal treatises then current.

Robertson's Elements of _Navigation passed through six editions between 1755 and 1796. It contains good teaching on arithmetic, geometry, spherics, astronomy, geography, winds and tides, also a small useful table for correcting the middle time between the equal altitudes of the sun, - all good, as is also the remark that "the greater the moon's meridian altitude the greater the tides will be." He states that Lacaille recommends equal altitudes being observed and worked separately, in order to find the time from noon, and the mean of the results taken as the truth. There is a sound article on chronology, the ancient and modern modes of reckoning time. A long list of latitudes, longitudes, and times of high water finishes vol. i. The second volume is said by the author to treat of navigation mechanical and theoretical ; by the former he means seamanship. He gives instructions for all imaginary kinds of sailings, for marine surveying and making Mercator's chart. There are two good traverse tables, one to quarter points, the other to every 15 minutes of the arc ; the distance to each is 120 miles. There is a table of meridional parts to minutes, which is more minute than customary. Book ix., upon what is now called "the day's work," or dead-reckoning, appears to embrace all that is necessary. A great many methods, we are told, were then used for measuring a ship's rate of sailing, but among the English the log and line with a half-minute glass were generally used. Bouguer and Lacailleproposed a log with a diver to avoid the drift motion (1753 and 1760). Robertson's rule of computing the equation of equal altitudes is as good as any used at the present day. He gives also a description of an equal-altitude instrument, having three horizontal wires, probably such acone as was used at Portsmouth for testing Harrison's timekeeper. The mechanical difficulties must have been great in preserving a perpendicular stem and a truly horizontal sweep for the telescope. It gave place to the improved sextant and artificial horizon. The second edition of Robertson's work in 1764 contains an excellent dissertation on the rise and progress of modern navigation by Dr James Wilson, which has been greatly used by all subsequent writers.

Don Jorge Juan's Conyendio de Navegaci on, for the use of mid.

shipmeu, was published at Cadiz in 1757. Chapter i. explains what pilotage is, practical and theoretical. He speaks of the change m variation, "which sailors have not believed and do not believe now.' He described the lead, log, and sand-glass, the latter corrected by a pendulum, charts plane and spherical. Supposing his readers tc be versed in trigonometry, he will explain what latitude and longi• tude are, and show a method for finding the latter different from what has been taught. He will show the error of middle latitude sailing, and show that the longitude found by it is always less than the truth. (It is strange that while reckoning was so rough and imperfect in many respects they should strain at such a trifle as that is in low latitudes.) He promised to find the difference of longitude without a departure (a similar rule to that of Bond). After speaking of meridional parts, he offered to explain the English method, which was discovered by Edmund Halley, but omits the principles upon which Halley founded his theory, as it was too embarrassing. (He was not the first.) He gives instructions for currents and leeway, tables of declination, a few stars, meridional parts, &c. It is worthy of remark that, after giving a form for a log-book, he added that this had not been previously kept by any one, but he thought it should not be trusted to memory. He only required the knots, fathoms, course, wind, and leeway to be marked every two hours. Every hour is quite long enough, and that is often divided now. He gave a sketch of Hadley's quadrant, iu shape like those in use fifty years back, but without a clamping screw or tangent screw. Back glasses were much valued in those days, - the force of habit, no doubt. The book is quite free from all extraneous rubbish.

The introduction of timekeepers by which Greenwich time can be carried to any part of the world, and the longitude found with ease, simplicity, and certainty, is due to the invention of John Harrison. The idea of keeping time at sea was no novelty. HUYGENS (q.v.) made pendulum watches for the purpose prior to 1665, at which date Major Holmes communicated to the Royal Society (Phil. Trans., i. 13) the fact of his having tried two of them on the coast of Guinea. He sailed from St Thomas, set his watches, sailed west 700 or 800 leagues, without changing course, then steered towards the coast of Africa N.N.E. 200 or 300 leagues. The masters of the other ships under his charge, fearing the want of water, wished to steer for Barbados. Holmes, on comparing the calculations, found them to differ from him from 80 to over 100 leagues. He considered that they were only 30 leagues from the Cape Verd Islands, where they arrived next afternoon. The vague manner of estimating distance is worth notice. William Derham published a scientific description of various kinds of timekeepers in The Artificial Clock-Maker, in 1700, with a table of equations from Flamsteed to facilitate comparison with the sun-dial. In 1714 Henry Sully, an Englishman, published a treatise at Vienna, on finding time artificially. He went to France, and spent the rest of his life in trying to make a timekeeper for the discovery of the longitude at sea. In 1716 he presented a watch of his own make to the Academy of Sciences, which was approved ; and ten years later he went to Bordeaux to try his marine watches, and died before embarking. Julien le Roy was his scholar, and perfected many of his inventions in watchmaking.

Harrison's great invention was the principle of compensation through the unequal contraction of two metals, which he first applied in the invention of the compensation pendulum, still in use, and then modified so as to fit it to a watch, devising at the same time a means by which the watch retains its motion while being wound up. To what has been said in the article HARRISON on his successive attempts, and the success of the trial journey to Jamaica in 1761-62, it may be added that by the journal of the House of Commons we find that the error of the watch (as if there were only one) was ascertained by equal altitudes at Portsmouth and Barbados, the calculations being made by Short. The watch came greatly within the limits of the Act. At Jamaica it was only in error five seconds (assuming that the longitude previously found by the transit of Mercury could be so closely depended on, which as we now know, was not the case; the observations were too few in number, and taken with an untrustworthy instrument). Short found the whole error from November 6, 1761, till April 2, 1762, to be P" 548•5 =18 geographical miles in the latitude of Portsmouth. He considered that a position determined by a transit of Mercury was liable to an error of 308 only, and by Jupiter's best satellite to 3m 44'. During the passage home in the " Merlin " sloop-of-war the timekeeper was placed in the after part of the ship, because it was the dry-est place, and there it received violent shocks which retarded its motion. It lost on the voyage home Pi' 498=16 geographical miles.

One might have supposed that Harrison had now secured the prize ; but there were powerful competitors who hoped to gain it by lunars, and a bill was passed through the House in 1763 which left an open chance for a lunarian during four years. A second West Indies trial of the watch took place between November 1763 and March 1764, in a voyage to Barbados, which occupied four months ; during which time it is said, in the preamble to Act 5 Geo. III. 1765, not to have erred 10 geographical miles in longitude. We only find in the public records the equal altitudes taken at Portsmouth and at Bridgetown, Barbados. William Harrison assumed an average rate of 18 a-day gaining, as he anticipated that it would go slower by 18 for every 10° increase in temperature. The longitude of Bridgetown was determined by Maskelyne and Green by nine emersions of Jupiter's first satellite, against five of Bradley's and two at Greenwich Observatory, to be 3h 54m 20' west of Greenwich. In February 1765 the commissioners of longitude expressed an opinion that the trial was satisfactory, but required the principles to be disclosed and other watches made. Half the great reward was paid to Harrison under Act of Parliament in this year, and he and his son gave full descriptions and drawings, upon oath, to seven persons appointed by the commissioners of longitude.' The other half of the great reward was promised to Harrison when he had made other timekeepers to the satisfaction of the commissioners, and provided he gave up everything to them within six months. The second half was not paid till 1773, after trials had been made with five watches. These trials were partly made at Greenwich by Maskelyne, who, as we shall see, was a great advocate of lunars, and was not ready to admit more than a subsidiary value to the watch. A bitter controversy arose, and Harrison in 1767 published a book in which he charges Maskelyne with exposing his watch to unfair treatment. The feud between the astronomer-royal and the watchmakers continued long after this date.

Even after Harrison had received his £20,000, doubts were felt as to the certainty of his achievement, and fresh rewards were offered in 1774 both for timekeepers and for improved lunar tables or other methods. But the tests proposed for timekeepers were very discouraging, and the watchmakers complained that this was due to Maskelyne. A fierce attack on the astronomer's treatment of himself and other watchmakers was made by Mudge in 1792, in A Narrative of Facts, addressed to the first lord of the Admiralty, and Maskelyne's reply does not convey the conviction that full justice was done to timekeepers. Maskelyne at this date still says that he would prefer an eclipse of a bright star by the moon and a number of correspondent observations by transits of the moon compared with those of fixed stars, made by two astronomers at remote places, to any timekeeper. The details of these controversies, and of subsequent improvements in timekeepers, need not detain us here. In England the names of Arnold and Earnshaw are prominent, each of whom received, up to 1805, £3000 reward from the commissioners of longitude. It was Arnold who introduced the name chronometer.' The French emulated the English efforts for the production of good timekeepers, and favourable trials were made between 1768 and 1772 with watches by Le Roy and Berthoud.

Meantime the steady progress of astronomy both by the multiplication and increased accuracy of observations, and by corresponding advances in the theory, had made it possible to construct greatly improved tables. In observations of the moon Greenwich still took the lead ; and it was here that Halley's successor Bradley made his two grand discoveries of aberration and nutation which have added so much to the precision of modern astronomy. Kepler's Rudolphine tables of 1627 and Street's tables of 1661, which had held their ground for almost a century, were rendered obsolete by the observations of Halley and his successor. At length, in 1753, in the second volume of the Commentarii of the Academy of Gottingen, Tobias Mayer printed his new solar and lunar tables, which were to have so great an influence on the history of navigation. Mayer afterwards constructed and submitted to the English Government in 1755 an improved body of MS. tables. Bradley found that the moon's place by these tables was generally correct within 1', so that the error in a longitude found by them would not be much more than half a degree if the necessary observations could be taken accurately on shipboard. Thus the lunar problem seemed to have at length become a practical one for mariners, and in England it was taken up with great energy by N. Maskelyne" the father," as he has been called, "of lunar observations."

In 1761 Maskelyne was sent to St Helena to observe the transit of Venus. On his voyage out and home he used Mayer's printed tables for lunar determinations of the longitude, and from St Helena he wrote a letter to the Royal Society (Phil. Trans., vol. lii. p. 558, 1762), in which he described his observations made with Hadley's quadrant of 20 inches radius, made by Bird, and the glasses ground by Dollond. He took the observations both ways to avoid the errors. The arc and index were of brass, the frame mahogany ; the vernier was subdivided to minutes. The telescope was 6 inches long, magnified four times, and inverted. Very few seamen in that day possessed so good an instrument. He considered that ship's time should be ascertained within twelve hours, as a good common watch will scarcely vary above a minute in that time. This shows that he must have intended the altitude of sun or star to be calculated - which would lead to new errors. He considered that his observations would each give the longitude within 1 degrees. On 1'ebruary 11th he took ten; the extremes were a little over one degree apart.

On his return to England Maskelyne prepared the British Mariner's Guide (1763), in which he undertakes to furnish complete and easy instructions for finding the longitude at sea or on shore, within a degree, by observing the distance between the moon and sun, or a star, by Hadley's quadrant. How far that promise was fulfilled, and the practicability of the instructions, are points worth consideration, as the book took a prominent place for some years. The errors which he said were inseparable from the dead-reckoning "even in the hands of the ablest and most skilful navigators," amounting at times to 15 degrees, appear to be overestimated.

On the other hand, the lunar equations, which were from Mayer's tables, would, he believed, always determine the longitude within a degree, and generally to half a degree, if applied to careful observations. He recommends the two altitudes and distance being taken simultaneously when practicable. The probable error in a meridian altitude he estimated at one or two minutes, and in a lunar distance two minutes (equal to one degree of longitude). He then gave clear rules for finding the moon's position and distance by ten equations, too laborious for seamen to undertake. Admitting the requisite calculations for finding the moon's place to be difficult, he desired to see the moon's longitude and latitude computed for every twelve hours, and hence her distance from the sun and from a proper star on each side of her carefully calculated for every six hours, and published beforehand.

In 1765 Maskelyne became astronomer-royal, and was able to give effect to his own suggestion by organizing the publication of the Nautical Almanac. The same Act of 1765 which gave Harrison his first £10,000 gave the commissioners authority and funds for this undertaking. Mayer's tables, with his MS. improvements up to his death in 1762, were bought from his widow for £3000 ; £300 was granted to the famous mathematician Euler, on whose theory of the moon Mayer's later tables were formed ; and the first Almanac, that for 1767, was published in the previous year, at the cost and under the authority of the commissioners of longitude. This was not the first almanac in the country, perhaps by a hundred, as that name was applied to small periodical works, frequently of a frivolous character, - though the later and better description gave the sun's declination and moon's meridional passage approximately. In 1696 the French nautical almanac for the following year appeared, an improvement on what had been before issued by private persons, but it did not attempt to give lunar distances.' In the English Nautical Almanac for 1767 we find everything necessary to render it worthy of confidence, and to satisfy every requirement at sea. The great achievement was that of giving the distance from the moon's centre to the sun, when suitable, and to about seven fixed stars, every three hours. The mariner has only to find the apparent time at ship, and clear his own measured distance from the effects of Parallax and refraction (for which at the end of the book are given Lyon's and Dun-thorn's methods), and then by simple proportion, or proportional logarithms, find the time at Greenwich. The calculations respecting the sun and moon were made from Mayer's last manuscript tables under the inspection of Maskelyne, and were so continued till 1804.3 The calculations respecting the planets are from Halley's tables, and those of Jupiter's satellites from tables made by required ; and greater accuracy at that time was not desirable, or at least would not have been appreciated.

Page 1 of each month gave the Sundays and holidays, four phases of the moon, and positions of sun, moon, and planets in the signs of the zodiac; page 2, sun's longitude, right ascension in time, declination, and equation for noon each day ; page 3, sun's semidiameter, time of passing the meridian, hourly motion of the sun, logarithm of sun's distance, and place of the moon's node, for every sixth day ; also eclipses of Jupiter's satellites, time of immersion ; page 4, the positions of the four principal planets for every sixth day ; page 5, the configuration of Jupiter's satellites at 11 P.m. of every day; page 6, the moon's longitude and latitude for noon and midnight of every day; page 7, the moon's age, passage over the meridian, right ascension, and declination at noon and midnight ; page 8, the moon's semidiameter, horizontal parallax, and logistic logarithm - each at noon and midnight; pages 9 to 12, the moon's centre from the sun and seven stars for every three hours, while within about 116 degrees. Then follow tables of refraction, moon's parallax in altitude, a catalogue of stars, with their right ascension and declination, table for the " dip " of the sea horizon, and several other useful things, many of which are omitted in modern Nautical Almanacs, as they are included in and more properly belong to the permanent rules and requirements of navigation.

Various useful rules and tables were appended to early volumes of the Almanac. Thus the volume for 1771 contains a method and table for determining the latitude by two altitudes and the elapsed time first published by Cornelius Downes of Amsterdam in 1740.1 At the end of the Almanac for 1772 Maskelyne and Whichell gave three special tables for clearing the lunar distance ; still their rule is neither short nor easily remembered. An improvement of Dunthorn's solution is also given, and a problem in Mercator's sailing by Halley solved by Israel Lyons,' viz., the latitude of the point of departure given, distance sailed, and change of longitude, - required the course steered. In the edition for 1773 a new table for equations of equal altitudes is given by W. Whale. In those for 1797 and 1800 tables are added by John Brinkley for rendering the calculations for double altitudes easier.

From 1777 to 1788 inclusive, the moon's place was calculated from improved tables by Charles Mason, recently from vol. iii. of Professor Vince's Astronomy.

lambre's new tables. In 1827 the positions of sixty of the principal stars were given for every tenth day, from the tables of Maskelyne and Dr Pearson. Since 1824 the work has been printed three and latterly four years in advance. The price was 5s. till 1855; but the Almanacs for that and subsequent years have been issued at 2s. 6d.

A book of Tables Requisite to be Used with the Nautical Ephemeris was published byMaskelyne at the same time as the first Almanac, and tell thousand copies were quickly sold. A second edition, prepared by W. Wales, appeared in 1781, an octavo of 237 pages, in the preface of which it is stated, with apparent truth, that it contains everything necessary for computing the latitude and longitude by observation. There are in all twenty-three tables, the traverse table and table of meridional parts alone being deficient as compared with modern works of the kind; dead-reckoning Maskelyne did not touch. He gave practical methods for working several problems ; the lunar especially is an improvement on those by Lyons and Dun-thorn, though a rule there given for clearing the distance, called Dunthorn's improved method, is remarkably short. The half sum of three logarithms gives an arc, and the half sum of other two gives half the true distance. The objection is the use of special logarithms. Maskelyne's rule for finding the latitudes by two altitudes and the elapsed time is also good, but with the same objection. The third edition of the Tables was issued in 1802. It has been said that Maskelyne neglected the planets ; be that as it may, he established the positions of sixty of the principal stars, and completed many other things. He had but one assistant, whereas there are now eight, and the Nautical Almanac is under another department.

As the necessary calculations for clearing the lunar distance from the effects of parallax and refraction were considered difficult to seamen, many efforts were made to shorten the process. Among others Whichell, master of the Loyal Naval Academy, Portsmouth, conceived a plan whereby it could be taken from a table by inspection. In October 1765 the commissioners of longitude awarded him £100 to enable him to complete and print 1000 copies of his table. On the following April they gave him £200 more. The work was continued on the same plan by Shepherd, the Plumian professor of astronomy, Cambridge, with some additions by the astronomer-royal. The total cost of the ponderous 4to volume up to the time of publication in June 1772 was £3100, after which £200 more was paid to the Rev. Thomas Parkinson and Israel Lyons for examining the errata. It is a very large and expensive volume, - very ill-adapted for ship's use. Considerable sums were paid by the commissioners from time to time for other tables to facilitate navigation - not always very judiciously. It is sufficient to mention here the tables of Michael Taylor and the still esteemed tables of Mendoza, published in 1815. Here also may be mentioned a useful table by Stevens (1780) for finding the latitude by the altitude of the pole star, and Crosswell's tables for facilitating the computation of lunars - partly new and partly after Maskelyne. These appear to be the first tables in which half the logarithmic sine, &c., is given, to save the trouble of halving a sum of four or more logarithms.

The plan of the Nautical Almanac was soon imitated by other nations. In France the Academie Royale de Marine had all the lunar distances translated from the British Nautical Almanac for 1773 and following years, retaining the Greenwich time for the three-hourly distances. The tables were considered excellent, and national pride was satisfied by their having been formed on the plan proposed by .Lacaille. They did not imitate the mode given for clearing the distance, considering their own better.

Though the Spaniards were leaders in the art of navigation during the 16th and 17th centuries, it was not till November 4, 1791, that their first nautical almanac was printed at Madrid, having been previously calculated at Cadiz for the year 1792. They acknowledge borrowing from the English and French. The lunar distances were reduced from Greenwich meridian to that of Cadiz, by subtracting 25m 95. It is in larger and better print than the French almanac. In the book for 1803 the meridian of the royal observatory at Isla de Leon is placed 24m 475.5 west of Greenwich. In the English almanac for 1883 it is given as 24m 495.6; therefore they were very near the truth in 1791. The almanac for 1810, published at Madrid in 1807, was the first in which the lunar distances were reduced to the meridian of Isla de Leon - that is, giving the distance to the even hours 3, 6, 9, 12, &.c. The Spanish almanacs for 1813 to 1816 were published in Fleet Street, the first-named only one year in advance. From 1820 to 1832 they were good octavos and clear type. Soon after that time they appeared in folio, giving nearly all the information found in the English almanac, but not neglecting the saints' days and festivals. The excellent Berlin A stronomisches faltrbuch began to appear in 1776, the American Ephemeris in 1849. These two ephemerides and the French Connaissance des Temps are independent and valuable works, and for astronomers at least in some respects superior to the English Almanac.

After Maskelyne's death the correctness and reputation of the Nautical Almanac underwent a serious decline. The matter came before parliament in 1818, when the board of longitude was reconstructed, and the old Acts consolidated. Dr T. Young was appointed secretary to the commissioners, and superintendent of the Almanac. Ten years later, in 1828, the board was swept away, the Almanac was placed under the Admiralty, and Young, with Faraday as a chemist and Sabine as a practical observer, were appointed scientific advisers to the Admiralty, which ever since has spent a certain annual sum on scientific research. The Almanac still gave cause for dissatisfaction ; a memorial to the House of Commons, dated January 28, 1829, states that the Nautical Almanac was for the good of astronomy as well as navigation, and that it is so declared in the first Almanac in 1767 ; that in 1818 fifty-eight errors were discovered, and a similar number in the Almanac for 1830, and that it had not kept pace with navigation or astronomy ; that it did not give the moon's distance from the four principal planets as the Portuguese and Danish ephemerides did, nor did it give the positions of those planets ; that there was no list of the occultating stars which were ascertained to be visible in Halley's time, but were neglected after the invention of Hadley's sextant (they were in the Milan ephemerides); and that the tables of the sun were not correct. This was supported by a paper signed by J. F. W. Herschel, read at the board of longitude April 5, 1827, which stated that the moon's meridian passage was not given at all, that of the sun roughly to the nearest minute. The right ascension and declination of the larger planets were not given with accuracy, as they should have been, as their theory was perfect. The moon's right ascension in time and hourly motion should have been given, also the time of semi-diameter passing the meridian, for use with moon-culminating stars. Young replied to this memorial and maintained that the fifty-eight errors were exaggerated; forty of them were in reality only one in the moon's place, which would put a ship out 5 miles, and which was corrected in the next year's book, "which every accurate navigator is bound to consult, to guard against possible minute accident." The errors of 1830 were, he says, of less importance : the French Connaissance des Temps of 1821 was corrected by the English Almanac; some errors were found in Taylor's logarithms ; the error in the solar tables, said to be 15 seconds, was really only one. The ultimate result of these controversies was the appearance of the new and reformed Nautical Almanac in 1834. It may be added that the last remnant of the old laws, the protection of the Almanac against competition by a penalty, was abolished by an Act passed August 6, 1861. The number -of copies of the Nautical. Almanac (for 1851) printed in