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History. entitled the Haven-finding Art. In this ancient tract is also described the method by which our sailors estimate the rate of a ship in her course, by an instrument called the log. This was so named from the piece of wood or log which floats in the water, whilst the time is reckoned during which the line that is fastened to it is veering out. The inventor of this contrivance is not known; but it was first described in an account of an East India voyage published by Purchas in 1607, from which time it became famous, and was much taken notice of by almost all writers on navigation in every country. It still continues to be used as at first, although many attempts have been made to improve it, and contrivances proposed to supply its place, many of which have succeeded in quiet water, but proved useless in a stormy sea.

In the year 1581 Michael Coignet, a native of Antwerp, published a treatise, in which he animadverted on Medina. In this he showed, that as the rhumbs are spirals, making endless revolutions about the poles, numerous errors must arise from their being represented by straight lines on the sea charts; but although he hoped to find a remedy for these errors, he was of opinion that the proposals of Nonius were scarcely practicable, and therefore in a great measure useless. In treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the cross staff with three transverse pieces, which he admitted were then in common use amongst the sailors. He likewise described some instruments of his own invention; but all of them are now laid aside, excepting perhaps his nocturnal. He constructed a sea table to be used by such as sailed beyond the sixtieth degree of latitude; and at the end of the book is delivered a method of sailing upon a parallel of latitude by means of a ring dial and a twenty-four hour glass. The same year the discovery of the dipping-needle was made by Robert Norman. In his publication on that subject he maintains, in opposition to Cortes, that the variation of the compass was caused by some point on the surface of the earth, and not in the heavens; and he also made considerable improvements on the construction of compasses themselves, showing especially the danger of not fixing, on account of the variation, the wire directly under the fleur de lis, as compasses made in different countries have it placed differently. To this performance of Norman's is prefixed a discourse on the variation of the magnetical needle, by William Burrough, in which he shows how to determine the variation in many different ways, and also points out many errors in the practice of navigation at that time, speaking in very severe terms concerning those who had published upon it.

During this time the Spaniards continued to publish treatises on the art. In 1585 an excellent Compendium was published by Roderico Zamorano, and contributed greatly towards the improvement of the art, particularly in the sea charts. Globes of an improved kind, and of a much larger size than those formerly used, were now constructed, and many improvements were made in other instruments; nevertheless, the plane chart continued still to be followed, though its errors were frequently complained of. Methods of removing these errors had indeed been sought after; and Gerard Mercator seems to have been the first who found the true method of effecting this, so as to answer the purposes of seamen. He represented the degrees both of latitude and longitude by parallel straight lines, but gradually augmented the space between the former as they approached the pole. Thus the rhumbs, which otherwise ought to have been curves, were now also extended into straight lines; and thus a straight line drawn between any two places marked upon the chart formed an angle with the meridians, expressing the rhumb leading from the one to the other. But although in 1569 Mercator published a universal map constructed in this manner. it does not appear that he was

acquainted with the principles upon which this proceeded; History. and it is now generally believed, that the true principles on which the construction of what is called Mercator's Chart depends, were first discovered by Edward Wright, an Englishman.

Wright supposed, but, according to the general opinion, without sufficient grounds, that this enlargement of the degrees of latitude was known and mentioned by Ptolemy, and that the same thing had also been spoken of by Cortes. The expressions of Ptolemy alluded to relate, indeed, to the proportion between the distances of the parallels and meridians; but instead of proposing any gradual widening between the parallels of latitude in a general chart, he speaks only of particular maps, and advises not to confine a system of such maps to one and the same scale, but to plan them out by a different measure, as occasion might require; with this precaution, however, that the degrees of longitude in each should bear some proportion to those of latitude, and this proportion was to be deduced from that which the magnitude of the respective parallels bore to a great circle of the sphere. He added, that, in particular maps, if this proportion be observed with regard to the middle parallel, the inconvenience will not be great, although the meridians should be straight lines parallel to each other. But here he is understood only to mean, that the maps should in some measure represent the figures of the countries for which they are drawn. In this sense Mercator, who drew maps for Ptolemy's tables, understood him; thinking it, however, an improvement not to regulate the meridians by one parallel, but by two, one distant from the northern, the other from the southern extremity of the map, by a fourth part of the whole depth; by which means, in his maps, although the meridians are straight lines, yet they are generally drawn inclining to each other towards the poles. With regard to Cortes, he speaks only of the number of degrees of latitude, and not of the extent of them; nay, he gives express directions that they should all be laid down by equal measurement in a scale of leagues adapted to the map.

For some time after the appearance of Mercator's map it was not rightly understood, and it was even thought to be entirely useless, if not detrimental. However, about the year 1592 its utility began to be perceived; and seven years afterwards Wright printed his famous treatise entitled The Correction of certain Errors in Navigation, where he fully explained the reason of extending the length of the parallels of latitude, and the uses thereof to navigators. In 1610 a second edition of Wright's book was published, with improvements. An excellent method was proposed of determining the magnitude of the earth; and at the same time it was judiciously proposed to make our common measures in some proportion to a degree on its surface, that they might not depend on the uncertain length of a barleycorn. Amongst his other improvements may be mentioned the Table of Latitudes for Dividing the Meridian computed to Minutes, whereas it had been only divided to every tenth minute. He also published a description of an instrument which he calls the sea rings, by which the variation of the compass, the altitude of the sun, and the time of the day, may at once readily be determined in any place, provided the latitude is known. He also showed how to correct the errors arising from the eccentricity of the eye in observing by the cross staff. In the years 1594, 1595, 1596, and 1597, he amended the tables of the declinations and places of the sun and stars from his own observations made with a six-feet quadrant, a sea quadrant to take altitudes by a forward or backward observation, and likewise with a contrivance for the ready finding of the latitude by the height of the pole-star, when not upon the meridian. To this edition was subjoined a translation of Zamorano's Compendium, above mentioned, in which he corrected some


History mistakes in the original, adding a large table of the variation of the compass observed in different parts of the world, in order to show that it was not occasioned by any magnetical pole.

These improvements soon became known abroad. In 1608 a treatise, entitled Hypomnemata Mathematica, was published by Simon Stevin for the use of Prince Maurice. In the portion of the work relating to navigation, the author treated of sailing on a great circle, and showed how to draw the rhumbs on a globe mechanically; he also set down Wright's two tables of latitudes and of rhumbs, in order to describe these lines more accurately; and even pretended to have discovered an error in Wright's table. But Stevin's objections were fully answered by the author himself, who showed that they arose from the rude method of calculating made use of by the former.

In 1624 the learned Willebrordus Snellius, professor of mathematics at Leyden, published a treatise of navigation on Wright's plan, but somewhat obscurely; and as he did not particularly mention all the discoveries of Wright, the latter was thought by some to have taken the hint of all his discoveries from Snellius. But this supposition has been long ago refuted; and Wright's title to the honour of those discoveries remains unchallenged.

Having shown how to find the place of the ship upon his chart, Wright observed that the same might be performed more accurately by calculation; but considering, as he says, that the latitudes, and especially the courses at sea, could not be determined so precisely, he forbore setting down particular examples; as the mariner may be allowed to save himself this trouble, and only to mark out upon his chart the ship's way, after the manner then usually practised. However, in 1614, Raphe Handson, amongst the nautical questions which he subjoined to a translation of Pitiscus's Trigonometry, solved very distinctly every case of navigation, by applying arithmetical calculations to Wright's Tables of Latitudes, or of Meridional Parts, as it has since been called. Although the method discovered by Wright for finding the change of longitude by a ship sailing on a rhumb is the proper way of performing it, Handson also proposes two methods of approximation without the assistance of Wright's division of the meridian line. The first was computed by the arithmetical mean between the cosines of both latitudes; and the other by the same mean between the secants, as an alternative when Wright's book was not at hand; although this latter is wider of the truth than the former. By the same calculations also he showed how much each of these compends deviates from the truth, and also how widely the computations on the erroneous principles of the plane chart differ from them all. The method generally used by our sailors, however, is commonly called the middle latitude, which, although it errs more than that by the arithmetical mean between the two cosines, is preferred on account of its being less operose; yet in high latitudes it is more eligible to use that of the arithmetical mean between the logarithmic cosines, equivalent to the geometrical mean between the cosines themselves—a method since proposed by John Bassat. The computation by the middle latitude will always fall short of the true change of longitude, that by the geometrical mean will always exceed; but that by the arithmetical mean falls short in latitudes of about 45°, and exceeds in lesser latitudes. However, none of these methods will differ much from the truth when the change of latitude is sufficiently small.

and tangents to every minute of the quadrant, which he History. published in 1620. In this work he applied to navigation, and other branches of mathematics, his admirable ruler known by the name of Gunter's Scale,1 on which are described lines of logarithms, of logarithmic sines and tangents, of meridional parts, &c.; and he greatly improved the sector for the same purposes. He also showed how to take a back observation by the cross staff, by which the error arising from the eccentricity of the eye is avoided. He likewise described another instrument, of his own invention, called the cross bow, for taking altitudes of the sun or stars, with some contrivances for more readily finding the latitude from the observation. The discoveries concerning logarithms were carried into France in 1624 by Edmund Wingate, who published two small tracts in that year at Paris. In one of these he taught the use of Gunter's scale; and in the other, that of the tables of artificial sines and tangents, as modelled according to Napier's last form, erroneously attributed by Wingate to Briggs.

About this period logarithms were invented by John Napier, Baron of Merchiston in Scotland, and proved of the utmost service to the art of navigation. From these Edmund Gunter constructed a table of logarithmic sines

Gunter's scale was projected into a circular arch by the Reverend William Oughtred in 1633; and its uses were fully shown in a pamphlet entitled the Circles of Proportion, where, in an appendix, several important points in navigation are well treated. It has also been made in the form of a sliding ruler.

The logarithmic tables were first applied to the different cases of sailing, by Thomas Addison, in his treatise entitled Arithmetical Navigation, printed in the year 1625. He also gave two traverse tables, with their uses; the one to quarter points of the compass, and the other to degrees. Henry Gellibrand published his discovery of the changes of the variation of the compass, in a small quarto pamphlet, entitled A Discourse Mathematical on the Variation of the Magnetical Needle, printed in 1635. This extraordinary phenomenon he found out by comparing the observations which had been made at different times near the same place by Burrough, Gunter, and himself, all persons of great skill and experience in these matters. This discovery was likewise soon known abroad; for Athanasius Kircher, in his treatise entitled Magnes, first printed at Rome in the year 1641, informs us that he had been told of it by John Greaves, and then gives a letter of the famous Marinus Mersennus, containing a very distinct account of the same.

As altitudes of the sun are taken on shipboard by observing his elevation above the visible horizon, to obtain from these the sun's true altitude with correctness, Wright observed it to be necessary that the dip of the visible horizon below the horizontal plane passing through the observer's eye should be brought into the account, which cannot be calculated without knowing the magnitude of the earth. Hence he was induced to propose different methods for finding this; but he complains that the most effectual was out of his power to execute, and therefore he contented himself with a rude attempt, in some measure sufficient for his purpose. The dimensions of the earth deduced by him corresponded very well with the usual divisions of the logline; nevertheless, as he did not write an express treatise on navigation, but only for correcting such errors as prevailed in general practice, the log-line did not fall under his notice. Richard Norwood, however, put in execution the method recommended by Wright as the most perfect for measuring the dimensions of the earth, with the true length of the degrees of a great circle upon it; and in 1635 he actually measured the distance between London and York; from which measurement, and the summer solstitial altitudes of the sun observed on the meridian at both places, he found a degree on a great circle of the earth to contain 367,196 English feet, equal to 57,300 French


History. fathoms or toises; which is very exact, as appears from many measurements that have been made since that time. Of all this Norwood gave a full account in his treatise called the Seaman's Practice, published in 1657. He there showed the reason why Snellius had failed in his attempt; and he also pointed out various uses of his discovery, particularly for correcting the gross errors hitherto committed in the divisions of the log-line. But necessary amendments have been little attended to by sailors, whose obstinacy in adhering to established errors has been complained of by the best writers on navigation. This improvement, however, has at length made its way into practice; and few navigators of reputation now make use of the old measure of forty-two feet to a knot. In this treatise Norwood also describes his own excellent method of setting down and perfecting a sea reckoning, by using a traverse table, which method he had followed and taught for many years. He likewise shows how to rectify the course, by taking into consideration the variation of the compass; as also how to discover currents, and to make proper allowance on their account. This treatise, and another on Trigonometry, were continually reprinted, as the principal books for learning scientifically the art of navigation. What he had delivered, especially in the latter of them, concerning this subject, was abridged as a manual for sailors, in a very small work called an Epitome; which useful performance has gone through a great number of editions. No alterations were ever made in the Seaman's Practice till the twelfth edition in 1676, when the following paragraph was inserted in a smaller character:-" About the year 1672, Monsieur Picart has published an account in French concerning the measure of the earth, a breviate whereof may be seen in the Philosophical Transactions, No. 112, wherein he concludes one degree to contain 365,184 English feet, nearly agreeing with Mr Norwood's experiment ;" and this advertisement is continued through the subsequent editions as late as the year 1732.


About the year 1645, Bond published, in Norwood's Epitome, a very great improvement of Wright's method, from a property in his meridian line, whereby the divisions are more scientifically assigned than the author himself was able to effect. It resulted from this 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 third edition of Gunter's works, printed in 1653, where he observed that the logarithmic tangents from 45° upwards increase in the same manner as the secants do added together, 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 to be 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 of the general theorem, that the degrees of latitude bear to one degree (or sixty minutes, which in Wright's table stand for the meridional parts of one degree) 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 here there was still wanting the demonstration of this general theorem, which was at length supplied by James Gregory of Aberdeen, in his Exercitationes Geometrica, printed at London in 1668; and afterwards more concisely demonstrated, together with a scientific determination of the divisor, by Dr Halley, in the Philosophical

Transactions for 1695 (No. 219), 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 Roger Cotes, in his Logometria, first published in the Philosophical Transactions for 1714 (No. 388). It is, moreover, added in Gunter's book, that ifth 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 set down 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 12633, when the logarithmic tables consist of eight places of figures besides the index.


In an edition of a book called the Seaman's Kalendar, Bond declared that he had discovered the longitude 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. 40), he published a table of the variation for forty-nine years to come. he acquired such reputation, that his treatise entitled The Longitude Found, was, in the year 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 the year 1678 another treatise, entitled The Longitude not Found, made its appearance; and as Bond's hypothesis did not answer its author's sanguine expectations, the solution of the difficulty was undertaken by Dr Halley. The result of his speculation was, that the magnetic needle is influenced by four poles; but this wonderful phenomenon seems hitherto to have eluded all our researches. (See MAGNETISM.) In 1700, however, Dr Halley 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 1644 and 1756, by Mountain and Dodson. 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, new improvements were daily made, and everything 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, not a perfect sphere, but a spheroid, with the shortest diameter passing through the poles, a tract was published in 1741 by the Reverend Dr Patrick Murdoch, wherein he accommodated Wright's sailing to such a figure; and the same year Colin Maclaurin, in the Philosophical Transactions (No. 461), gave a rule for determining the meridional parts of a spheroid; which speculation is farther treated of in his book of Fluxions, printed at Edinburgh in 1742, and in Delambre's Astronomy (t. iii., ch. xxxvi.).

Amongst the later discoveries in navigation, that of finding the longitude, both by lunar observations and by timekeepers, is the principal. It is owing chiefly to the rewards offered by the British Parliament that this has attained the present degree of perfection. We are indebted to Dr Maskelyne for putting the first of these methods in practice, and for other important improvements in navigation. The time-keepers constructed by Harrison for this express


Practice of purpose were found to
Naviga parliamentary reward.


answer so well that he obtained the These have been improved by ArThese have been improved by Arnold, Earnshaw, and many others, so as now to be almost

in common use.

The works which have latterly appeared on navigation are Preliminthose on the longitude and navigation by Mackay, Inman, ary PrinRiddle, Norie, Jeans, and others; and these contain every ciples. necessary requisite to form the practical navigator.






The art of navigation depends upon mathematical and
astronomical principles. The problems in the various
modes of sailing are resolved either by trigonometrical cal-
culations, or by tables or rules formed by the assistance of
plane and spherical trigonometry. By mathematics the
necessary tables are constructed and rules investigated for
performing the more difficult parts of navigation.

The places of the sun, moon, and planets, and fixed
stars, are deduced from observation and calculation, and
arranged in tables, the use of which is absolutely necessary
in reducing observations taken at sea for the purpose of
ascertaining the latitude and longitude of the ship and the
variation of the compass. The investigation of the rules
required for this purpose belongs properly to the science
of ASTRONOMY, to which the reader is referred. A few tables
are given at the end of this article, but as the other tables
necessary for the practice of navigation are to be found in
almost every treatise on this subject, it seems unnecessary
to insert them in this place. The subject naturally divides
itself under two heads:-First, The methods of conducting a
ship from one port to another by help of rules, in which the
log-line and compass are alone required, which is Naviga-1°
tion properly so called. Second, The method of ascertain-
ing the ship's latitude and longitude, and variation of com-
pass, by means of observations on the heavenly bodies; and
the rules for that purpose deduced from astronomy, in order
to correct the ship's place, and the courses derived from
the former method, to which the name of Nautical Astro-
nomy is generally applied. Although the reader is referred
to the respective articles on the sciences on which naviga-
tion is founded in this work for complete information, we
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

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 First Meridian. The situation of the equator is fixed; but that of the first meridian is arbitrary, and therefore different nations assume different first meridians. In Great Britain we assume that to be the first meridian which passes through the Royal Observatory at Greenwich.

The equator is a great circle on the earth's surface, every point of which is equally distant from the two poles or the extremities of the imaginary axis about which the earth makes her diurnal rotation. It therefore divides the earth into two equal parts, called the Northern and the Southern

Hemispheres, according as the North or the South Pole
lies within them.

The latitude of a place is its distance from the equator, reckoned on a meridian in degrees, minutes, and seconds, and decimal parts of seconds (if necessary), being either north or south, according as it is the Northern or Southern Hemisphere. Hence it appears that the latitudes of all places are comprised within the limits 0° and 90° N., and 0° and 90° S.

The first meridian, which is a great circle passing through the poles, also divides the earth into two equal portions, called the Eastern and Western Hemispheres, according as they lie to the right or left of the first meridian; the spectator being supposed to be looking towards the north.

The longitude of a place is the arc of the equator intercepted between the first meridian and the meridian of the given place reckoned in degrees, minutes, and seconds; and is either east or west as the place lies in the Eastern or Western Hemisphere respectively to the first meridian. The longitude of all places on the earth's surface is comprised within the limits of 0° and 180° E., and 0° and 180° W. On the supposition that the earth is a sphere, the length of all arcs of great circles upon it subtending an angle of at the centre are equal; hence l' of latitude or longitude is equal to one geographical or nautical mile, of which a degree contains 60. Hence intervals of latitude and longitude, reduced to minutes and parts of minutes, also represent the same number of nautical miles and parts of a nautical mile.

In the practice of navigation, the latitude and longitude of the place which a ship leaves, are called the latitude and longitude from; and the latitude and longitude of the place at which it has arrived, are called the latitude and longitude in.


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Let QR...V be a portion of the equator, P the pole, and PAQ, PBR, PCS......PFV be meridians supposed very near to one another, passing through points A, B, C, D, E, F, the line AF being the path traced out by a vessel in passing from A to F, such that it makes equal angles with every meridian over which it passes. From B, C, D, &c., let BH, CI, DK, EL. &c., be drawn perpendicular to the

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Prelimin- two meridians between which they respectively lie; or, in ary Prin- other words, be arcs of small circles or parallels of latitude ciples. through the points B, C, &c. These are consequently all parallel to one another, and to FG the whole arc of the parallel at F included between the extreme meridians PAQ and PFV.

The constant angle at which the line AF is inclined to the successive meridians, viz., BAP, CBP, DCP, &c., is called the course. Also, if the small circles or parallels at B, C, &c., be continued to the meridian PAQ, the portion of this meridian intercepted between any two consecutive parallels, as cd, will be equal to CK, the distance between the parallels through C and D ; and so on for all. Hence the sum of these distances, AH+BI+CK+DL+EM= AG; which is called the true difference of latitude, or true diff. lat. from A to F.

The corresponding arcs of parallels at different latitudes. intercepted between the same meridians are not equal, but gradually decrease from the equator to the poles. Hence the sum of the arcs BH÷CI+DK+EL+ FM is less than QV, the intercepted arc of the equator, but greater than FG, the arc of the highest parallel intercepted between PAQ and PFV.

In navigation, each of the triangles ABH, BCI, &c., is considered as a plane triangle; and as each of them is right-angled, and contains, besides, one constant angle, viz., the course, the other angle in each must also be constant ; and all the triangles will be equiangular and similar. Hence we have

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BH + CI+ DK + EL + FM is called the departure ;
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
A to F, is called the distance.

dep. dist. x sin course by (11.)

5. Given the distance and departure, to find the course

and true difference of latitude.

Sin BAC=

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Or in logarithms,

log. true diff. lat.=log. dist. + L cos course - 10, where L means tabular logarithm, i.e., logarithm increased by 10; and log. dep. = log. dist. + L sin course – 10.


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Cos BAC=



or cosine course = true diff. lat. dist. (VII.)


or sin course dep. ÷ distance.



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And having found the course, we have

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And then we have
true diff. lat. dist. x cos course.

6. Given the true difference of latitude and departure,
to find the course and distance.

Tan BAC=


or tan course=departure ÷ true diff. lat. . (IX.)

dist. = true diff. lat. x sec course by (III.);
or dist. = dep.
x cosec course by (v.)


(III.) . (IV.)


. (vi.)

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Length of Arc of 1° of Parallels of Latitude.

We have already stated that the lengths of the parallels
of latitude diminish as the lati-
tude increases. In fact, it de-
creases in the ratio of cosine la-
titude to unity.

Let EQ be the equator, PCP'
the polar diameter of the earth
passing through C, and LML'any
parallel of latitude. Let the angle
ECL, or the latitude, be l, and let
LM and EF be the arcs of the
parallel and of the equator in-
tercepted between the meridians
PEP and PFP'. Then angle ECF = angle LOM, because
they both measure the angle between the planes of the two
meridians. Hence

Fig. 3.

arc LM: arc EF::OL:CE:: OL: CL, because CE = CL;

=arc EF x sin OCL

or arc LMarc EF x
=arc EF x cos LCE=arc EF x cos l.

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Hence if FE be the length of an arc 1° of longitude at the equator, or 60 miles, LM the length of an arc 1° of longitude in latitude 60 x cos l.


ary Prin ciples.

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