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Now the difference between 17° 35′ and 6° 12′ is 11° 23': consequently the ship would be at Rikefiord in Iceland when the prevailing mode of computation made her 11° 23′ to the eastward of that place.

18. As tides, currents, and heave of the sea frequently affect the ship's velocity and direction, it is necessary that the navigator should investigate their nature and effect: the solution of this proposition is called CURRENT SAILING, and comprehends three cases.

(1). When the ship sails in the direction of the current. Then it is obvious that her velocity will be augmented by the velocity of the current. For instance; if the ship's apparent velocity, or rate of sailing as given by the log, be 8 knots, and the drift of the current be 2 knots, the ship's absolute velocity is equal to their sum, that is, 10 knots.

(2). When the ship sails directly against the current: in which case her absolute velocity will be equal to the difference between the apparent velocity and the velocity of the current. If these velocities be equal, the ship will remain stationary: if the velocity of the ship be greater than that of the current, ber absolute motion will be a-head: but if it be less, she will make stern-way. For example, if the ship's rate of sailing, as given by the log, be 6 knots, she will remain stationary in a current whose drift is 6 knots: her absolute motion a-head will be 2 knots, if the drift of the current be 4 and her sternway will be 2 knots, if the drift of the current be 8.

(3). When the ship's course is oblique to the setting of the current: then her true course and distance will be compounded of the apparent course and distance, and of the setting and drift of the current. And the distance made good in a given time will be represented by the diagonal of a parallelogram, of which the apparent distance run by the ship and the drift of the current in that time are the sides. For example, suppose,

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2.6785096. 3.1478308. 3.5845962,

A ship from latitude 48° 30'N. longitude 15°24′W. sails S. W. 8 miles an hour for 24 hours in a current setting W. b. S. 3 miles an hour: to find the latitude and longitude come to.

Through the point A (Fig. 7.) from which the ship sailed, draw the indefinite meridian AG, the SW rhumb AB, and the W. b. S. rhumb Ac, making AB = 192 (24 x 8) and AC=72 (24×3) miles; complete the parallelogram ABDC, and draw the diagonal AD.

Writers on mechanics have shewn that the diagonal AD is the equivalent of the two forces whose ratio is expressed by AB, AC, and whose directions coincide with these lines. (See Gregory's Mechanics, Vol. I. art. 41; et alibi frequenter.) But writers on navigation consider, that as the current sets in the direction of the line AC, which is pa rallel to BD, it will neither accelerate nor retard the ship's motion towards the line BD; that is, the wind will bring her to the line BD in the same time as if the current did not act: nor will the wind which biows in the direction AB, parallel to CD, either accelerate or retard the ship's motion towards the line CD, that is, the current will carry her to that line in the same time, as if the wind did not act hence the wind acting alone would bring the ship to BD, in the same time as the current acting alone would carry her to CD; consequently, the ship at the end of that time will be found in both these lines, that is, in D, the point where they intersect.

Hence, the angle GAD is the course, AD the distance, and AE the space intercepted between A and E (the point where the perpendicular demitted from D upon the meridian AG meets the latter line) is the difference of latitude.

In order to ascertain the values of these quæsita, and likewise to determine the difference of longitude FG, we may conceive that the ship sailed from A to B, and from thence to D, and solve the compound course by the last article, thus:

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Longitude in

15 24 W.

20°26 W.

Therefore the latitude come to is 46° N. and longitude 20° 261 W.

We might here have introduced oblique and windward sailing; but conceive, that although the first be a fertile source of exercises in plane trigonometry, yet its utility is principally limited to marine surveying, and that the latter ought to be referred to seamanship.

19. The shortest distance, cæteris paribus, is his primary consideration who wishes to perform a journey in the shortest possible time: but when the advantage in point of space is counterbalanced by serious obstacles, the traveller hesitates: thus is the mariner situated in the case before us; for,

although the requisite operations in Mercator's sailing may be performed with facility, and the results are rigidly accurate, yet the space passed over by the ship which sails from A to B, agreeably to that hypothesis, is seldom the shortest, for the arc of a great circle intercepted between those points is the shortest distance between them; but a thumb coincides with an arc of a great circle only when the ship sails either upon the equator or upon a meridian (art. 3): and consequently in every other case the length of the segment of the thumb intercepted between the points A and B exceeds the intercepted arc of a great circle. Hence if a ship could be conducted upon the arc of a great circle with the same facility as upon a thumb, that mode of sailing would be highly preferable.

We have already reinarked, that the angle which a great circle makes with the meridians is continually varying; therefore in order that a ship may sail from A to B upon that arc, she must change her course at every point, or in every moment of her progless, which is manifestly impracticable; for, previous to a ship's course being known, it must continue constant for some space of time, and whilst it continues constant, the ship describes a rhumb: it is therefore impossible to conduct a ship upon the arc of a great circle, the meridians and equator excepted; yet, wind, shoals, and land permitting, she may be frequently brought upon it, and always kept near it; and this, because of the intention, is called

GREAT CIRCLE SAILING.

It is obvious that the utility of great circle sailing must be chiefly confined to extensive tracts of ocean in which the monsoons prevail, since it enables the navigator, under those circumstances, to shorten the space he must necessarily pass over: hence, notwithstanding the numerous varieties this species of sailing admits of, we shall confine our researches to the artifice by which a ship is frequently brought upon and always kept near the arc of a great circle.

In Fig. 8. Pl. 115, the port sailed from, that bound to, and the elevated pole, are respectively represented by A, B, S, and SD is a perpendicular demitted from s, upon the great circle described through ▲ and 3. We have seen that a momentaneous change of the course is impracticable, but To find the 83° 39' 55 31

Logarithmic versed sine of angle ASB =
Logarithmic sine of AS

Logarithmic sine of BS

Sum, rejecting twice radius

Α

the ship may describe rhumbs between A and a, a and b, b and c, &c. where a, b, c, &c. are asIf the distance of signed points in the arc AB. those points be small, the difference between the intercepted segments of the rhumb and are will be inconsiderable. For the circular arc in some measure represents the chord of the rhumb, and the difference between an arc of 10° and its chord is only 0017694, yet this is not to be understood as a fair criterion of the difference of the intercepted circular and spiral segments; for that difference will be much smaller when the navigation is near the equator, or the course near the meridian in any latitude, than when the course is wide and the latitude high.

Now, in order to obtain data for conducting the ship in the manner proposed, in the triangle ASB, the sides As, BS, the complements of the latitudes of A and B, together with the difference of their longitudes, measured by the angle ASB are given, hence AB and the angles SAB, SBA becomes known. Then, in the right-angled spherical triangle'SDA, the hypothenuse SA, and the angle SAD, are given to find the perpendicular SD and the angle ASD.

Having described a meridian through the points $, a: because the angle Asa is an assigned quantity, the angle Dsa is known; and therefore in the right-angled spherical triangle Dsa, the perpendicular SD and angle Dsa are given to find sa, the complement of the latitude of the point a. Thus may the latitudes of the several points b, c, d, &c. be ascertained; and consequently, from these and the assumed longitudes the courses and distances between them obtained by art. 10. For au example,

If it were required to conduct a ship, agreeably to this hypothesis, from the Cape of Good Hope, in latitude 34° 29′ S. longitude 18° 23′ E. to Bencoolen, whose latitude is 3° 49′ S. and longitude 102° 2' E. varying her course at every tenth degree of longitude: to find the latitudes and longitudes of the points (a, b, c, &c.) where her course must be altered, and also the courses and distances between them?

The Cape, Bencoolen, and the south pole, being represented by A, B, and s, respectively. In the triangle ASB, AS = 55° 31', BS = 86° 11', and the included angle ASB = 83° 39′ are given. distance AB.

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Natural versed sine of the difference of the sides BS, AS

9.8642125 =

Sum or natural versed sine of the distance AB = 82° 36′ 29′′ Then, sine AB 82° 36′ 29′′: sine angle ASB 83° 39' sine As 55° 31': sine angle ABS 55° 42′ (the angle of position at the Cape): : sine SB 86° 11': sine angle SAB 8934 (the angle of position at Bencoolen).

Hence, radius: sine SB 86° 11':: sine angle ABS 55° 42′ sine of the perpendicular SD 30′ 51′′.

:

55°

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scribed, making the angles Dsa, Bsb, DSC, &c. 11° 21° 31°, &c. as in the first column of the following table, with these angles, and the perpendicular SD the several latitudes where the ship alters her course, are found by saying, as Radius Cotangent SD 55° 30′ 51′′ Cosine of polar angle Dse 10°

Cotangent of lat. of. a 38° 58′ 30′′

Radius
Cotangent SD 55° 30'
51′′
Cosine of polar angle Ds b 21°

otangent of lat. b 32° 39′ 19′′

10-0000000

9.8366335 9.9919466

9.8285801

10.0000000

9 8366335 9.9919466

9.8067852

The latitudes of the points a, b, c, &c. being are deduced the successive courses the ship must obtained as in the third column, and their longi- steer, and distances she ought to run, as speci tudes known; from these, by Mercator's sailing, fied in the eighth and ninth columns.

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To find the distance between A and B by Mercator's sailing. Merid. diff. lat. 1477-39: diff. long. 5019 :: radius: tang. course 68° 28′ 37′′ Difference Cosine course 68° 28′ 37′′: diff. lat. 1840:: radius: distance

Which is

Having here aspired at accuracy only to exhibit the true difference of the intercepted arcs and rhumbs, yet we may be allowed to observe once for all, that these rigorous calculations are as inconvenient as unuecessary to the practical navigator.

We have hitherto considered the earth as a sphere, and but for the inaccuracy which unavoidably exists in the modes of estimating the ship's course and distance, the doubts whether the northern and southern hemispheres be similar figures, the uncertainty of the ratio of the equatorial and polar diameters, and particularly the trivial effect which the ratio of greatest inequality that has been assigned to the axes would produce in a ship's dead reckoning, we would here have considered the method of estimating a ship's relative situation agreeably to the spheroidal hypothesis. For these reasons, however, together with the restricted view to which we are indispensably limited, such researches might be justly deemed nugatory. Those who wish to se the subject treated in as plain a manner perhaps as

its nature will admit, may consult Robertson's Navigation, book viii. sect. 8.

20. Were the application of the principles which we have elucidated always practicable, the ship's place might be determined at any time to the same degree of precision as the data upon which it depends. But incidents may often occur to baffle the most accomplished, experienced, and indefatigable navigator. Thus circumstanced, be has recourse to celestial observations to ascertain his latitude and longitude.

As these problems are independent of each other, we shall first inquire how the latitude may be found For this purpose, let Fig. 9. be a ste reographic projection of the sphere upon the plane of the meridian HZR; where HR is the horizon, z its pule or the observer's zenith, £g the equator, p its elevated pole, and the observed place of the sun or a star; then will Hor RO be the altitude of the object when on the meridian, its zenith distance, and E the declination.

Now the latitude is equal to the angle which the

miles more than the length of the arc A B.

4956-48

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equator makes with the prime vertical, as well as to the height of the pole above the horizon: conSequently, when the arcs HO E are given, the atitude Ez RP becomes known. For it is obvious that Ez is equal to the sum of the arcs z

QE E when they are of the same name, and to their difference when of contrary names: except when is between the horizon and elevated pole, in which case the sum of (RO) the altitude, and (P) the polar distance, gives the latitude.

Example January 1st, 1811, the meridian altitude of the sun's lower limb was 34° 15′ 30′′, the observer being N. of the sun and the height of his eye 18 feet, longitude by estimation 60 W. Required the latitude?

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21. Hence, it is obvious that the determination of the latitude, if we could always observe the meridian alitude of an object whose declination is known, would be an easy problem; but clouds very frequently prevent this observation. The mariner therefore requires other methods of determining the latitude; and the one which is best adapted, and has been most approved, is that which gives the latitude from the sun's declination two altitudes, and the interval of times between the observations. The requisite operation is necessarily rather long and complicated, and may at first sight appear very formidable; but, as in other instances, a little acquaintance with it causes its terrors to disappear.

Example.

O's true altitudes.

To illustrate the direct method of solution, let z (Fig. 9) be the zenith, P the pole, PSPs the sun's polar distance at the middle time between the observations, and ss the places of the sun corrected for semidiameter, dip, refraction, and parallax. Then in the isosceles triangle SPS, the equal sides SP. SP, and included angle SPS, which measures the interval between the observations, are given to find ss and the angle Pss; and in the triangle szs, the three sides are given to find the angle zss; then zSP PSS zss: hence in the triangle zSP, the sides zs. SP, and included angle ZSP, are given to find zp, the complement of the latitude corresponding to the time to which the declination is reduced.

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sine 9.5732441

PSS=

85° 4′ 20′′

cotangent 8.9351032

2

900

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If the interval between the observations be considerable, it may not only be necessary to reduce the altitude at the second observation to what it would have been if the observations had been taken at the same place (vide Vince's Practical Astron. p. 50), but likewise to consider PS. PS equal to the true polar distances at the respecttive observations. (Vide Vince's Trig, art. 254). This problem has engaged the attention of several eminent mathematicians, whose researches have been principally directed to an approximate method called Douwe's, employing as an element

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of calculation the latitude by account. This method, in its original form, was sometimes tedious, requiring a repetition of the operation, and when repeated not always leading to right conclusions. On this account Dr. Brinkley invented a supplementary process of computation, by which the latitude found by Douwe's method might be conveniently corrected. About the same time that this process, with its demonstration and tables, was published in the Nautical Almanac for 1797, a similar solution of the problem by M. Mendoza Rios was published in the Connoissance des Temps:

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