the following paragraph was inserted in a smaller character: About 1672 M. Picart published an account in French, concerning the measure of the earth, a breviate whereof may be seen in the Philosophical Transactions, No. cxii.; wherein he concludes 1° to contain 365,184 English feet, nearly agreeing with Mr. Norwood's experiment;' and this advertisement is continued in the subsequent editions as late as 1732. About 1645 Mr. Bond published, in Norwood's Epitome, a very great improvement in Wright's method; it was deduced from the theorem, that these divisions are analogous to the excesses of the logarithmic tangents of half the respective latitudes, augmented by 45° above the logarithm of the radius. This he afterwards explained more fully in the edition of Gunter's works, printed in 1653; where, after observing that the logarithmic tangents from 45° upwards increase in the same manner that the secants added together do; if every half degree be accounted as a whole degree of Mercator's meridional line; his rule for computing the meridional parts belonging to any two latitudes, supposed on the same side of the equator, is to the following effect: Take the logarithmic tangent, rejecting the radius, of half each latitude, augmented by 45°; divide the difference of those numbers by the logarithmic tangent of 45° 30′, the radius being likewise rejected; and the quotient will be the meridional parts required, expressed in degrees.' This rule is the immediate consequence from the general theorem, that the degrees of latitude bear to 1° (or 60', which in Wright's table stand for the meridional parts of 1°) the same proportion as the logarithmic tangent of half any latitude augmented by 45° and the radius neglected, to the like tangent of half a degree augmented by 45°, with the radius likewise rejected.

But still there was wanting the demonstration of this general theorem, and it was at length supplied by Mr. James Gregory of Aberdeen, in his Exercitationes Geometricæ, printed at London in 1668; and it was afterwards more concisely demonstrated, together with a scientific determination of the divisor, by Dr. Halley in the Philosophical Transactions for 1695, No. ccxix., from the consideration of the spirals into which the rhumbs are transformed, in the stereographic projection of the sphere upon the plane of the equinoctial; and which is rendered still more simple by Mr. Roger Cotes, in his Logometria, first published in the Philosophical Transactions for 1714, No. ccclxxxviii. It is added in Gunter's book, that if one-twentieth of this division, which does not sensibly differ from the logarithmic tangent of 45° 1′ 30′′ (with the radius subtracted from it), be used, the quotient will exhibit the meridional parts expressed in leagues; and this is the divisor used in Norwood's Epitome. After the same manner the meridional parts will be found in minutes, if the like logarithmic tangent of 45° 1′ 30′, diminished by the radius, be taken; that is, the number used by others being 12,633, when the logarithmic tables consist of eight places of figures besides the index.

In an edition of the Seaman's Calendar, Mr. Bond declared that he had discovered the longi

tude by having found out the true theory of the magnetic variation; and, to gain credit to his assertion, he foretold that at London, in 1657, there would be no variation of the compass, and from that time it would gradually increase the other way; which happened accordingly. Again, in the Philosophical Transactions for 1668, No. xl., he published a table of the variation for fortynine years to come.. Thus he acquired such reputation, that his treatise, entitled The Longitude Found, was, in 1676, published by the special command of Charles II., and approved by many celebrated mathematicians.

It was not long, however, before it met with opposition; and, in 1678, another treatise, entitled The Longitude not Found, made its appearance; and, as Mr. Bond's hypothesis did not in any manner answer its author's sanguine expectations, the affair was undertaken by Dr. Halley, who, in 1700, published a general map, with curve lines expressing the paths where the magnetic needle had the same variation; which was received with universal applause. But, as the positions of these curves vary from time to time, they should frequently be corrected by skilful persons; as was done in 1744 and 1756 by Mr. William Mountaine, and Mr. James Dodson, F. R.S. In the Philosophical Transactions for 1690 Dr. Halley also gave a dissertation on the monsoons; containing many very useful observations for such as sail to places subject to these winds.

After the true principles of the art were settled by Wright, Bond, and Norwood, the authors on navigation became so numerous that it would be impossible to enumerate them; and every thing relative to it was settled with an accuracy not only unknown to former ages, but which would have been reckoned utterly impossible. The earth being found to be a spheroid, and not a perfect sphere, with the shortest diameter passing through the poles, Dr. Murdoch published a tract, in 1742, in which he accommodated Wright's sailing to such a figure; and, in the same year, the celebrated Maclaurin gave a rule for determining the meridional parts on a spheroid; and he extended his speculations on the subject farther in his work on Fluxions.

This theoretical refinement has not however in any instance been reduced to practice, as the data for determining the plan of a ship at sea can never be obtained with such precision as would justify a practical man in introducing it as a matter of correction.

Among the latter improvements in the science of navigation the methods of finding the longitude by lunar observations and time-keepers are the principal. To such perfection are these methods now brought, that it has been observed by a gentleman of distinguished nautical skill, whose situation imposes on him the duty of examining the logs of all ships belonging to one of the first trading companies in the world, that the longitudes of ships are often determined more exactly than their latitudes.

We may notice also an important improvement in the method of finding the latitude by two altitudes, and the elapsed time, given by Mr. Riddle in his Treatise on Navigation, an improvement

which has tended to bring that useful problem into more general use among practical seamen. The method of deducing the iongitude from occultations has also been greatly simplified. See LONGITUDE.

For the perfection which the lunar method has attained we are chiefly indebted to the late Dr. Maskelyne; and the highest credit is due to the British parliament for the encouragement which they have given to all who have usefully labored in supporting any department of this important branch of knowledge.

At present it may be safely affirmed that no great improvements in the science are to be looked for. The tables of the planetary motions appear nearly adequate to all the wants, and certainly to all the reasonable wishes, of the navigator; though we are happy to understand that professor Airy of Cambridge is engaged in still farther improving them. As an art, navigation may be considered to have nearly reached its limit of perfection; for further refinements in the theory of astronomy will by no means insure a corresponding increase of accuracy in its practical application; for there exists a limit in the size of the instruments which can be managed on shipboard, and in the imperfections of every thing which is to be accomplished by a being of powers so limited as man.

The motion of a ship in the water depends on the action of the wind on its sails, and is regulated by the direction of the helm. There is always a great resistance on the fore part of the ship when in motion, and, when this resistance becomes sufficient to balance the force of the wind on the sails, the motion of the ship is no longer accelerated, but becomes uniform. This maximum velocity depends on the strength of the wind; but as the resistance increases with the velocity, whatever may be the force of the wind, there is a limit to the velocity of the ship; for the sails and ropes can bear but a certain force; and, when the resistance of the water becomes equal to their strength, the velocity cannot be increased, and the tackle gives way.

The direction of a ship's motion depends on the situation of her sails with regard to the wind. The most natural position is when she runs directly before it; but this cannot often be done, from the variable nature of the winds, and the situations of the places to which the ship may be bound. When the wind therefore is unfavorable the sails and rudder must be so placed that the ship's way may make an angle with the direction of the wind.

The ship moves forward under such circumstances because the water resists the side more than the forepart as much as the length of the ship exceeds the breadth; and this proportion is so considerable that the ship moves in the direction in which the resistance is least, and sometimes very swiftly. But if the angle made by the keel, with the direction of the wind, be too acute, the ship cannot be kept in that position; and a barge cannot be kept nearer the direction of the wind than within about 67° 30′, though small sloops may sometimes lie and sail within about 50°. But in such circumstances the velocity of the vessel is greatly retarded, both on

account of the obliquity of her motion, and of what is called ner leeway. This is occasioned by the yielding of the water on the leeside of the ship, in consequence of which the vessel moves in a sort of diagonal direction, between the direction of the wind, and that in which it is wished that she should go.

It would be a matter of extreme difficulty to determine from theoretical principles what the leeway in any given case would amount to; as it depends on the strength of the wind, the roughness of the sea, the velocity, the shape, and the trim of the vessel.

When the wind is light the resistance on the lee side bears a great proportion to the strength of the wind, and it therefore yields very little. But the water having once begun to yield will continue to do so for some time, and the leeway will increase till the resistance on the lee side balances the force of the wind on the other, when it will become uniform. The leeway will be less as the velocity of the ship is greater; and in a strong gale, when the ship makes little headway, the leeway will be greatest of all.

When there is a rough sea, the whole water of the ocean, to a considerable depth, acquires a motion in a particular direction. The rolling waves carry the ship out of her course, and the deviation is in proportion to their velocity and magnitude. From all these causes it arises that there is very great difficulty in determining the actual course of a ship at sea; and it becomes in consequence a matter of the utmost importance to determine her place as often as possible by celestial observations.

In many places of the ocean there are currents which run with considerable velocity. They occasion errors of the same nature as leeway, only that they affect the distance as well as the course. Whenever a current is perceived, its velocity and direction ought to be determined and allowed for.

Another source of embarrassment to the mariner is the continually changing variation of his compass. In few situations the points of that invaluable instrument correspond with what they ought to indicate on the horizon, and even in the same place the variation itself is in a state of variation. The astronomical methods of determining the variation are however both numerous and simple.

But it has recently been discovered that the whole mass of iron in a ship acts on the needle of the compass as one great magnet placed in its vicinity. As the centre of this mass is always in the forepart of the vessel, an ingenious philosopher, Mr. Barlow of Woolwich, conceived that its effect might be counteracted by placing a small mass behind the compass, so near it, and so situated with respect to it, that it might act as a counterbalance, by its great influence, to the general effect of the mass. This happy idea has been put to the test, and found to answer completely the anticipations which were entertained respecting it.

It is of importance to navigators in long voyages to reach their port by the shortest practicable route. The shortest distance between two points on the surface of a sphere is along a great circle intercepted between them. But it is no

easy matter to keep a ship on a great circle, as the only means of guiding her in any particular direction at sea is the compass, by which she is kept on a rhumb line, which, unless the ship is sailing exactly north or south, or, on the equator, exactly east or west, does not coincide with a great circle.

But a small portion of a rhumb line may be conceived to be identical in direction with a great circle; and, if the distance be divided into small portions, the course at each point may readily be computed by spherical trigonometry, and the ship kept, though not exactly on the arc of the great circle on which it is desirable that she should sail, yet sufficiently near it for all practical purposes.

PRINCIPLES OF NAVIGATION.

The earth is nearly a globe of 7916 English miles in diameter, and it revolves on an imaginary line called its axis, from west to east in twenty-four hours; and it is this rotation that causes the apparent diurnal motion of the heavenly bodies from east to west.

That the art of navigation may be successfully practised, it is necessary that the mariner should be furnished with accurate maps of the seas through which he sails, and of the coasts and harbours which he may have occasion to visit; and be well acquainted both with the use of all instruments necessary for determining the ship's place from celestial observations, and the methods of deducing the desired conclusions from observations when taken.

Great circles passing through the poles of the earth are called meridians, and that great circle which is equidistant from both poles is called the equator, the equinoctial, or the line; and less circles, parallel to the equator, are called parallels of latitude. The meridian passing over any place is called the meridian of that place, and the portion of a meridian between any place and the equator is called the latitude of that place, and it is called north or south latitude, according as the place is north or south of the equator.

The difference of latitude between two places is an arc of the meridian intercepted between their parallels of latitude. Hence, when the latitudes of two places are both north or both south, their difference is the difference of latitude; but when one is north and the other south their sum is the difference of latitude.

It is customary to call the meridian of some remarkable place or observatory the first meridian; and the angle included between that first meridian, and the meridian of any other place is called the longitude of that place, or of any place under the same meridian. These polar angles are measured by the arcs which they intercept on the equator; and hence in navigation the longitude of a place may also be said to be, or to be measured by, the arc of the equator intercepted between the first meridian and the meridian of that place; and it is considered as east or west, according as the place is eastward or westward of the first meridian. Englishmen refer to the meridian of Greenwich observatory as the first meridian; French to that of the

observatory at Paris; Spaniards to that of Ca diz; but the Danes to that of Greenwich.

The difference of longitude between two places is the angle at the pole contained by the meridians of those places, in the arcs of the equator which they intercept; and hence when the longitudes are of the same denomination, their difference, but when of contrary denominations their sum is the difference of longitude. A curve on the globe which cuts every meridian which it meets on the same angle is called a loxodromic or rhumb line; a ship at sea being guided by the compass is steered upon a curve of this kind. The angle which a rhumb line makes with the meridian is called the course between any two places through which that rhumb passes; and the arc of a rhumb line between two places is called their nautical distance, to distinguish it from the least distance, which is the arc of a great circle passing through both places.

If a ship be steered due north or due south her distance and difference of latitude are the same.

Meridional distance is an arc of the parallel of latitude arrived at, intercepted between the meridian left, and that arrived at; and departure is the sum of all the intermediate meridional distances, computed on the supposition that the distance is divided into indefinitely small equal parts.

If a ship sail due east or west she will eithe sail on the equator or on some parallel of latitude, and her meridian distance, departure, and distance sailed, will all be the same.

The bearing between two places, or the bearing from one place to another, on the same parallel of latitude is east or west, on the same meridian north or south, and in all other situations on an oblique rhumb line, continually approaching the pole.

OF PLANE SAILING.

Proposition 1.-In sailing on a rhumb line the differences of latitude are exactly proportional to the distances sailed.

For in Plate I. fig. 1, if PK, PL, PM, PN, and PO, be meridians, AE a rhumb line passing through A and E, and cutting every meridian which it meets at the same angle, FE and AI parallels of latitude; and if AB, BC, CD, &c., portions of the rhumb line, be considered to be equal, indefinitely small, and their number indefinitely great, the triangles A Bb, BC e, &c., may be considered as indefinitely small identical plane triangles; whence AE will be the same multiple of A B that the sum of Ab, Bc, Cd, &c., is of Bb. But the sum of Ab, B c, &c., is equal to the whole difference of latitude A For EI. Again, by reasoning in a similar way, we find that A E is the same multiple of AB that the sum of Bc, Cc, &c., is of Bb; and the sum of Bb, Cc, &c., is what has been denominated the definition.

Hence, as AB b is a right-angled plane triangle, a straight line equal to the curve A E, and one equal to the arc of the meridian AF, will form the base and perpendicular of a rightangled plane triangle as PQR, fig. 2, in which PQ will represent AE; PR, AF, and RQ,

the departure or the sum of Bb, Cc, &c., and the angle P will represent the angle EAF, the course. While the course remains the same it is evident the departure is greater than the meridional distance FE (see fig. 1) and less than A I, but nearly equal to RS, the meridional distance in the middle latitude, between the latitude sailed from, and the latitude arrived at. When the places are near the equator, or when their parallels are not very distant from each other, the nautical conclusions drawn from a supposition that the departure is equal to the meridian distance in the middle latitude, are very nearly correct; but in high latitudes, or in a long run, when the course is not near some of the cardinal points, this principle gives results deviating more from the truth than is desirable. If a ship sail on several courses she makes a less departure near the pole, and a greater one near the equator, than if she sail on a direct course; but in such small distances as a day's run, the difference is almost insensible. In problems solved on the principles of this proposition the conclusions are the same as if the earth were a plane, and all the meridians parallel to each other; and it is hence that it is called plane sailing, whose principles are therefore correct as far as difference of latitude, course, distance, and departure are concerned.

Example.-1. If a ship sail from Oporto N. W. W. 315 miles. required her departure, and the latitude which she has arrived at?

By construction.-Draw the vertical line of A B fig. 3, to represent the meridian of Oporto; with the centre A, and chord of 60°, describe the arc D E, on which from the line of rhumbs longitude of D E = 41 points, the given course. Draw A C, and make it equal to 315 from any convenient scale of equal parts. From C draw CB perpendicular to A B; then AB and BC will represent the difference of latitude and departure, and, if measured on the scale from which AC was taken, it will be found that AB 2116 and BC= 233.4 miles, and the departure is westerly as the course is westerly.

By Gunter's scale.-Extend on the line of sine rhumbs from 8 points to 41, and that extent applied from 315 on the line of numbers will reach towards the left to 233-4, the departure. Again, extend on the line of sine rhumbs from 8 points to 33 points, the complement of the course; and that extent will reach on the line of numbers from the distance 315, to the difference of latitude 211-6

By inspection in a table of difference of latitudes and departures.-Seek the given course in the table, and opposite the distance will be found the difference of latitude and departure. If the distance is beyond the limits of the table, seek the difference of latitude and departure corresponding to the several parts of it, and their sums will be the whole difference of latitude and departure.

In the present example, with course 44 points and dist. 300, we have diff. lat. 2015, dep. 222-3; with course 4 points and dist. 15, we have diff. lat. 10-1, and dep. 11'1; hence the whole diff. lat. is 2116, and dep. 233-4, as before.

By computation.-As rad. dist. A C, 315 :: cos. A, the course 41 points: A B the diff. lat. 2116.

By Gunter's scale.-Extend on the line o numbers from 71 to 76, that extent will reach on the line of tangents from radius, or tan. 45° to 47°, the course. Extend from the complement of this course to radius on the line of sines, and the extent will reach from the diff. lat. 71, towards the right to the distance 104 on the line of numbers.

By inspection.-Seek in a table of difference of latitude and departure till 76 dep. and 71 diff. lat. correspond in their proper columns, and the corresponding course and dis ance vill be found as above.