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History. entitled the Haven-finding Art. In this ancient tract is acquainted with the principles upon which this proceeded; History .

also described the method by which our sailors estimate and it is now generally believed, that the true principles on
the rate of a ship in her course, by an instrument called which the construction of what is called Mercator's Chart
the log. This was so named from the piece of wood or log depends, were first discovered by Edward Wright, an
which floats in the water, whilst the time is reckoned during Englishman.
which the line that is fastened to it is veering out. The Wright supposed, but, according to the general opi-
inventor of this contrivance is not known; but it was first nion, without sufficient grounds, that this enlargement of
described in an account of an East India voyage published the degrees of latitude was known and mentioned by Pto-
by Purchas in 1607, from which time it became famous, and lemy, and that the same thing had also been spoken of by
was much taken notice of by almost all writers on naviga- Cortes. The expressions of Ptolemy alluded to relate, in-
tion in every country. It still continues to be used as at first

, deed, to the proportion between the distances of the paral-
although many attempts have been made to improve it, and lels and meridians; but instead of proposing any gradual
contrivances proposed to supply its place, many of which widening between the parallels of latitude in a general chart,
have succeeded in quiet water, but proved useless in a he speaks only of particular maps, and advises not to con-
stormy sea.

fine a system of such maps to one and the same scale, but In the year 1581 Michael Coignet, a native of Antwerp, to plan them out by a different measure, as occasion might published a treatise, in which he animadverted on Medina. require ; with this precaution, however, that the degrees of In this he showed, that as the rhumbs are spirals, making longitude in each should bear some proportion to those of endless revolutions about the poles, numerous errors must latitude, and this proportion was to be deduced from that arise from their being represented by straight lines on the which the magnitude of the respective parallels bore to a sea charts; but although he hoped to find a remedy for great circle of the sphere. He added, that, in particular these errors, he was of opinion that the proposals of Nonius maps, if this proportion be observed with regard to the were scarcely practicable, and therefore in a great measure middle parallel

, the inconvenience will not be great, aluseless. In treating of the sun's declination, he took notice though the meridians should be straight lines parallel to of the gradual decrease in the obliquity of the ecliptic; he each other. But here he is understood only to mean, that also described the cross staff with three transverse pieces, the maps should in some measure represent the figures of which he admitted were then in common use amongst the the countries for which they are drawn.

In this sense sailors. He likewise described some instruments of bis Mercator, who drew maps for Ptolemy's tables, understood own invention ; but all of them are now laid aside, except- him; thinking it, however, an improvement not to reguing perhaps his nocturnal. He constructed a sea table to late the meridians by one parallel, but by two, one distant be used by such as sailed beyond the sixtieth degree of from the northern, the other from the southern extremity latitude; and at the end of the book is delivered a method of the map, by a fourth part of the whole depth; by which of sailing upon a parallel of latitude by means of a ring dial means, in his maps, although the meridians are straight and a twenty-four hour glass. The same year the discovery lines, yet they are generally drawn inclining to each other of the dipping-needle was made by Robert Norman. towards the poles. With regard to Cortes, he speaks only In his publication on that subject he maintains, in opposition of the number of degrees of latitude, and not of the extent to Cortes, that the variation of the compass was caused by of them; nay, he gives express directions that they should some point on the surface of the earth, and not in the hea- all be laid down by equal measurement in a scale of vens; and he also made considerable improvements on the leagues adapted to the map. construction of compasses themselves, showing especially For some time after the appearance of Mercator's map the danger of not fixing, on account of the variation, the it was not rightly understood, and it was even thought to wire directly under the fleur de lis, as compasses made in be entirely useless, if not detrimental. However, about different countries have it placed differently. To this per- the year 1592 its utility began to be perceived ; and seven formance of Norman's is prefixed a discourse on the varia- years afterwards Wright printed his famous treatise ention of the magnetical needle, by William Burrough, in titled The Correction of certain Errors in Navigation, which he shows how to determine the variation in many where he fully explained the reason of extending the length different ways, and also points out many errors in the prac- of the parallels of latitude, and the uses thereof to navitice of navigation at that time, speaking in very severe gators. In 1610 a second edition of Wright's book was terms concerning those who had published upon it. published, with improvements. An excellent method was

During this time the Spaniards continued to publish trea- proposed of determining the magnitude of the earth; and at tises on the art. In 1585 an excellent Compendium was the same time it was judiciously proposed to make our compublished by Roderico Zamorano, and contributed greatly mon measures in some proportion to a degree on its surface, towards the improvement of the art, particularly in the sea that they might not depend on the uncertain length of a charts. Globes of an improved kind, and of a much larger barleycorn. Amongst his other improvements may be mensize than those formerly used, were now constructed, and tioned the Table of Latitudes for Dividing the Meridian many improvements were made in other instruments; never- computed to Minutes, whereas it had been only divided to theless, the plane chart continued still to be followed, though every tenth minute. He also published a description of an its errors were frequently complained of. Methods of re- instrument which he calls the sea rings, by which the varimoving these errors had indeed been sought after ; and ation of the compass, the altitude of the sun, and the time Gerard Mercator seems to have been the first who found of the day, may at once readily be determined in any place, the true method of effecting this, so as to answer the pur- provided the latitude is known. He also showed how to poses of seamen. He represented the degrees both of lati- correct the errors arising from the eccentricity of the eye in tude and longitude by parallel straight lines, but gradually observing by the cross staff. In the years 1594, 1596, augmented the space between the former as they approached 1596, and 1597, he amended the tables of the declinations the pole. Thus the rhumbs, which otherwise ought to have and places of the sun and stars from his own observations been curves, were now also extended into straight lines; made with a six-feet quadrant, a sea quadrant to take altiand thus a straight line drawn between any two places tudes by a forward or backward observation, and likewise marked upon the cliart formed an angle with the meridians, with a contrivance for the ready finding of the latitude by expressing the rhumb leading from the one to the other. the height of the pole-star, when not upon the meridian. But although in 1569 Mercator published a universal map To this edition was subjoined a translation of Zamorano's constructed in this manner. it does not appear that he was Compendium, above mentioned, in which he corrected some

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History mistakes in the original, adding a large table of the varia- and tangents to every minute of the quadrant, which he History.

tion of the compass observed in different parts of the published in 1620. In this work he applied to navigation,
world, in order to show that it was not occasioned by any and other branches of mathematics, his admirable ruler
magnetical pole.

known by the name of Gunter's Scale,' on which are de-
These improvements soon became known abroad. In scribed lines of logarithms, of logarithmic sines and tan-
1608 a treatise, entitled Hypomnemata Mathematica, was gents, of meridional parts, &c.; and he greatly improved
published by Simon Stevin for the use of Prince Maurice. the sector for the same purposes. He also showed how to
In the portion of the work relating to navigation, the au- take a back observation by the cross staff, by which the
thor treated of sailing on a great circle, and showed how error arising from the eccentricity of the eye is avoided.
to draw the rhumbs on a globe mechanically; he also set He likewise described another instrument, of his own in-
down Wright's two tables of latitudes and of rhumbs, in vention, called the cross bow, for taking altitudes of the
order to describe these lines more accurately; and even sun or stars, with some contrivances for more readily find-
pretended to have discovered an error in Wright's table. ing the latitude from the observation. The discoveries
But Stevin's objections were fully answered by the author concerning logarithms were carried into France in 1624 by
bimself, who showed that they arose from the rude method Edmund Wingate, who published two small tracts in that
of calculating made use of by the former.

year at Paris. In one of these he taught the use of Gunter's In 1624 the learned Willebrordus Snellius, professor of scale; and in the other, that of the tables of artificial sines mathematics at Leyden, published a treatise of naviga- and tangents, as modelled according to Napier's last form, tion on Wright's plan, but somewhat obscurely; and as he erroneously attributed by Wingate to Briggs. did not particularly mention all the discoveries of Wright, Gunter's scale was projected into a circular arch by the the latter was thought by some to have taken the hint of all Reverend William Oughtred in 1633; and its uses were his discoveries from Snellius. But this supposition has been fully shown in a pamphlet entitled the Circles of Proporlong ago refuted; and Wright's title to the honour of those tion, where, in an appendix, several important points in discoveries remains unchallenged.

navigation are well treated. It has also been made in the Having shown how to find the place of the ship upon form of a sliding ruler. his chart, Wright observed that the same might be per- The logarithmic tables were first applied to the different formed more accurately by calculation ; but considering, cases of sailing, by Thomas Addison, in his treatise enas he says, that the latitudes, and especially the courses titled Arithmetical Navigation, printed in the year

1625. at sea, could not be determined so precisely, he forbore He also gave two traverse tables, with their uses; the one setting down particular examples; as the mariner may be to quarter points of the compass, and the other to degrees. allowed to save himself this trouble, and only to mark out Henry Gellibrand published his discovery of the changes upon his chart the ship’s way, after the manner then usually of the variation of the compass, in a small quarto pamphlet, practised. However, in 1614, Raphe Handson, amongst entitled A Discourse Mathematical on the Variation of the the nautical questions which he subjoined to a translation Magnetical Needle, printed in 1635. This extraordinary of Pitiscus's Trigonometry, solved very distinctly every phenomenon he found out by comparing the observations case of navigation, by applying arithmetical calculations to which had been made at different times near the same Wright's Tables of Latitudes, or of Meridional Parts, as it place by Burrough, Gunter, and himself, all persons of has since been called. Although the method discovered by great skill and experience in these matters. This discovery Wright for finding the change of longitude by a ship sail- was likewise soon known abroad; for Athanasius Kircher, ing on a rhumb is the proper way of performing it, Hand- in his treatise entitled Magnes, first printed at Rome in son also proposes two methods of approximation without the year 1641, informs us that he had been told of it by the assistance of Wright's division of the meridian line. John Greaves, and then gives a letter of the famous The first was computed by the arithmetical mean between Marinus Mersennus, containing a very distinct account of the cosines of both latitudes ; and the other by the same the same. mean between the secants, as an alternative when Wright's As altitudes of the sun are taken on shipboard by obbook was not at hand; although this latter is wider of the serving his elevation above the visible horizon, to obtain truth than the former. By the same calculations also he from these the sun's true altitude with correctness, Wright showed how much each of these compends deviates from observed it to be necessary that the dip of the visible horithe truth, and also how widely the computations on the zon below the horizontal plane passing through the oberroneous principles of the plane chart differ from them server's eye should be brought into the account, which all. The method generally used by our sailors, however, cannot be calculated without knowing the magnitude of the is commonly called the middle latitude, which, although it earth. Hence he was induced to propose different methods errs more than that by the arithmetical mean between the for finding this ; but he complains that the most effectual two cosines, is preferred on account of its being less oper- was out of his power to execute, and therefore he contented ose ; yet in high latitudes it is more eligible to use that himself with a rude attempt, in some measure sufficient for of the arithmetical mean between the logarithmic cosines, his purpose. The dimensions of the earth deduced by him equivalent to the geometrical mean between the cosines corresponded very well with the usual divisions of the logthemselves—a method since proposed by John Bassat. line; nevertheless, as he did not write an express treatise The computation by the middle latitude will always fall on navigation, but only for correcting such errors as preshort of the true change of longitude, that by the geome- vailed in general practice, the log-line did not fall under trical mean will always exceed; but that by the arithme- his notice. Richard Norwood, however, put in execution tical mean falls short in latitudes of about 45°, and exceeds the method recommended by Wright as the most perfect in lesser latitudes. However, none of these methods will for measuring the dimensions of the earth, with the true differ much from the truth when the change of latitude is length of the degrees of a great circle upon it; and in sufficiently small.

1635 he actually measured the distance between London About this period logarithms were invented by John and York; from which measurement, and the summer solNapier, Baron of Merchiston in Scotland, and proved of stitial altitudes of the sun observed on the meridian at both the utmost service to the art of navigation. From these places, he found a degree on a great circle of the earth Edmund Gunter constructed a table of logarithmic sines to contain 367,196 English feet, equal to 57,300 French



History. fathoms or toises; which is very exact, as appears from Transactions for 1695 (No. 219), from the consideration

many measurements that have been made since that time. of the spirals into which the rhumbs are transformed in the
Of all this Norwood gave a full account in his treatise stereographic projection of the sphere upon the plane of
called the Seaman's Practice, published in 1657. He the equinoctial, and which is rendered still more simple
there showed the reason why Snellius had failed in his by Roger Cotes, in his Logometria, first published in the
attempt ; and he also pointed out various uses of his dis- Philosophical Transactions for 1714 (No. 388). It is,
covery, particularly for correcting the gross errors hitherto moreover, added in Gunter's book, that if th of this divi-
committed in the divisions of the log-line. But necessary sion, which does not sensibly differ from the logarithmic
amendments have been little attended to by sailors, whose tangent of 45° 1' 30", with the radius subtracted from it,
obstinacy in adhering to established errors has been com- be used, the quotient will exhibit the meridional parts ex-
plained of by the best writers on navigation. This im- pressed in leagues ; and this is the divisor set down in
provement, however, has at length made its way into prac- Norwood's Epitome. After the same manner, the meridi-
tice; and few navigators of reputation now make use of the onal parts will be found in minutes, if the like logarithmic
old measure of forty-two feet to a knot. In this treatise tangent of 45° 1' 30", diminished by the radius, be taken ;
Norwood also describes his own excellent method of set- that is, the number used by others being 12633, when the
ting down and perfecting a sea reckoning, by using a tra- logarithmic tables consist of eight places of figures besides
verse table, which method he had followed and taught for the index.
many years. He likewise shows how to rectify the course, In an edition of a book called the Seaman's Kalendar,
by taking into consideration the variation of the compass; Bond declared that he had discovered the longitude by
as also how to discover currents, and to make proper allow- having found out the true theory of the magnetic variation ;
ance on their account. This treatise, and another on and to gain credit to his assertion, he foretold, that at Lon-
Trigonometry, were continually reprinted, as the principal don in 1657 there would be no variation of the compass,
books for learning scientifically the art of navigation. What and from that time it would gradually increase the other
he had delivered, especially in the latter of them, concerning way; which happened accordingly. Again, in the Philo-
this subject, was abridged as a manual for sailors, in a very sophical Transactions for 1668 (No. 40), he published a
small work called an Epitome ; which useful performance table of the variation for forty-nine years to come. Thus
has gone through a great number of editions. No alterations he acquired such reputation, that his treatise entitled The
were ever made in the Seaman's Practice till the twelfth Longitude Found, was, in the year 1676, published by the
edition in 1676, when the following paragraph was inserted special command of Charles II., and approved by many ce-
in a smaller character :—“About the year 1672, Monsieur lebrated mathematicians. It was not long, however, before
Picart has published an account in Frencli concerning the it met with opposition ; and in the year 1678 another trea-
measure of the earth, a breviate whereof may be seen in tise, entitled The Longitude not Found, made its appear-
the Philosophical Transactions, No. 112, wherein he con- ance; and as Bond's hypothesis did not answer its author's
cludes one degree to contain 365,184 English feet, nearly sanguine expectations, the solution of the difficulty was un-
agreeing with Mr Norwood's experiment;" and this adver- dertaken by Dr Halley. The result of his speculation was,
tisement is continued through the subsequent editions as that the magnetic needle is influenced by four poles ; but
late as the year 1732.

this wonderful phenomenon seems hitherto to have eluded
About the year 1645, Bond published, in Norwood's all our researches. (See MAGNETISM.) In 1700, however,
Epitome, a very great improvement of Wright's method, Dr Halley published a general map, with curve lines ex-
from a property in his meridian line, whereby the divisions pressing the paths where the magnetic needle had the same
are more scientifically assigned than the author himself variation ; which was received with universal applause.
was able to effect. It resulted from this theorem, that But as the positions of these curves vary from time to
these divisions are analogous to the excesses of the loga- time, they should frequently be corrected by skilful persons,
rithmic tangents of half the respective latitudes ang- as was done in 1644 and 1756, by Mountain and Dodson.
mented by 45° above the logarithm of the radius. This in the Philosophical Transactions for 1690, Dr Halley
he afterwards explained more fully in the third edition of also gave a dissertation on the monsoons, containing many
Gunter's works, printed in 1653, where he observed that very useful observations for such as sail to places subject to
the logarithmic tangents from 45° upwards increase in the these winds.
same manner as the secants do added together, if every After the true principles of the art were settled by
half degree be accounted as a whole degree of Mercator's Wright, Bond, and Norwood, new improvements were
meridional line. His rule for computing the meridional daily made, and everything relative to it was settled with
parts belonging to any two latitudes, supposed to be on an accuracy not only unknown to former ages, but which
the same side of the equator, is to the following effect:- would have been reckoned utterly impossible. The earth
“Take the logarithmic tangent, rejecting the radius, of half being found to be, not a perfect sphere, but a spheroid,
each latitude, augmented by 45° ; divide the difference with the shortest diameter passing through the poles, a
of those numbers by the logarithmic tangent of 45° 30', tract was published in 1741 by the Reverend Dr Patrick
the radius being likewise rejected, and the quotient will be Murdoch, wherein he accommodated Wright's sailing to
the meridional parts required, expressed in degrees.” This such a figure ; and the same year Colin Maclaurin, in the
rule is the immediate consequence of the general theorem, Philosophical Transactions (No. 461), gave a rule for
that the degrees of latitude bear to one degree (or sixty determining the meridional parts of a spheroid ; which
minutes, which in Wright's table stand for the meridional speculation is farther treated of in his book of Fluxions,
parts of one degree) the same proportion as the logarithmic printed at Edinburgh in 1742, and in Delambre’s Astro-
tangent of half any latitude augmented by 45°, and the nomy (t. iii., ch. xxxvi.).
radius neglected, to the like tangent of half a degree aug- Amongst the later discoveries in navigation, that of find-
mented by 45°, with the radius likewise rejected. But ing the longitude, both by lunar observations and by time-
here there was still wanting the demonstration of this keepers, is the principal. It is owing chiefly to the rewards
general theorem, which was at length supplied by James offered by the British Parliament that this has attained the
Gregory of Aberdeen, in his Exercitationes Geometrica, present degree of perfection.

We are indebted to Dr
printed at London in 1668; and afterwards more con- Maskelyne for putting the first of these methods in prac-
cisely demonstrated, together with a scientific determi- tice, and for other important improvements in navigation.
nation of the divisor, by Dr Halley, in the Philosophical The time-keepers constructed by Harrison for this express

Practice of purpose were found to answer so well that he obtained the The works which have latterly appeared on navigation are PreliminNaviga- parliamentary reward. These have been improved by Ar- those on the longitude and navigation by Mackay, Inman, ary Printion. nold, Earnshaw, and many others, so as now to be almost Riddle, Norie, Jeans, and others; and these contain every

ciples. in common use.

necessary requisite to form the practical navigator.



Hemispheres, according as the North or the South Pole

lies within them.
CONTAINING THE VARIOUS METHODS OF SAILING. The latitude of a place is its distance from the equator,
The art of navigation depends upon mathematical and reckoned on a meridian in degrees, minutes, and seconds,
astronomical principles. The problems in the various and decimal parts of seconds (if necessary), being either
modes of sailing are resolved either by trigonometrical cal- north or south, according as it is the Northern or Southern
culations, or by tables or rules formed by the assistance of Hemisphere. Hence it appears that the latitudes of all
plane and spherical trigonometry. By mathematics the places are comprised within the limits 0° and 90° N., and
necessary tables are constructed and rules investigated for 0° and 90° S.
performing the more difficult parts of navigation.

The first meridian, which is a great circle passing
The places of the sun, moon, and planets, and fixed through the poles, also divides the earth into two equal
stars, are deduced from observation and calculation, and portions, called the Eastern and Western Hemispheres, ac-
arranged in tables, the use of which is absolutely necessary cording as they lie to the right or left of the first meridian ;
in reducing observations taken at sea for the purpose of the spectator being supposed to be looking towards the
ascertaining the latitude and longitude of the ship and the north.
variation of the compass. The investigation of the rules

The longitude of a place is the arc of the equator inter-
required for this purpose belongs properly to the science cepted between the first meridian and the meridian of the
of ASTRONOMY, to which the reader is referred. A few tables given place reckoned in degrees, minutes, and seconds ;
are given at the end of this article, but as the other tables and is either east or west as the place lies in the Eastern or
necessary for the practice of navigation are to be found in Western Hemisphere respectively to the first meridian. The
almost every treatise on this subject, it seems unnecessary longitude of all places on the earth's surface is comprised
to insert them in this place. The subject naturally divides within the limits of 0° and 180° E., and 0° and 180° W.
itself under two heads:- First, The methods of conducting a

On the supposition that the earth is a sphere, the length ship from one port to another by help of rules, in which the of all arcs of great circles upon it subtending

an angle of
log-line and compass are alone required, which is Naviga- 1° at the centre are equal ; hence l' of latitude or longi-
tion properly so called. Second, The method of ascertain- tude is equal to one geographical or nautical mile, of which a
ing the ship’s latitude and longitude, and variation of com- degree contains 60. Hence intervals of latitude and longi-

, by means of observations on the heavenly bodies ; and tude, reduced to minutes and parts of minutes, also repre-
the rules for that purpose deduced from astronomy, in order sent the same number of nautical miles and parts of a
to correct the ship’s place, and the courses derived from nautical mile.
the former method, to which the name of Nautical Astro-

In the practice of navigation, the latitude and longitude
nomy is generally applied. Although the reader is referred of the place which a ship leaves, are called the latitude and
to the respective articles on the sciences on which naviga- longitude from ; and the latitude and longitude of the place
tion is founded in this work for complete information, we at which it has arrived, are called the latitude and longitude in.
shall, nevertheless, endeavour to make our explanation of the
several rules as complete as possible, even at the risk of re-
peating somewhat the substance of portions of our other







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1. Latitude and Longitude.
The situation of a place, or any object on the earth's
surface, is estimated by its distance from two imaginary
lines on that surface intersecting each other at right angles.
The one of these is called the Equator, and the other the

Fig. 1.
First Meridian. The situation of the equator is fixed;
but that of the first meridian is arbitrary, and therefore

2. Definitions of Terms used in Navigation, and
different nations assume different first meridians. In Great

Britain we assume that to be the first meridian which passes Let QR... V be a portion of the equator, P the pole, and
through the Royal Observatory at Greenwich.

PAQ, PBR, PCS......PFV be meridians supposed very
The equator is a great circle on the earth's surface, every near to one another, passing through points A, B, C, D, E, F,
point of which is equally distant from the two poles or the the line Af being the path traced out by a vessel in pass-
extremities of the imaginary axis about which the earth ing from A to F, such that it makes equal angles with
makes her diurnal rotation. It therefore divides the earth every meridian over which it passes. From B, C, D, &c.,
into two equal parts, called the Northern and the Southern let BH, CI, DK, EL, &c., be drawn perpendicular to the

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Prelimin- two meridians between whico they respectively lie; or, in Or in logarithms,
ary Prin- other words, be arcs of small circles or parallels of latitude

log. true diff. lat.=log. dist. + L cos course – 10,
through the points B, C, &c. These are consequently all
parallel to one another, and to FG the whole arc of the where L means tabular logarithm, i.e., logarithm increased
parallel at F included between the extreme meridians by 10; and log. dep. = log. dist. +L sin course – 10.
PAQ and PFV.

2. Given course (BAC) and true difference of latitude
The constant angle at which the line AF is inclined to (AB), to find distance and departure.
the successive meridians, viz., BAP, CBP, DCP, &c., is

AC=AB x sec BAC
called the course. Also, if the small circles or parallels at

BC=AB x tan BAC;
B, C, &c., be continued to the meridian PAQ, the portion

or dist.=true diff. lat. x sec course

· (111.) of this meridian intercepted between any two consecutive

dep.=true diff. lat. x tan course

· (iv.)
parallels, as cd, will be equal to CK, the distance between
the parallels through C and D; and so on for all. Hence 3. Given course and departure, to find distance and true
the sum of these distances, AH + BI + CK+DL + EM= difference of latitude.
AG; which is called the true difference of latitude, or true

AC=BC x cosec BAC
diff. lat. from A to F.

AB=BC x cot BAC;
The corresponding ares of parallels at different latitudes

or dist. = dep. x cosec course

(v.) intercepted between the same meridians are not equal, but and true diff. lat.=dep. x cot course

gradually decrease from the equator to the poles. Hence

4. Given distance and true difference of latitude, to find
the sum of the arcs BH + CI+DK+EL+ FM is less than
QV, the intercepted arc of the equator, but greater than

course and departure.

FG, the arc of the highest parallel intercepted between

Cos BAC=
PAQ and PFV.
BH + CI+ DK + EL + FM is called the departure ;

or cosine course = true diff. lat., dist. (VII.)
the arc QV of the equator is the difference of longitude ; And the course having been found, we get
and AF, the curve described by the vessel in passing from

dep.=dist. x sin course by (11.)
A to F, is called the distance.

5. Given the distance and departure, to find the course In navigation, each of the triangles ABH, BCI, &c., is

and true difference of latitude. considered as a plane triangle; and as each of them is

right-angled, and contains, besides, one constant angle, viz.,

Sin BAC=
the course, the other angle in each must also be constant ;
and all the triangles will be equiangular and similar.

or sin course = dep. - distance . (viii.)
Hence we have

And then we have
AH : BH: AB :: AH : BH : AB

true diff. lat.=dist. x cos course.
BI : CI : BC :: AH: BH: AB

6. Given the true difference of latitude and departure, CK : DK : CD:: AH : BH : AB

to find the course and distance.
&c. &c. &c.

EM : FM: EF :: AH : BH: AB.

Tan BAC =
And since, when any number of quantities are in con-

or tan course=departure = true diff. lat. . (1x.)
tinued proportion, as the first consequent is to its antece-
dent, so are all the consequents to all the antecedents; And having tound the course, we have
we have

dist. true diff. lat. x sec course by (111.);
AH+ BI+CK+ &c.: BH +GI+DK+ c. :AB+ BC+CD+ &c. or dist. = dep. x cosec course by (v.)

:: AH: BH : AB.
But AH + BI + CK + &c.= AG the true diff. lat.

Length of Arc of 1° of Parallels of Latitude.
BH + CI + DK+&c.= the departure.
AB + BC + CD + &c.= AF or the distance.

We have already stated that the lengths of the parallels

of latitude diminish as the lati-
Hence the true difference of latitude, departure, and dis-

tude increases. In fact, it des
tance, may be considered as the sides of a right-angled creases in the ratio of cosine la-
triangle, similar to each of the small triangles ; the angle titude to unity.
of which, therefore, between the true

Let EQ be the equator, PCP
difference of latitude and distance, is

the polar diameter of the earth the course.

passing through C, and LML'any Take AB (fig. 2)=the true diff. lat.

parallel of latitude. Let the angle Draw BC at right angles to it=de

ECL, or the latitude, bel, and let parture. Join AC. Then AC is the

LM and EF be the arcs of the
distance, and BAC is the course.

From this it appears, that when any

parallel and of the equator in-
tercepted between the meridians

Fig. 3.
two of the four quantities, true difference

PEP and PFP'. Then angle ECF=angle LOM, because of latitude, departure, distance, and

they both measure the angle between the planes of the two
course are given, the remaining two

meridians. Hence
can be found by solving the right. A
angled triangle ABC.

Fig. 2.

arc LM: arc EF::OL:CE:: OL:CL, because CE = CL; 1. Given course (BAC) and distance (AC), to find true

OL or arc LM=arc EF X

= arc EF x sin OCL
difference of latitude (AB), and departure (BC).


= arc EF x cos LCE=arc EF * cos l.
BC=AC sin BAC,

Hence if FE be the length of an arc lo of longitude at the i.e., true diff. lat. (in miles) = dist. x cos course . (1.) equator, or 60 miles, LM the length of an arc 1° of longitude

and departure = dist. x sin course (11.) in latitude 1-= 60 x cos l.











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