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 magnetic 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 expensive; 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.
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 and tangents to every minute of the quadrant, which he 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, 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.
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 log-line; 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 57,800 French
---
1 See Gunter's Scale. 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 $45^\circ$ 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 $45^\circ$ 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 $45^\circ$; divide the difference of those numbers by the logarithmic tangent of $45^\circ$, 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 $45^\circ$, and the radius neglected, to the like tangent of half a degree augmented by $45^\circ$, 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 Geometricae*, 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 1696 (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 if $\frac{1}{3}$th of this division, which does not sensibly differ from the logarithmic tangent of $45^\circ$ $1'30''$, 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 $45^\circ$ $1'30''$, 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. Thus 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 time-keepers, 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 NAVIGATION.
BOOK I.
CONTAINING THE VARIOUS METHODS OF SAILING.
The art of navigation depends upon mathematical and astronomical principles. The problems in the various modes of sailing are resolved either by trigonometrical calculations, 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 Navigation properly so called. Second, The method of ascertaining the ship's latitude and longitude, and variation of compass, 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 Astronomy is generally applied. Although the reader is referred to the respective articles on the sciences on which navigation 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 repeating somewhat the substance of portions of our other articles.
CHAP. I.—PRELIMINARY PRINCIPLES.
SECT I.—ON LATITUDE AND LONGITUDE; DEFINITION OF TERMS USED IN NAVIGATION; AND GENERAL EXPLANATIONS.
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 1° at the centre are equal; hence 1° 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.
2. Definitions of Terms used in Navigation, and Explanations.
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 two meridians between which they respectively lie; or, in other words, be arcs of small circles or parallels of latitude 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.
BH + CI + DK + EL + FM is called the departure; the arc QV of the equator is the difference of longitude; and AF, the curve described by the vessel in passing from A to F, is called the distance.
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
\[ \begin{align*} AH : BH : AB &= AH : BH : AB \\ BI : CI : BC &= AH : BH : AB \\ CK : DK : CD &= AH : BH : AB \\ &\vdots \\ EM : FM : EF &= AH : BH : AB. \end{align*} \]
And since, when any number of quantities are in continued proportion, as the first consequent is to its antecedent, so are all the consequents to all the antecedents; we have
\[ \begin{align*} AH + BI + CK + &\vdots = AG \text{ the true diff. lat.} \\ BH + CI + DK + &\vdots = \text{the departure.} \\ AB + BC + CD + &\vdots = AF \text{ or the distance.} \end{align*} \]
Hence the true difference of latitude, departure, and distance, may be considered as the sides of a right-angled triangle, similar to each of the small triangles; the angle of which, therefore, between the true difference of latitude and distance, is the course.
Take AB (fig. 2) = the true diff. lat. Draw BC at right angles to it = departure. Join AC. Then AC is the distance, and BAC is the course.
From this it appears, that when any two of the four quantities, true difference of latitude, departure, distance, and course are given, the remaining two can be found by solving the right-angled triangle ABC.
1. Given course (BAC) and distance (AC), to find true difference of latitude (AB), and departure (BC).
\[ \begin{align*} AB &= AC \cos BAC \\ BC &= AC \sin BAC, \end{align*} \]
i.e., true diff. lat. (in miles) = dist. × cos course \quad (I)
and departure = dist. × sin course \quad (II)
Or in logarithms,
\[ \log \text{ true diff. lat.} = \log \text{ dist.} + L \cos \text{ course} - 10, \]
where \(L\) means tabular logarithm, i.e., logarithm increased by 10; and log. dep. = log. dist. + L sin course - 10.
2. Given course (BAC) and true difference of latitude (AB), to find distance and departure.
\[ \begin{align*} AC &= AB \times \sec BAC \\ BC &= AB \times \tan BAC; \end{align*} \]
or dist. = true diff. lat. × sec course \quad (III)
dep. = true diff. lat. × tan course \quad (IV)
3. Given course and departure, to find distance and true difference of latitude.
\[ \begin{align*} AC &= BC \times \csc BAC \\ AB &= BC \times \cot BAC; \end{align*} \]
or dist. = dep. × csc course \quad (V)
and true diff. lat. = dep. × cot course \quad (VI)
4. Given distance and true difference of latitude, to find course and departure.
\[ \cos BAC = \frac{AB}{AC}; \]
or cosine course = true diff. lat. - dist. \quad (VII)
And the course having been found, we get
dep. = dist. × sin course by (II)
5. Given the distance and departure, to find the course and true difference of latitude.
\[ \sin BAC = \frac{BC}{AC}; \]
or sin course = dep. - distance \quad (VIII)
And then we have
true diff. lat. = dist. × cos course.
6. Given the true difference of latitude and departure, to find the course and distance.
\[ \tan BAC = \frac{BC}{AB}; \]
or tan course = departure - true diff. lat. \quad (IX)
And having found the course, we have
dist. = true diff. lat. × sec course by (III);
or dist. = dep. × csc course by (V)
Length of Arc of 1° of Parallels of Latitude.
We have already stated that the lengths of the parallels of latitude diminish as the latitude increases. In fact, it decreases in the ratio of cosine latitude 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 intercepted between the meridians PEP and PEP'. Then angle ECF = angle LOM, because they both measure the angle between the planes of the two meridians. Hence
\[ \begin{align*} \text{arc LM} &= \text{arc EF} :: OL : CE :: OL : CL, \text{ because } CE = CL; \\ \text{or arc LM} &= \text{arc EF} \times \frac{OL}{CL} = \text{arc EF} \times \sin OCL \\ &= \text{arc EF} \times \cos LCE = \text{arc EF} \times \cos l. \end{align*} \]
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 \(l = 60 \times \cos l\). Middle Latitude.
The departure is less than VQ (fig. 1), the intercepted arc of the equator, or than the intercepted arc of the parallel through A; but it is greater than FG. But since the arc of the parallel gradually decreases from A to F, there is some point intermediate in position between A and F (y), the intercepted arc of the parallel of which will be exactly equal to the departure. The exact determination of this point is not very easy. Various methods have been proposed to determine this latitude nearly, with as little trouble as possible: first, by taking the arithmetical mean of the two latitudes for that of the mean latitude; secondly, by using the arithmetical mean of the cosines of the latitudes; thirdly, by using the geometrical mean of these cosines; and, lastly, by employing the latitude deduced from the mean of the meridional parts of the two latitudes. The first of these methods is the one usually employed. It has the merit of great simplicity; and as all the rules in navigation are approximate only, it may perhaps be depended on as much as any of these. Hence y may be considered as the middle point between A and F; and dep. = xyz.
But \(xyz = VQ \cos \text{lat. of } y\);
or dep. = diff. long. \(x \cos y\), where \(y\) is the arithmetical mean of the latitudes of A and F \(= \frac{1}{2}(l + l')\), if
\(l = \text{latitude of } A,\)
\(l' = \text{latitude of } F.\)
This is commonly called the middle latitude. Hence we have
dep. = diff. long. \(x \cos \text{middle latitude}.\)
By equation (i) we have
true diff. lat. = dist. \(x \cos \text{course}.\)
Whence it appears that if the middle latitude be considered as a course, and the departure as a true difference of latitude, the corresponding distance will be the difference of longitude. And conversely, treating the middle latitude as a course, and the difference of longitude as a distance, the corresponding true difference of latitude will be the departure.
Mercator's Chart—Meridional Parts.
The chart used at sea for tracing the ship's track exhibits the surface of the earth on a plane, in which the meridians are parallel, and consequently the distance between them throughout their length equal to the equatorial distance, instead of gradually decreasing as the latitude increases. In other words, FG is increased so as to become equal to VQ. Now, in order that on this chart all points may occupy the same relative position with respect to each other that the points corresponding to them do on the surface of the globe, the distance AF and the distance AG must be increased in the same proportion that BH + CI + DK + &c., i.e., the departure has been increased.
The distance AG so increased is called the meridional difference of latitude, or mer. diff. lat.; and the chart constructed on this principle is called Mercator's Chart. If for any latitude the meridional difference of latitude between this point and the equator be expressed in miles or minutes, the number of miles so expressed is called the meridional parts for that latitude. A table of meridional parts for every minute of latitude from 0° up to 90°, is given in every collection of nautical tables.
Construction of Table of Meridional Parts.
This table may be constructed approximately, by dividing the whole meridian from 0° to 90° into intervals of 1', and supposing the increase of the arc of the parallel of latitude, and consequently that of the arc of latitude, to take place at the end of the successive minutes.
Now we have seen that an arc of any parallel = corresponding arc of equator \(x \cos \text{lat.};\) and since the arcs of the successive parallels have all become equal to the correspond-
ing arc of the equator, they have all been increased in the ratio of sec lat. to 1.
Hence, if the length of 1' of the meridian be 1, and \(a, b, c, d,\) the corresponding increased lengths between 0 and 1', between 1' and 2', 2' and 3', &c.,
\[ \begin{align*} a &= 1 \times \sec 1' \\ b &= 1 \times \sec 2' \\ c &= 1 \times \sec 3' \\ d &= 1 \times \sec 3' \\ &\vdots \end{align*} \]
And \(a + b + c, &c. = \sec 1' + \sec 2' + \sec 3' + &c.;\) or the meridional parts in any arc of the meridian is equal to the sum of the secants of all the successive angles, differing by 1', from 1' up to the given latitude.
The true investigation is as follows:—Let \(m\) be the circular measure of the angle subtended at the centre by the meridional parts in the arc between the latitudes 0 and \(l;\) and let \(l\) become \(l + \delta l,\) and let \(\delta m\) be the corresponding increase of \(m.\) Then \(\delta m\) is proportional to the secant of \(l + \delta l;\)
\[ \frac{\delta m}{\delta l} = \sec (l + \delta l). \]
And ultimately taking the limit
\[ \frac{dm}{dl} = \sec l; \]
\[ \therefore m = \int_0^1 \sec ld l = \int_0^1 \frac{1}{\cos l} dl \]
\[ = \int_0^1 \frac{\cos^2 \frac{l}{2} - \sin^2 \frac{l}{2}}{\cos^3 \frac{l}{2} - \sin^3 \frac{l}{2}} dl = \int_0^1 \left(1 + \tan^2 \frac{l}{2}\right) dl \]
\[ = \int_0^1 \frac{d \cdot \tan \frac{l}{2}}{1 - \tan^2 \frac{l}{2}} = \log \frac{1 + \tan \frac{l}{2}}{1 - \tan \frac{l}{2}} \]
\[ = \log \left(\frac{\cos \left(45 - \frac{l}{2}\right)}{\cos 45 + \frac{l}{2}}\right). \]
Let \(45 - \frac{l}{2} = \frac{l_1}{2},\) or \(90 - l = l_1;\)
\[ 45 + \frac{l}{2} = 90 - \frac{l_1}{2}; \]
and \(m = \log \frac{\cos \frac{l_1}{2}}{\sin \frac{l_1}{2}} \cot \frac{l_1}{2}\)
\[ = 2 \cdot 3025851 \log_{10} \cot \frac{l_1}{2}. \]
Now \(m = \frac{\text{arc radius}}{\text{radius (in minutes)}} = \frac{\text{arc radius (in miles)}}{\text{meridional parts for lat. } l}\)
\[ = \frac{57 \cdot 29577 \times 60}{\text{meridional parts for lat. } l} \]
\[ = 57 \cdot 29577 \times 60 \times 2 \cdot 3025851 \times \log_{10} \cot \frac{l_1}{2}. \]
In logarithms—
\[ \log \text{mer. parts for lat. } l = 3 \cdot 8984895 + \log_{10} (\text{L cot } \frac{1}{2} \text{colat.} - 10). \]
Whence we deduce the following
Rule for Finding the Meridional Parts.
Diminish the tabular logarithm of the cotangent of one-half the colatitude by 10. Find the logarithm of the remainder, and add to it the constant logarithm 3.8984895. The result is the logarithm of the meridional parts for the given latitude. Ex.—Required the meridional parts for latitude 65° and 65° 20'.
\[ L \cot 12° 30' - 10 = 0.6542448 \\ \text{Log. } 0.6542448 = 1.8157403 \\ \text{Constant log.} = 3.8984895 \\ = 3.7142998 \\ \therefore \text{Meridional parts,} = 5178.80 \]
\[ L \cot 12° 20' - 10 = 0.6602609 \\ \text{Log. } 0.6602609 = 1.8197155 \\ \text{Constant log.} = 3.8984895 \\ = 3.7182050 \\ \therefore \text{Meridional parts,} = 5226.43 \]
Relations between Meridional Difference of Latitude and Difference of Longitude, Course, &c.
It appears that on Mercator's chart the difference of longitude, meridional difference of latitude, and increased distance, form the sides of a right-angled triangle, and are proportional to the departure, true difference of latitude, and distance, in the triangle ABC. The two triangles are therefore similar.
If, then, in AB produced (fig. 4) AB' be taken equal to mer. difference of latitude, and BC' be drawn parallel to BC, meeting AC produced in C'; the sides of the triangle ABC will be the meridional difference of latitude, difference of longitude, and increased distance.
Hence, in equations (i.), (ii.), (iii.), (iv.), (v.), (vi.), (vii.), and (viii.), we may substitute meridional difference of latitude for true difference of latitude, difference of longitude for departure, and increased distance for distance; and the equations will still hold.
The relation principally required is that which connects the meridional difference of latitude, difference of longitude, and course;
or, diff. long. = mer. diff. lat. × tan course.
Parallel Sailing.
When the course is 90°, or the ship sails in a parallel of latitude, the equations (i.), &c., give no result when applied to find the distance.
In this case we must apply the formula,
arc of parallel = corresponding arc of equator × cos lat.
Here the arc of the parallel corresponds to the distance, and the arc of equator is difference of longitude, and we have for parallel sailing
\[ \text{dist.} = \text{diff. long.} \times \cos \text{lat.} \]
from which, any two of the quantities being given, the third may be found. Also, if \(d\) and \(d'\) be distances corresponding to the same difference of longitude in parallels \(l\) and \(l'\),
we have \(d = \text{diff. long.} \cos l\)
\(d' = \text{diff. long.} \cos l'\);
or \(d = d' \cdot \cos l\);
which enables us to find the distance on one parallel corresponding to a given distance on another, the difference of longitude being the same.
Middle-Latitude Sailing.
Since departure = diff. of longitude × cos. mid. latitude, if in equations (ii.), (iv.), (vi.), (viii.), we substitute this value for departure, we shall obtain the following equations:
\[ \text{diff. long.} \times \cos \text{mid. lat.} = \text{dist.} \times \sin \text{course} \\ \text{diff. long.} \times \cos \text{mid. lat.} = \text{true diff. lat.} \times \tan \text{course} \\ \text{true diff. lat.} = \text{diff. long.} \times \cos \text{mid. lat.} \times \cot \text{course} \\ \text{sin course} = \text{diff. long.} \times \cos \text{mid. lat.} + \text{dist.} \]
from which all the rules for middle-latitude sailing may be derived.
Traverse Tables.
A table in which the true difference of latitude and departure, corresponding to certain distances for every course, expressed in points and degrees, are laid down, is called a traverse table. It is very useful to enable the seaman to solve the several problems which occur in navigation by simple inspection. It must of course be calculated on some of the principles laid down in this chapter.
It is evident that as this table contains the relations of the sides and angles of a right-angled triangle, the solution of any right-angled triangle, whose sides represent any other quantities, will be given by it, by making the requisite changes.
Thus, the course remaining the same, if, for difference of latitude we look out in the table the meridional difference of latitude, the corresponding departure will be the difference of longitude; also, the difference of longitude can be found from the departure by these tables, by looking out the middle latitude as a course, and the departure as a true difference of latitude; then the corresponding distance is the difference of longitude.
SECT. II.—LONGITUDE AND LATITUDE.
PRON. I.—Given latitude from, and latitude in; to find the true difference of latitude.
Rule.—Under latitude in, with its proper name, i.e., N. or S., place latitude from. Subtract the greater from the less, if of the same name, and reduce to minutes. The result is the true difference of latitude required, and is of the same or different name with latitude in, according as latitude in is greater or less than latitude from. If they are of different names, add and affect with the name of latitude in.
Ex. 1.—A vessel sails from the Lizard, Lat. 49° 58' N., to Cape St Vincent, Lat. 37° 3' N.; what is the true diff. of latitude?
Latitude in........................................... 37° 3' N. Latitude from........................................ 49° 58' N. True diff. latitude.................................. 12° 55' 8" = 775 miles S.
Ex. 2.—A vessel sails from New York, Lat. 40° 42' N., to Liverpool, Lat. 53° 25' N.; find the true diff. of latitude.
Latitude in........................................... 53° 25' N. Latitude from........................................ 40° 42' N. True diff. latitude.................................. 12° 43' 8" = 763 miles N.
Ex. 3.—A ship sailed from Funchal, Lat. 22° 39' N., to the Cape of Good Hope, Lat. 34° 29' S.; what is the true diff. of latitude?
Latitude in........................................... 34° 29' S. Latitude from........................................ 32° 38' N. True diff. latitude.................................. 67° 7' 8" = 4027 miles S.
PRON. 2.—Given the latitude from, and true difference of latitude; to find latitude in.
Rule.—If the latitude from and true difference be of the same name, add them (the true difference of latitude being turned, if necessary, into degrees and minutes); the sum is the latitude in of the same name. If of unlike names, under latitude from place the true difference of latitude, subtract the less from the greater; the result, with the name of the greater, is the latitude in.
Ex. 1.—A ship sailed from the Lizard, 49° 58' N., and made good, in a northerly direction, 207 miles; what is the latitude in?
Latitude from........................................ 49° 58' N. True diff. latitude.................................. 3° 27' N. Latitude in........................................... 53° 25' N.
Ex. 2.—A ship in Lat. 57° 18' N. sailed due S. 3789 miles; what is the latitude in?
Latitude from........................................ 57° 18' N. True diff. latitude.................................. 63° 9' S. Latitude in........................................... 51° 51' S. Prob. 3.—To find the meridional difference of latitude; having given latitude from, and latitude in. Take the meridional parts for the two latitudes from the table for meridional parts; subtract if the names be alike, and add if the names be unlike. The result is the meridional difference of latitude, and is N. or S. according as the latitude in is N. or S. of latitude from.
Ex. 1.—Required the meridional diff. of latitude in sailing from Cape Finisterre, Lat. 42° 52' N., to Port Praya, in the island of Santiago, Lat. 14° 54' N.
| Latitude in | 14° 54' N. | Mer. parts | 904 N. | |------------|------------|------------|--------| | Latitude from | 42° 52' N. | Mer. parts | 2832 N. | | Mer. diff. lat. | 1948 S. |
Ex. 2.—To find the meridional diff. of latitude in sailing from Lat. 6° 35' N. to 8° 17' S.
| Latitude in | 8° 17' S. | Mer. parts | 499 S. | |------------|-----------|------------|--------| | Latitude from | 6° 35' N. | Mer. parts | 335 N. | | Mer. diff. lat. | 834 S. |
Prob. 4.—To find the difference of longitude; having given longitude in, and longitude from.
Rule.—Under longitude in place longitude from; subtract, if of like name; reduce the result to minutes, and call it E. or W., according as longitude in is E. or W. of longitude from. Add, if of unlike names, and attach the name E. or W., according as the longitude in is E. or W. of longitude from. If the longitude found by this rule exceed 180° it must be subtracted from 360°, and affected with the contrary name.
Ex. 1.—A ship sails from Liverpool, Long. 2° 59' W., to New York, Long. 73° 59' W.; required the diff. of longitude.
| Longitude in | 73° 59' W. | |--------------|------------| | Longitude from | 2° 59' W. | | Diff. longitude | 71° 0' W. = 4260 miles W. |
Ex. 2.—A ship sails from Masekelyne's Isles, in Long. 167° 59' E., to Olinda, in Long. 35° 54' W.; find the diff. of longitude.
| Longitude in | 35° 54' W. | |--------------|------------| | Longitude from | 167° 59' E. | | Diff. longitude | 132° 5' E. = 9367 miles E. |
Prob. 5.—To find the longitude in; having given longitude from, and difference of longitude.
Rule.—Under longitude from place difference of longitude. If of like names, add, and the result is the longitude in of the same name as the longitude from; if of unlike names, subtract the less from the greater. The result, with the name of the greater, is the longitude in.
Ex. 1.—A ship from Long. 9° 54' E. sailed westerly till the difference of longitude was 1398 miles; what is the longitude in?
| Longitude from | 9° 54' E. | |----------------|-----------| | Diff. longitude | 23° 18' W. | | Longitude in | 13° 24' W. |
Ex. 2.—The longitude sailed from is 25° 9' W., and diff. of longitude 112° 6' W.; find longitude in.
| Longitude from | 25° 9' W. | |----------------|-----------| | Diff. longitude | 18° 46' W. | | Longitude in | 43° 55' W. |
Sect. III.—Of Measuring a Ship's Run in a Given Time.
The method commonly used at sea to find the distance sailed in a given time is by means of a log-line and half-minute glass. (A description of these is given under the articles Log, and Log-Line, which see.)
The interval between two consecutive knots on the line—also technically called a knot—is supposed to be the same fraction of a nautical mile (6080 feet) that half a minute is of an hour. Hence the proper length of a knot is 6080 feet nearly. But although the line and glass be at any time perfectly adjusted to each other, yet as the line shrinks after being wet, and as the weather has a considerable effect on the glass, it will be necessary to examine them from time to time; and the distance given by them must be corrected accordingly. The distance sailed, therefore, may be affected by an error in the glass or in the line, or in both. The true distance may, however, be found as follows:
Prob. 1.—The distance sailed by the log, and the seconds run by the glass, being given; to find the true distance run, the line being supposed right.
Let the number of seconds in which the glass runs out be \( n \), and let \( d \) and \( d' \) be the true distance and distance by log respectively. Then evidently the longer the time the glass is running, the less is the distance by log compared with the true distance, and conversely. Hence we have the proportion
\[ \frac{d}{d'} : : \frac{30}{n} : : \text{number of seconds the glass is running}; \]
or \( d = \frac{d' \times 30}{n} \).
Rule.—Multiply the distance given by log by 30, and divide by the number of seconds the glass is running; the result is the true distance run.
Ex. 1.—The hourly rate of sailing by the log is 9 knots, and the glass is found to run out in 35"; required the true rate of sailing.
\[ \frac{9}{30} = \frac{36}{270} = 7.7 = \text{true rate of sailing}. \]
Ex. 2.—The distance sailed by the log is 73 miles, and the glass runs out in 26"; required the true distance.
\[ \frac{73}{30} = \frac{26}{190} = 84.2 = \text{the true distance}. \]
Prob. 2.—Given the distance sailed by the log, and the measured interval between two adjacent knots on the line; to find the true distance, the glass running exactly 30".
Here evidently the true distance is greater or less than the distance by log, as the measured interval is greater or less than 51 feet.
Let \( a \) be the measured interval between the knots in feet, and \( d \) and \( d' \) the true, and measure distances as before.
Then \( 51 : a :: d' : d \),
or \( 51 \text{ feet} : \text{measured distance in feet} :: \text{distance by log} : \text{true distance} \).
\[ \text{True distance} = \text{distance by log} + \frac{51}{\text{measured distance}}. \]
Rule.—Multiply the distance given by log by the measured length of a knot, and divide by 51; the quotient is the true distance.
Ex. 1.—The hourly rate of sailing by the log is 5 knots, and the interval between knot and knot measures 53 feet; required the true rate of sailing.
\[ \frac{53}{5} = \frac{51}{265} = 5.19 = \text{true rate of sailing}. \]
Ex. 2.—The distance sailed is 85 miles, by a log line which measures 42 feet to a knot; required the true distance run.
\[ \frac{85}{42} = \frac{51}{3570} = 70 = \text{true distance run}. \]
Prob. 3.—Given the length of a knot, the number of seconds run by the glass in half a minute, and the distance... To find the true distance we must evidently compound the ratios given in problems 1 and 2, and we have
\[ n \times 51 : 30a \times d : d ; \]
or \( d = d' \times \frac{30a}{51n} = \frac{d' \cdot 10a}{17n}. \)
**Rule.**—Multiply the distance given by the log by 10 times the measured distance between the knots, and divide by 17 times the number of seconds the glass is running.
**Ex.**—The distance sailed by the log is 159 miles, the measured length of a knot is 42 feet, and the glass runs out in 33°; required the true distance.
Distance by the log.................. 159 10 times length of a knot........... 420 17 times number of seconds run by the glass.............. 636
\[ \text{true distance} = \frac{159 \times 420}{636} = 119.037 \text{ miles} \]
**SECT. IV.—ON COURSES AND CORRECTIONS OF COURSES.**
**Mariner's Compass.**
A ship is enabled to keep her course at sea by means of an instrument called the mariner's compass. It consists of a magnetic steel bar attached to the under side of a card, divided into points and quarter points, and supported by a fine pin, on which it turns freely within a box covered with glass. By reason of the directive property of the magnet, the north point, which is commonly denoted by a fleur de lis, is readily known. The circumference of the card is generally divided into thirty-two points, which, in the best compasses, are again subdivided into half points and quarters. These are reckoned sufficient for nautical purposes. On the inside of the box is drawn a dark vertical line called lubber's point. This point, or rather line, and the pin on which the card turns, are in the same line or plane with the keel of the ship; and hence the point on the circumference of the card opposite to lubber's point shows the angle which the ship's course makes with the magnetic meridian, called the course of the ship.
The annexed diagram (fig. 5) gives a general view of the compass. (For a full explanation of its magnetic properties, see Magnetism.) The names of the points, and the angles which they form with the meridian, are given in fig. 6, and as thus represented the instrument is called the steering compass.
The azimuth compass is the same instrument more nicely made. The circumference of the card is divided into degrees and parts by a vernier, and is fitted up with sights vanes to take amplitudes and azimuths, for the purpose of determining the variation of the compass by observation. The variation is then applied to the magnetic course shown by the steering compass, whence the true course, with respect to the meridian, becomes known,
Besides the variation, the needle is also affected by the dip, which is likewise fully explained in the article Magnetism, as well as Mr Barlow's method of correcting the effects of local attraction, arising from the effects of the iron, guns, &c., in the vessel itself.
The compass course generally differs from the true course on account of three causes:—1. The variation of the compass; 2. The deviation of the compass; 3. The leeway. We shall now explain how these errors are to be applied.
1. **The Variation of the Compass.**—This is fully explained under the article Magnetism, which see. The mode of ascertaining its amount will be given hereafter.
**Prob. I.**—To find the true course, having given compass course.
**Rule 1.**—Allow easterly variation to the right, and allow westerly variation to the left.
**Ex. 1.**—The compass course is W.N.W., and variation 3½ pts. W.; find the true course.
| Compass course | Pts. qrs. | |----------------|-----------| | Variation | 3 |
True course........... 9 1 left of N., or W. by S.½ S.
**Ex. 2.**—The compass course is S.W.½ W., and variation 2½ E.; find the course.
| Compass course | Pts. qrs. | |----------------|-----------| | Variation | 2 |
True course........... 7 1 right of S., or W. by S.½ W.
**Ex. 3.**—The compass course is N.W., the variation is 3½ E.; required the true course.
| Compass course | Pts. qrs. | |----------------|-----------| | Variation | 3 |
True course........... 0 3 left of N., or N.½ W.
**Prob. II.**—Given the true course, to find the compass course.
**Rule 2.**—Allow easterly variation to the left, and westerly to the right.
**Ex. 1.**—The true course is N.N.E.½ E., and variation 1½ W.; find the compass course.
| True course | Pts. qrs. | |-------------------|-----------| | Variation | 1 |
Compass course........ 4 0 right of N., or N.E.
**Ex. 2.**—The true course is N.½ E., the variation 3½ E.; required the compass course. 2. The Deviation of the Compass.—This error arises from the effects of local attraction, and varies with every different position of the ship's head. Several methods are employed in order to ascertain its amount. That most commonly adopted is to place a compass on shore, out of reach of the ship's attraction, and to take the bearing of the ship's compass, or some other object in the same direction with it; while at the same time the bearing of the compass on shore is taken on board. If now 180° be added to the bearing of the shore compass, so as to bring it round to the opposite point, the difference between this augmented bearing and the bearing at the ship's compass will be the amount of deviation for that position of the ship's head.
Suppose the ship's head is N., and that the reading off at the shore compass is S. 17° 15' W., and that the reading off at the ship's compass is N. 20° E. Adding 180° to the bearing of the shore compass, we get S. 197° 15' W., or N. 17° 15' E.; and subtracting this from the bearing of the ship's compass, N. 20° E., we get the deviation equal to 2° 45' E., when the ship's head is N. The ship is now turned round, so that the head points successively to every point of the compass, and the deviation for each position found as before.
A table is then made, showing the deviation for every point of the compass. The deviation so found is treated exactly as the variation,—i.e., in correcting the compass course to find the true course, easterly deviation is allowed to the right, and westerly deviation to the left; and conversely, to find the compass course from the true course, easterly deviation is allowed to the left, and westerly deviation to the right. Hence it appears, that when both variation and deviation are given, we may consider the latter as a correction of the former—to be added to it if of the same name, and to be subtracted from it if of the opposite name.
Ex. 1.—The compass course is S.W. 4 W., variation 1 1/2 E., and deviation 1/2 W.; what is the true course?
Compass course ................. 4 3 right of S. Variation .......... 1 3 right Deviation .......... 2 left
True course .................. 6 0 right of S., or W.S.W.
Ex. 2.—The compass course is W. 1/2 N., the variation 2 1/2 W., and deviation 3/4 W.; what is the true course?
Compass course ................. 7 2 left of N. Variation .......... 2 2 left Deviation .......... 0 3 left
True course .................. 10 3 left of N., or S.W. by W. 1/2 W.
Ex. 3.—The true course is N.N.W. 4 W., variation 1 1/2 E., and deviation 1 W.; required the compass course.
True course .................. 2 3 left of N. Variation .......... 1 3 left Deviation .......... 0 1 right
Compass course ................. 4 1 left of N., or N.W. 4 W.
The following table is taken from the monthly examination-papers at the Royal Naval College, Portsmouth, and will serve as a specimen of the tables which ought to be made for all ships:
| Direction of Ship's Head | Deviation of Compass | |--------------------------|---------------------| | N. | 2° 45' E. | | N. by E. | 4 57 | | N.N.E. | 7 30 | | N.E. by N. | 9 0 | | N.E. | 10 0 | | N.E. by E. | 10 55 | | E.N.E. | 10 40 | | E. by N. | 9 55 | | E. | 8 50 | | E. by S. | 7 15 | | E.S.E. | 6 35 | | S.E. by E. | 3 40 | | S.E. | 1 50 | | S.E. by S. | 0 20 | | S.S.E. | 0 56 | | S. by E. | 2 20 |
| Direction of Ship's Head | Deviation of Compass | |--------------------------|---------------------| | S. | 3° 0' W. | | S. by W. | 4 20 | | S.S.W. | 5 0 | | S.W. by S. | 6 7 | | S.W. | 7 9 | | S.W. by W. | 7 27 | | W.S.W. | 7 50 | | W. by S. | 8 20 | | W. | 8 50 | | W. by N. | 8 10 | | W.N.W. | 6 50 | | N.W. by W. | 5 40 | | N.W. | 4 50 | | N.W. by N. | 3 20 | | N.N.W. | 1 40 | | N. by W. | 1 10 |
3. Leeway.—The effect of the action of the wind upon the sails and hull of a ship is sometimes to produce a motion of the ship in a direction at right angles to that of the head or apparent course, as well as in this latter direction. The true course, therefore, is not that given by the compass, but that which is due to the composition of the two velocities of the ship,—viz., that in the direction of its head, and that at right angles to this direction. To obtain the true course from the compass course, therefore, we must add or subtract the angle of leeway, which is the angle between the compass course and the true course.
If the wind be on the right of a person on board ship who is looking towards the head, the real course is then evidently to the left of the direction of the ship's head,—i.e., of the apparent course. If the wind be on his left hand, the true course is to the right. In the former case the ship is said to be on the starboard tack, and in the latter on the port tack. Whence is derived the rule for obtaining the true course from the compass course.
Rule.—If the ship is on the starboard tack, allow leeway to the left; if on the port tack, allow leeway to the right. Conversely, to obtain the compass course from the true course on the starboard tack, allow leeway to the right; if on the port tack, allow it to the left.
There are many circumstances which prevent laying down accurate rules for the allowance of leeway. The construction of different vessels, their trim with regard to the nature and quantity of cargo, the position and area of sail set, the velocity of the ship and the swell of the sea, are all susceptible of great variation, and very much affect the leeway.
The following rules are usually given for the purpose:
1. When a ship is close-hauled under all sail, the water smooth, and with a light breeze, allow no leeway. 2. When the top-gallant sails are handed, allow one point. 3. Under close-reefed topsails allow two points. 4. When one topsail is handed, allow two points and a half. 5. When both topsails are handed, allow three points. 6. When the fore-course is handed, allow four points. 7. When under mainsail only, allow five points. 8. Under balanced mizen, allow six points. 9. Under bare poles, allow seven points.
These rules, however, are not much to be depended upon. A very good method of estimating the leeway is to observe the bearing of the ship's wake as frequently as may be judged necessary, which may be conveniently enough done by drawing a small semicircle on the taffrail, with its diameter at right angles to the ship's length, and dividing its circumference into points and quarters. The angle contained between the semi-diameter which points right aft, Plane sailing and that which points in the direction of the wake, is the leeway. But the best and most rational way of finding the leeway is to have a compass or semicircle on the taffrail, as before described, with a low crutch or swivel in its centre; after heaving the log, the line may be slipped into the crutch just before it is drawn in, and the angle it makes on the limb with the line drawn right aft, will show the leeway very accurately.
Ex. 1.—A ship's apparent course is S.S.W. 4 W., leeway 2½ points, the wind being S.E. ¼ E.; required the true course. In this case the wind is on the left of the vessel, or it is on the port tack, and leeway must be allowed to the right.
| Apparent course | 3 pts. | 3 right of S. | |-----------------|-------|---------------| | Leeway | 2 | 2 right of S. | | True course | 5 | 1 right of S., or S.W. by W. ¼ W. |
Ex. 2.—The apparent course is N.N.W., leeway 1½ points, and wind E.N.E. Here the vessel is on the starboard tack.
| Apparent course | 2 pts. | 0 left of N. | |-----------------|--------|--------------| | Leeway | 1 | 2 left of N. | | True course | 3 | 2 left of N., or N.W. by N. ¼ W. |
Ex. 3.—The true course of a ship is S.E. ¼ S., leeway 2½ points, and wind N.N.E.; required the compass course. Here the ship is on the port tack.
| True course | 3 pts. | 2 left of S. | |-----------------|--------|--------------| | Leeway | 2 | 2 right of S. | | Compass course | 1 | 0 left of S., or S. by E. |
**CHAP. II.—ON PLANE SAILING.**
Plane sailing is the art of navigating a ship upon principles deduced from the notion of the earth's being an extended plane. On this supposition, the meridians are considered parallel lines. The parallels of latitude are at right angles to the meridians; the lengths of the degrees on the meridians, equator, and parallels of latitude are everywhere equal; and the degrees of longitude are reckoned on the parallels of latitude as well as on the equator; and consequently the departure and difference of longitude are equal. In fact, in the right-angled triangle ABC (fig. 7), where AB is the true difference of latitude, BAC the course, AC the distance, and BC the departure, which is assumed equal to the difference of longitude; all the problems in sailing are solved by the relations of the sides and angles of the single right-angled triangle ABC. Except, however, for a small portion of the earth's surface near the equator, the departure cannot be assumed equal to the difference of longitude without very considerable error; and the longitude in cannot be at all depended on when found by this method. If, however, the departure be an element, this method is correct.
In fig. 7, A is the place from which the ship sails; AB the meridian, and equal to the true difference of latitude; BC perpendicular to the meridian, and equal to the departure. It is always possible and easy to construct a right-angled triangle when two parts, of which one is a side, beside the right angle, are given. Consequently problems in navigation may always be solved by construction, with the aid of the rule and compasses. In making constructions for this purpose, it is only necessary to attend to the following convention:—Let the upper part of the paper or plan on which the drawing is to be made represent the north; then the lower part will be south, the right-hand side east, and the left-hand side west. This convention we have already tacitly assumed in treating of the corrections of the courses.
**To make a Construction.**
A north and south line is to be drawn, to represent the meridian of the place from which the ship sailed; and the upper or lower end of this line is to be marked as the position of the place, according as the course is southerly or northerly. From this point as centre, with the chord of 60° (on the rule), an arc of a circle is to be described from the meridian, towards the right or left, according as the course is easterly or westerly; and the course, taken from the line of chords if given in degrees, but from the line of rhumbs if expressed in points of the compass, is to be laid on this arc, beginning from the meridian. A straight line drawn through this point and the point sailed from is the direction of the distance, which, if given, must be laid down on this line, beginning at the point sailed from. A straight line is to be drawn from the extremity of the distance perpendicular to the meridian; and hence the true difference of latitude and the departure will be found.
If the true difference of latitude be given, it is to be laid down on the meridian, beginning at the point from which the ship sailed; and a straight line drawn through the extremity of the difference of latitude, perpendicular to the meridian, to meet the distance produced, will limit the figure, and enable us to find the parts required.
If the departure be given, it is to be laid off on a parallel, and the line drawn through its extremity will limit the distance.
If the distance and true difference of latitude be given, through the extremity of the true difference of latitude draw a straight line perpendicular to the meridian; extend a pair of compasses to the given distance, place one of its points at the place from which the ship sailed, and let the other point be in the perpendicular line first drawn; join this point with the point from, and the triangle is determined, and the course and departure found.
If the departure and distance are given, with the point from as centre and the distance as radius, describe an arc of a circle. Let the departure be laid off on a parallel, so that one point is in the meridian and the other in the circle just described. Join this latter point with the point from, and the triangle is formed. The general mode of solving problems in plane sailing has already been given in chap. I, sect. 2. The following examples will show how the formulae are to be applied:
**Obs.** It is to be distinctly understood that the above method cannot be applied to obtain the difference of longitude without very sensible error.
**Ex. 1.**—A ship from St. Helena, in Lat. 15° 53' S., sailed S.W. by S. 158 miles; required the latitude in, and departure.
*By Construction.*—Draw the meridian AB (fig. 8), and with the chord of 60° describe the arc, and make it equal to the rhumb of three points, and through it draw AC equal to 158 miles; from C draw CB perpendicular to AB; then AB applied to the scale from which AC was taken will be found to measure 131½', and BC 87½'.
*By Calculation.*—To find the true difference of latitude.
| Log cosine of course | 3 pts. | = 9·01885 | |---------------------|--------|------------| | Log distance | 158 m. | = 2·19866 | | Log true diff. lat. | | = 2·11851 |
Hence, true diff. lat. = 131½'.
To find departure.
| Log sin course | 3 pts. | = 9·74474 | |---------------------|--------|------------| | Log distance | 158 m. | = 2·19866 | | Log departure | 87½ | = 1·94340 | By Inspection.—In the traverse table, the difference of latitude answering to the course 3 points, and distance 158 miles, in a distance column, is 131°4', and departure 87°8'.
By Gunter's Scale.—The extent from 8 points to 5 points, the complement of the course on the line of sine rhumbs (marked S.R.) will reach from the distance 158 to 131°4', the difference of latitude on the line of numbers; and the extent from 8 points to 3 points on sine rhumbs will reach from 158 to 87°8', the departure on numbers.
Latitude St Helena ........................................... 16°56' N. Diff. latitude .................................................. 2°11' S. Latitude in ...................................................... 18°6' S.
Ex. 2.—A ship from St George's, in Lat. 38°45' N., sailed S.E.18°, and the latitude by observation was 35°7' N.; required the distance run, and departure.
Latitude St George's ........................................... 38°45' N. Latitude in ...................................................... 35°7' N. Diff. latitude .................................................. 3°38'=218 miles S.
By Construction.—Draw the portion of the meridian AB (fig.9) equal to 218 miles; from the centre A, with the chord of 60°, describe the arc mn, which make equal to the rhumb of 31 points; through An draw the line AC, and from B draw BC perpendicular to AB, and let it be produced till it meets AC in C. Then the distance AC being applied to the scale will measure 282 miles, and the departure BC 179 miles.
By Calculation.—To find the distance.
Log. sec of the course ........................................ 31 pts. = 10-11181 Log. true diff. latitude ....................................... 218 m. = 2-33846
Log. distance .................................................. 2-45527 Or distance .................................................... = 282
To find departure.
Log. tan of the course ........................................ 34 pts. = 9-91417 Log. true diff. latitude ....................................... 218 m. = 9-33846
Log. departure ................................................ 2-25263 Or departure .................................................... = 178°9'
By Inspection.—Find the given difference of latitude 218 miles in latitude column, under the course of 31 points; opposite to which, in distance column, is 282 miles; in departure column 178°9'; the distance and departure required.
By Gunter's Scale.—Extend the compass from 44 points, the complement of the course, to 8 points on sine rhumbs; that extent will reach from the difference of latitude 218 miles to the distance 282 miles on numbers; and the extent from 4 points to the course 34 points on the line of tangent rhumbs (marked T.R.) will reach from 218 miles to 178°9', the departure on numbers.
Ex. 3.—A ship from Palma, in Lat. 28°37' N., sailed N.W. by W., and made 192 miles of departure; required the distance run, and the latitude come to.
By Construction.—Make the departure BC (fig.10) equal to 192 miles, draw BA perpendicular to BC, and from the centre C, with the chord of 60°, describe the arc mn, which make equal to the rhumb of 3 points, the complement of the course; draw a line through Cn, which produce till it meet BA in A. Then the distance AC being measured, will be equal to 231 m., and the difference of latitude AB will be 128°3 miles.
By Calculation.—To find the distance.
Log. cos of course ............................................. 5 pts. = 10-08015 Log. departure ................................................ 192 m. = 2-28330
Log. distance .................................................. 230-9 = 2-36345
To find the true difference of latitude.
Log. cot of course ............................................. 5 pts. = 9-84269 Log. departure ................................................ 192 m. = 2-28330
Log. true diff. latitude ....................................... 128°3 = 2-10819
By Inspection.—Find the departure 192 miles in its proper column above the given course 5 points; and opposite thereto is the distance 231 miles, and difference of latitude 128°3, in their respective columns.
By Gunter's Scale.—The extent from 5 points to 8 points on the line of sine rhumbs, being laid from the departure 192 on numbers, will reach to the distance 231 on the same line; and the extent from 5 points to 4 points on the line of tangent rhumbs will reach from the departure 192, to the difference of latitude 128°3 on numbers.
Latitude of Palma ............................................. 28°37' N. Diff. latitude .................................................... 2°8' N. Latitude in ....................................................... 30°45' N.
Ex. 4.—A ship from a place in Lat. 43°13' N., sails between the north and east 285 miles, and is then by observation found to be in Lat. 46°31' N.; required the course and departure.
Latitude sailed from ........................................ 43°13' N. Latitude by observation .................................... 46°31' N. Diff. of latitude ................................................ 3°18'=198 miles.
By Construction.—Draw the portion of the meridian AB (fig.11) equal to 198 miles; from B draw BC perpendicular to AB; then take the distance 285 miles from the scale, and with one foot of the compass in A describe an arc intersecting BC in C, and join AC. With the chord of 60° describe the arc mn, the portion of which contained between the distance and difference of latitude, applied to the line of chords, will measure 46°, the course; and the departure BC being measured on the line of equal parts, will be found equal to 205 miles.
By Calculation.—To find the course.
Log. true diff. latitude ....................................... (198) + 10 = 12-29669 Log. distance .................................................. 285 = 2-45484
Log. cos course ............................................... 46° = 9-84176 Or course is ..................................................... N.46°E.
To find the departure.
Log. sin course ................................................ 46° = 9-65693 Log. distance .................................................. 285 = 2-45484
Log. departure ................................................ 205 = 2-31177
By Inspection.—Find the given distance in the table in its proper column; and if the difference of latitude answering thereto is the same as that given, namely, 198, then the departure will be found in its proper column; and the course at the top or bottom of the page, according as the difference of latitude is found in a column marked lat. at top or bottom. If the difference of latitude thus found does not agree with that given, turn over till the nearest thereto is found to answer to the given distance. This is in the page marked 46 degrees at the bottom, which is the course, and the corresponding departure is 205 miles.
By Gunter's Scale.—The extent from the distance 285 to the difference of latitude 198 on numbers, will reach from 90° to 44°, the complement of the course on sines; and the extent from 90° to the course 46° on the line of sines being laid from the distance 285, will reach to the departure 205 on the line of numbers.
Ex. 5.—A ship from Fort-Royal, in the island of Grenada, in Lat. 12°9' N., sailed 260 miles between the south and west, and made 190 miles of departure; required the course and latitude come to.
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1 For the method of resolving the various problems in navigation by the Sliding Gunter, the reader is referred to Dr Mackay's Treatise on the Description and Use of that instrument. By Construction.—Draw BC perpendicular to AB, and equal to the given departure 190 miles; then from the centre C, with the distance 260 miles, sweep an arc intersecting AB in A, and join AC. Now describe an arc from the centre A with the chord of 60°, and the portion mn of this arc, contained between the distance and difference of latitude, measured on the line of chords, will be 47°, the course; and the difference of latitude AB, applied to the scale of equal parts, measures 177½ miles.
By Calculation.—To find the course.
Log. departure ........................................... 190 + 10 = 12-27875 Log. distance .................................................. 200 = -2-41497
L sin course .................................................... 46° 57' = 9-86378 Or course is ..................................................... S. 46° 57' W.
To find the true difference of latitude.
L cos course .................................................... 46° 57' = 9-83419 Log. distance .................................................. 200 = -2-41497
Log. true diff. latitude ....................................... 177-3 = 2-24916
By Inspection.—Seek in the traverse table until the nearest to the given departure is found in the same line with the given distance 200. This is found to be in the page marked 47° at the bottom, which is the course; and the corresponding difference of latitude is 177-3.
By Gunter's Scale.—The extent from the compass, from the distance 260 to the departure 190 on the line of numbers, will reach from 90° to 47°, the course on the line of sines; and the extent from 90° to 43°, the complement of the course on sines, will reach from the distance 260 to the difference of latitude 177½ on the line of numbers.
Latitude Fort-Royal ...................................... 12° 9' N. Difference of latitude ....................................... 177 = 2-678 Latitude in ..................................................... 9° 12 N.
Ex. 6.—A ship from a port in Lat. 7° 56' S., sailed between the south and east till her departure was 132 miles, and was then, by observation, found to be in Lat. 12° 3' S.; required the course and distance.
Latitude sailed from ..................................... 7° 56' S. Latitude in, by observation 12° 3' S. Difference of latitude ....................................... 4° 7 = 247
By Construction.—Draw the portion of the meridian AB equal to the difference of latitude 247 miles; from B draw BC perpendicular to AB, and equal to the given departure 132 miles, and join AC; then with the chord of 60° describe an arc from the centre A; and the portion mn of this arc, being applied to the line of chords, will measure about 28°; and the distance AC, measured on the line of equal parts, will be 280 miles.
By Calculation.—To find the course.
Log. departure ................................................ 132 + 10 = 12-19057 Log. true diff. latitude ....................................... 247 = 2-39270
L tan course .................................................... 28° 7' = 9-72787 Or course is ..................................................... S. 28° 7' E.
To find the distance.
L sec course .................................................... 28° 7' = 10-05454 Log. true diff. latitude ....................................... 247 = 2-39270
Log. distance .................................................... 280 = 2-44724
By Inspection.—Seek in the table till the given difference of latitude and departure, or the nearest thereto, are found together in their respective columns, which will be under 28°, the required course; and the distance answering thereto is 280 miles.
By Gunter's Scale.—The extent from the given difference of latitude 247 to the departure 132 on the line of numbers, will reach from 45° to 28°, the course on the line of tangents; and the extent from 62°, the complement of the course, to 90° on the sines, will reach from the difference of latitude 247 to the distance 280 on numbers.
The six problems whose solutions are illustrated above are all that occur in the solution of the right-angled triangle Mercator's whose sides represent the true difference of latitude, departure, and distance, and one of whose angles is the course.
Chap. III.—On Mercator's Sailing.
We have already explained the principle of Mercator's Chart, and have shown that in every problem of navigation there is a second right-angled triangle similar to that whose solutions formed the subject of investigation in chapter ii.; and the sides of which, corresponding to the true difference of latitude and departure, are the meridional difference of latitude and difference of longitude.
We have already given a rule for finding the meridional parts for any given latitude, on the supposition that the earth is a perfect sphere. If the earth's oblateness be taken into account, and the compression, i.e., the ratio of the difference of the equatorial and polar semi-diameters to the equatorial semi-diameter, be taken as \( \frac{1}{35} \), which it is very nearly, the meridional parts for latitude \( l \) will be given by the formula—
\[ \text{Mer. parts} = 7915705 \log_{10} \tan \left( \frac{45° + \frac{l}{2}}{2} \right) - 22-88 \sin l - 0-0508 \sin 3l - &c. \]
Let AD (fig. 14) be the meridian, BAC the course, AB the true difference of latitude, AD the meridional difference of latitude; then
DE is the diff. long., and
ED = DA × tan BAC; or,
diff. long. = mer. diff. lat. × tan course;
or log. diff. long. = log. mer. diff. lat. + L tan course = 10.
Also, by similar triangles, ADE and ABC, we have
\[ \frac{DE}{BC} : \frac{DA}{BA} = DE = BC \times DA \]
\[ \therefore \text{diff. long.} = \text{dep.} \times \text{mer. diff. lat.} \]
true diff. lat.; or,
log. diff. long. = log. dep. + log. mer. diff. lat. - log. true diff. lat.
Whence, from the departure and true and meridional differences of latitude, the difference of longitude may be found.
The following examples will illustrate the mode of using these formulæ:
Ex. 1. A ship sails from Cape Finisterre, Lat. 42° 52' N., Long. 9° 17' W., to Port Praya, in the island of Santiago, Lat. 14° 54' N., and Long. 23° 29' W.; required the course and distance.
Lat. from ..................................................... 42° 52' N. Lat. is .......................................................... 14° 54' N. Diff. of lat. ..................................................... 27° 58' S. Mer. parts ...................................................... 2852 Long. from ..................................................... 9° 17' W. Long. is .......................................................... 23° 29' W. Diff. long. ....................................................... 14° 12' = 852 W.
By Construction.—Draw the straight line AD (fig. 15), to represent the meridian of Cape Finisterre, upon which lay off AB, AD, equal to 1678 and 1948, the true and meridional differences of latitude. From D draw DE perpendicular to AD, and equal to the difference of longitude 852; join AE, and draw BC parallel to DE; then the distance AC will measure 1831 miles, and the course BAC 23° 37'.
By Calculation.—To find the course.
Log. diff. long. ................................................ 852 + 10 = 12-93044 Log. mer. diff. lat. .......................................... 1948 = 3-28599
L tan course .................................................. 23° 37' = 5-64085 Or course is .................................................... S. 23° 27' W. To find the distance.
| L sec course | 23° 27' | = 10-03798 | |--------------|---------|-------------| | Log. true diff. lat. | 1678 miles | = 3-22479 | | Log. distance | 1831 | = 3-22277 |
Ex. 2.—A ship from Cape Henlopen in Virginia, in Lat. 38° 47' N., Long. 75° 4' W., sailed 267 miles N.E. by N.; required the ship's present place.
By Construction.—With the course, and distance sailed, construct the triangle ABC (fig. 16), and the difference of latitude AB being measured, is 222 miles; hence the latitude in is 42° 29' N., and the meridional difference of latitude 276. Make AD equal to 293, and draw DE perpendicular to AD, and meeting AC produced in E; then the difference of longitude DE being applied to the scale of equal parts, will measure 196; longitude in is therefore 71° 43' W.
By Calculation.—To find the true difference of latitude.
| L cos course | 3 points | = 2-91965 | |--------------|----------|------------| | Log. distance | 267 miles | = 2-42651 | | Log. true diff. latitude | 222 N. | = 2-34636 | | Lat. from | 38° 47' N. | Mer. parts...2528 | | True diff. lat. | 3° 42' N. | | | Lat. in | 42° 29' N. | Mer. parts...2821 |
To find the difference of longitude.
| L tan course | 3 points | = 9-62489 | |--------------|----------|------------| | Log. mer. diff. lat. | 293 miles | = 2-46887 | | Log. diff. long. | 195° 8' E. | = 2-29176 | | Long. from | 75° 4' W. | Diff. long. | 3° 16' E. | | Long. in | 71° 48' W. | |
Ex. 3.—A ship from Port Canoe in Nova Scotia, in Lat. 45° 20' N., Long. 60° 55' W., sailed S.E. 3, and, by observation, was found to be in Lat. 41° 14' N.; required the distance sailed, and longitude come to.
Lat. Port Canoe | 45° 20' N. | Mer. parts...3058 | Lat. in, by observation | 41° 14' N. | Mer. parts...2720 | Diff. lat. | 4° 6' | Mer. diff. lat. | 338 |
By Construction.—Make AB (fig. 17) equal to 246, and AD equal to 338; draw AE, making an angle with AD equal to 31 points, and draw BC, DE perpendicular to AD. Now AC being applied to the scale, will measure 332, and DE, 306.
By Calculation.—To find the distance.
| L sec course | 31 points | = 10-13021 | |--------------|-----------|-------------| | Log. true diff. lat. | 246 miles | = 2-39093 | | Log. distance | 332 | = 2-52114 |
To find the difference of longitude.
| L tan course | 31 points | = 9-95729 | |--------------|-----------|-------------| | Log. mer. diff. lat. | 338 miles | = 2-52892 | | Log. diff. long. | 306° 3 E. | = 2-48621 | | Long. Port Canoe from | 60° 55' W. | Diff. long. | 5° 6 E. | | Long. in | 55° 49' W. | |
Ex. 4.—A ship sailed from Saltee, in Lat. 33° 58' N., Long. 6° 20' W., the corrected course was N.W. by W. 4° W., and departure 420 miles; required the distance run, and the latitude and longitude in.
By Construction.—With the course and departure construct the triangle ABC (fig. 18); now AC and AB being measured, will be found to be equal to 476 and 224 respectively; hence the latitude in is 37° 42' N., and meridional difference of latitude 276. Make AD equal to 276, and draw DE perpendicular thereto, meeting the distance produced in E; then DE applied to the scale will be found to measure 516. The longitude in is therefore 14° 56' W.
By Calculation.—To find the distance.
| L cos course | 51 points | = 10-05457 | |--------------|-----------|-------------| | Log. dep. | 420 miles | = 2-62325 | | Log. dist. | 476:2 | = 2-67782 |
To find true difference of latitude.
| L cot course | 51 points | = 9-72796 | |--------------|-----------|-------------| | Log. dep. | 420 miles | = 2-62325 | | Log. diff. latitude | 224:5 | = 2-35121 | | Latitude Saltee from... | 33° 58' N. | Mer. parts...2169 | | Diff. latitude | 3° 44' N. | | | Latitude in | 37° 42' N. | Mer. parts...2445 | | Mer. diff. lat. | 276 | |
To find the difference of longitude.
| L tan course | 51 points | = 10-27204 | |--------------|-----------|-------------| | Log. mer. diff. latitude | 276 miles | = 2-44091 | | Log. diff. longitude | 516:3 | = 2-71295 |
Or—
| Log. dep. | 420 | = 2-62325 | | Log. mer. diff. latitude | 276 | = 2-44091 | | Log. true diff. latitude | | = 2-35121 | | Log. diff. longitude | 516:3 | = 2-71295 |
Ex. 5.—A ship from St Mary's, in Lat. 36° 57' N., Long. 25° 9' W., sailed on a direct course between the north and east 1162 miles, and was then, by observation, in Lat. 40° 57' N.; required the course steered, and longitude come to.
Latitude of St Mary's... | 36° 57' N. | Mer. parts...2339 | Latitude in | 49° 57' N. | Mer. parts...3470 | Diff. latitude | 13° 0' N. | Mer. diff. lat...1081 N. |
780 N.
By Construction.—Make AB equal to 780, and AD equal to 1081; draw BC, DE, perpendicular to AD; make AC equal to 1162, and through A and C draw ACE. Then the course or angle A being measured, will be found equal 47° 50', and the difference of longitude DE will be 1194.
By Calculation.—To find the course.
| Log. true diff. latitude | 780+10 | = 12-89209 | |--------------------------|---------|-------------| | Log. dist. | 1162 | = 3-06521 | | L cos course | 47° 50' | = 9-82688 |
To find difference of longitude.
| L tan course | 47° 50' | = 10-04302 | |--------------|---------|-------------| | Log mer. diff. latitude | 1081 miles | = 3-03383 | | Log. diff. longitude | 1194 | = 3-07085 | | Longitude from | 25° 9' W. | Diff. longitude | 19° 54' E. | | Longitude in | 5° 15' W. | |
Ex. 6.—From Aberdeen, in Lat. 57° 9' N., Long. 2° 8' W., a ship sailed between the south and east till her departure was 146 miles, and Lat. in 53° 32' N.; required the course and distance run, and longitude in. Ex. 7.—A ship from Naples, in Lat. 40° 51' N., Long. 14° 14' E., sailed 253 miles on a direct course between the south and west, and made 173 miles of westing; required the course made good, and the latitude and longitude in.
By Construction.—With the distance and departure make the triangle ABC as formerly. Now the course BAC being measured by means of a line of chords, will be found equal to 43° 21', and the difference of latitude applied to the scale of equal parts will measure 183; hence the latitude in is 37° 48' N., and meridional difference of latitude 237. Make AD equal to 237, and complete the figure, and the difference of longitude DE will measure 224; hence the longitude in is 10° 39' W.
By Calculation.—To find the course.
Log. dep. ........................................... 173 + 10 = 12 23805 Log. dist. ............................................ 252 = 2 40140 L sin course ........................................ 43° 21' = 9 86164
To find the true difference of latitude.
L cot course ....................................... 43° 21' = 9 86164 Log. dist. ............................................ 252 miles = 2 40140 Log. true diff. latitude ......................... 183 - 2 = 2 26304
Latitude from (Naples) .... 40° 51' N. Mer. parts... 2600 True diff. latitude ............. 3 3 8. Latitude in .................. 37 48 N. Mer. parts... 2453 Mer. diff. lat. 237
To find the difference of longitude.
L tan course ...................................... 43° 21' = 9 97497 Log. mer. diff. latitude ....................... 237 miles = 2 37475 Log. diff. longitude ......................... 223 - 7 = 2 34972
Longitude from .................. 14° 14' E. Diff. longitude .................. 3 44 W. Longitude in .................. 10 39 E.
Ex. 8.—A ship from Terceira, in Lat. 38° 45' N., Long. 27° 6' W., sailed on a direct course, which, when corrected, was N. 32° E., and is found, by observation, to be in Long. 18° 24' W.; required the latitude come to, and distance sailed.
Longitude of Terceira .......... 27° 6' W. Longitude in .................. 18 24 W. Diff. longitude ............... 8 42 = 522
By Construction.—Make the right-angled triangle ADE, having the angle A equal to the course 32°, and the side DE equal to the difference of longitude 522; then AD will measure 835, which, added to the meridional parts of the latitude left, will give those of the latitude come to, 48° 40'; hence the difference of latitude is 601. Make AB equal thereto, to which let BC be drawn perpendicular; then AC applied to the scale will measure 708 miles.
By Calculation.—To find meridional difference of latitude.
L cot course ...................................... 32° 0' = 10 20421 Log. diff. longitude ....................... 522 miles = 2 71767 Log. mer. diff. latitude ..................... 835 2 N. = 2 92188
Latitude from Terceira .... 38° 45' N. Mer. parts... 2526 Mer. diff. lat. 835
Latitude in .................. 48 46 N. True diff. latitude ....... 10 1 N. = 601 miles N.
To find the distance.
L sec course ...................................... 32° 0' = 10 07158 Log. true diff. latitude ............... 601 miles = 2 77887 Log. dist. ........................................ 707 - 1 = 2 85045
CHAP. IV.—ON TRAVERSE SAILING, OR COMPOUND COURSES.
It is the first business of the navigator, when he is about to conduct a ship from one port to another, to calculate beforehand the course on which the vessel is to be steered, and the distance she must run on that course. If the sea is perfectly free from obstruction between the two ports, one course and one distance will suffice for this purpose. It very seldom happens, however, that the sea is free from obstruction; but rocks or shoals, islands or some part or a mainland, intervenes, and a change of course is thus rendered necessary. In this case, the course and distance of the vessel, supposing the navigation unobstructed, having been taken from the chart, the mariner will determine how many changes of course are necessary, and will proceed to calculate the several courses and distances which shall be equivalent to the one course and distance on which the vessel would sail if unobstructed. This calculation, it must be remarked, is very different from that of the course and distance actually made good on a given day, when, by reason of variation of winds and other causes, the course requires to be altered; although naturally the modes of making these calculations are similar. In the former case, however, the distances to be dealt with are very much greater, and the changes of course less frequent, than in the latter.
The investigations of this chapter are intended to guide the navigator in making his preliminary calculation; the mode of correcting the course and calculating the distance run in each day will form the subject of a subsequent investigation.
If a ship sail on two or more courses in a given time, the irregular track she describes is called a traverse; and to resolve a traverse is the method of reducing these several courses and distances run into a single course and distance.
Rule 1.—Make a table sufficiently large to contain the several courses, &c. Divide this table into six columns; the courses are to be put in the first, and the corresponding distances in the second column; the third and fourth columns are to contain the differences of latitude, and the two last the departures. The several courses and their corresponding distances being properly arranged in the table, find the true difference of latitude and departure answering to each in the traverse table, remembering that the true difference of latitude is to be put into a N. or S. column according as the course is in a northern or southern direction, and that the departure is to be put in E. or W. column according as the course is easterly or westerly. Add together these several quantities in each of the columns, and set the sum down at the bottom. The difference between the sums in the N. and S. columns will be the true difference of latitude made good, of the same name with the greater; and the difference between the sums of the E. and W. columns is the departure made good, of the same name with the greater sum.
Look in the traverse table for a true difference of latitude and departure agreeing as nearly as possible with those above; then the distance will be found on the same line, and the course at the top or bottom of the page, according as the true difference of latitude is greater or less than the departure, since in the former case the course is less than 45° or 4 points, and in the latter case greater.
Having found the latitude, find also the meridional difference of latitude; and to the course and meridional difference of latitude in a latitude column, the corresponding departure will be the difference of longitude, which, applied to the longitude from, will give the longitude in.
It is also easy to resolve a traverse by construction; and we now show how this may be done, although it is scarcely ever practised at sea.
Describe a circle with the chord of 60° as radius, and in it draw two diameters at right angles to each other, at whose extremities are to be marked the initials of the cardinal points, N. being uppermost.
Lay off each course on the circumference, reckoned from its proper meridian; and from the centre to each point draw lines, which are to be marked with the proper number of the course.
On the first radius lay off the first distance from the centre, and through its extremity, and parallel to the second radius, draw the second distance of its proper length; through the extremity of the second distance, and parallel to the third radius, draw the third distance of the proper length; and so on until all the distances are drawn.
A line drawn from the extremity of the last distance to the centre of the circle will represent the distance made good; and a line drawn from the same point perpendicular to the meridian, produced if necessary, will represent the departure; and the portion of the meridian intercepted between the centre and departure will be the difference of latitude made good.
To construct for the difference of longitude we must find by the table the meridional difference of latitude, and lay it off on the meridian, and then complete the triangle similar to that whose sides represent the true difference of latitude; distance and departure as usual.
Ex. 1.—A ship from Fayal, in Lat. 38° 32' N., and Long. 25° 36' W., sailed as follows:—E.S.E., 163 miles; S.W., 110 miles; S.E., 180 miles; and N. by E., 68 miles; required the latitude and longitude in, the course, and distance made good.
| Course | Dist. | Diff. of Lat. | Departure | |----------|-------|--------------|-----------| | | | N. | S. | E. | W. | | E.S.E. | 163 | ... | 62-4 | 150-6 | ... | | S.W. | 110 | ... | 69-8 | ... | 85-0 | | S.E. | 180 | ... | 144-5 | 167-2 | ... | | N. by E. | 68 | ... | 68-7 | ... | 13-3 | | | | 68-7 | 176-7 | 271-1 | ... | | | | 68-7 | 25-0 | ... | 85-0 | | S. 41° E.| 281 | 210-0 | 186-1 | ... | |
Latitude from 36° 32' N., Mer. parts 2509 True diff. latitude 3° 30' N. Latitude in 35° 2' N., Mer. parts 2247 Mer. diff. lat. 202
Now to course 41° E., and opposite 131, half the meridional difference of latitude in latitude column, stands 115 in a departure column, which doubled gives 230 for difference of longitude.
Longitude from 28° 36' W. Diff. longitude 3° 50' E. Longitude in 24° 46' W.
By Construction.—With chord of 60° describe the circle NESW (fig. 23), the centre of which represents the place the ship sailed from. Draw two diameters NS, EW, at right angles to each other, the one representing the meridian, and the other the parallel of latitude of the place sailed from. Take each course from the line of rhumbs, lay it off on the W. circumference from its proper meridian, and number it in order, 1, 2, 3, 4. Upon the first rhumb C lay off the first distance 163 miles from C to A; through it draw the second distance AB parallel to C2, and equal to 110 miles; through B draw BD equal to 180 miles, and parallel to C3; and draw DE parallel to C4, and equal to 68 miles. Now CE being joined, will represent the distance made good, which, applied to the scale, will measure 231 miles. The arc Sa, which represents the course, being measured on the line of chords, will be found equal to 41°. From E draw EF perpendicular to CS produced; then CF will be the difference of latitude, and FE the departure made good, which, applied to the scale, will be found to measure 210 and 186 miles respectively. On CE produced lay off to the scale CG equal to 262, the meridional difference of latitude; and through G draw GH parallel to FE, meeting CE produced in H. Then GH is the difference of longitude; and, when applied to the scale, will be found to measure 230 miles.
Although the above method is that usually employed at sea to find the difference of longitude, yet, as it has been already observed, it is not to be depended on, especially in high latitudes, long distances, and a considerable variation in the courses; in which case the following method becomes necessary:
Rule 2.—Complete the traverse table as before, to which annex five columns. Now, with the latitude from, and the several differences of latitude, find the successive latitudes, which are to be placed in the first of the annexed columns; in the second, the meridional parts corresponding to each latitude are to be put; and in the third, the meridional differences of latitude.
Then to each course, and corresponding meridional difference of latitude, find the difference of longitude by Ex. 4, chap. iii., which place in the fourth or fifth columns, according as the course is easterly or westerly; and the difference between the sums of these columns will be the difference of longitude made good upon the whole, of the same name with the greater.
Remarks.
1. When the course is north or south, there is no difference of longitude. 2. When the course is east or west, the difference of longitude cannot be found by Mercator's Sailing; in this case the following rule is to be used:
To the nearest degree to the given latitude taken as a course, find the distance answering to the departure in a latitude column; this distance will be the difference of longitude.
Ex. 2.—A ship from Lat. 78° 15' N., Long. 28° 14' E., sailed the following courses and distances, viz.:—W.N.W., 154 miles, S.W. 90, N.W. by W., 89, N. by E., 110, N.W. by N., 56, S. by E., 78. The latitude in is required, and the longitude, by both methods; the bearing and distance of Hecate's headland, in Lat. 79° 55' N., Long. 11° 55' E., is also required. **CHAP. V.—OF PARALLEL SAILING.**
When the course is 8 points or 90° from the meridian, —i.e., due E. or W.,—the true difference of latitude becomes = 0, and the rules we have investigated in chaps. iii. and iv. fail to give any result. In this case the ship sails on a parallel of latitude. We have already proved in chap. i., that, neglecting the earth's oblateness, the arc of a parallel, in any given latitude, intercepted between two meridians, is equal to the corresponding arc of the equator, in other words, the difference of longitude, multiplied by the cosine of the latitude.
Whence we derive these three formulae for parallel sailing:
\[ \begin{align*} \text{Distance} &= \text{diff. longitude} \times \cos \text{latitude}; \\ \cos \text{latitude} &= \frac{\text{distance}}{\text{diff. longitude}}; \\ \text{Diff. longitude} &= \text{distance} \times \sec \text{latitude}. \end{align*} \]
Problems in parallel sailing may be solved by construction; for it is evident that we have only to construct a right-angled triangle whose hypotenuse is the difference of longitude, one of the sides the distance, and the angle between this side and the hypotenuse the latitude. Also it is evident, that in a traverse table, if we consider the latitude a course, and the difference of longitude a distance, the distance will be a true difference of latitude.
Ex. 1.—Required the number of miles contained in a degree of longitude in latitude 55° 58'.
By Construction.—Draw the indefinite right line AB (fig. 24); make the angle BAC equal to the given latitude 55° 58', and AC equal to the number of miles contained in a degree of longitude at the equator, namely, 60; from C draw CB perpendicular to AB; and AB being measured on the line of equal parts, will be found equal to 33.5, the miles required.
By Calculation—
\[ \begin{align*} L \cos \text{lat} &= 55° 58' = 9.7476369 \\ \log \text{miles in a degree} &= 60 = 1.7781513 \\ \log \text{miles in a deg. in lat. } 55° 58' &= 33.5 = 1.5260873 \end{align*} \]
By Inspection.—To 56°, the nearest degree to the given latitude, and distance 60 miles, the corresponding difference of latitude is 33.6, which is the miles required.
By Gunter's Scale.—The extent from 90° to 34°, the complement of the given latitude on the line of sines, will reach from 60 to 33.6 on the line of numbers.
There are two lines on the other side of the scale, with respect to Gunter's line, adapted to this particular purpose, one of which is entitled chords, and contains the several degrees of latitude; the other, marked M. L., signifying miles of longitude, is the line of longitudes, and shows the number of miles in a degree of longitude in each parallel. The use of these lines is therefore obvious.
Ex. 2.—Required the compass course and distance from A to B.
Given lat. A = 17° 30' S. Long. A = 9° 12' E. lat. B = 17° 30' S. Long. B = 10° 42' E.
Variation 12° E., and deviation as in the table on p. 13.
The true course is due E.
Also, long. A = 10° 12' E. long. B = 10° 42' E.
Diff. long. = 1° 30' = 90 E.
Log. diff. long. = 90 = 1.954243
L cos lat. = 17° 30' = 9.979419
Log. dist. = 85.8 m. = 1.933662
To find compass course.
Pts. pts.
True course = 8° right of N. Variation = 1° left of N.
Deviation by table 9° 55' E., or 0° 3° left of N.
Compass course = 5° 2° r. of N., or N.E. by E. Ex. 3.—A ship sails from Treguler in France, Long. 3° 14' W., to Gaspey Bay, Long. 64° 27' W., the common Lat. being 48° 47' N.; required the distance run.
Longitude from ........................................... 3° 14' W. Longitude in .................................................. 64° 27' W.
L cos latitude .............................................. 48° 47' = 9-8188250 Log. diff. longitude ....................................... 3673 = 3-5650209
Log. distance run ......................................... 2420 = 3-3838459
Ex. 4.—A ship from Cape Finisterre, Lat. 42° 52' N., Long. 9° 17' W., sailed due W. 342 miles; required the longitude in.
By Construction.—Draw the straight line AB (fig. 25), equal to the given distance 342 miles, and make the angle BAO equal to 42° 52', the given latitude; from B draw BC perpendicular to AB, meeting AC in C; then AC applied to the scale will measure 466°, the difference of longitude required.
By Calculation—
L cosec lat. ................................................. 42° 52' = 10-13493 Log. distance ............................................... 342 = 2-53463
Log. diff. long. ............................................. 466° = 2-66896
Long. Cape Finisterre .................................... 9° 17' W. Diff. longitude ............................................. 7° 47' W.
Longitude in .................................................. 17° 4' W.
Ex. 5.—A ship sailed due E. 358 miles, and was found by observation to have differed her longitude 8° 42'; required the parallel of latitude.
By Construction.—Make the line AB (fig. 26) equal to the given distance; to which let BC be drawn perpendicular, with an extent equal to 52°, the difference of longitude; describe an arc from the centre A, cutting BC in C; then the angle BAC, being measured by means of the line of chords, will be found equal to 46°, the required latitude.
By Calculation—
Log. dist. .................................................. 358 + 10 = 12-55388 Log. diff. long. ............................................. 512 = 2-71767
L cos lat. ..................................................... 46° 42' = 9-83621
Ex. 6.—From two ports in Lat. 33° 58' N., distance 348 miles, two ships sail directly N. till they are in Lat. 48° 23' N.; required their distance.
By Construction.—Draw the lines CB, CE (fig. 27), making angles with CP equal to the complements of the given latitudes, namely, 56° 2' and 41° 37' respectively. Make BD equal to the given distance 348 miles, and perpendicular to CP. Now from the centre C, with the radius CB, describe an arc intersecting CE in E; then ED drawn from the point E, perpendicular to CP, will represent the distance required; which being applied to the scale, will measure 278° miles.
By Calculation, as under:—
Log. given distance ..................................... 348 miles = 2-54158 L cos lat. in .................................................. 48° 23' = 9-82226
L cos lat. from ............................................. 33° 58' = 12-36384
Log. distance required .................................. 278° 6 miles = 2-44510
Ex. 7.—Two ships, in Lat. 56° 0' N., distant 180 miles, sail due S.; and having come to the same parallel, are now 232 miles distant. The latitude of that parallel is required.
By Construction.—Make DB (fig. 28) equal to the first distance 180 miles, DM equal to the second 232, and the angle DBC equal to the given latitude 56°. From the centre C, with the radius CB, describe the arc BE; and through M draw ME parallel to CD, intersecting the arc BE in E. Join EC, and draw EF perpendicular to CD; then the angle FEO will be the latitude required; which, being measured, will be found equal to 43° 53'.
By Calculation, as under:—
L cos lat. from ............................................. 56° 0' = 9-74756 Log. distance on required parallel 232 miles = 2-35549
Log. distance on known parallel 180 = 2-25527
L cos latitude .............................................. 43° 53' = 9-85778
CHAP. VI.—OF MIDDLE-LATITUDE SAILING.
It has been already explained in chap. ii. that the departure is greater than the intercepted arc of the parallel of the higher latitude, and less than that of the parallel of the lower of two places between which a ship sails; but that there is an intermediate parallel, the arc of which is exactly equal to the departure. This parallel is supposed to pass through the middle point between the extreme latitudes; and hence the latitude of this point is called the middle latitude. The relations between course, distance, departure, and true difference of latitude are to be found as in chap. iii.; and the relation between the departure and difference of longitude is given by the above considerations, viz.—
Departure = diff. long. × cos mid. latitude. But departure = true diff. lat. × tan course.
Hence we get
True diff. lat. × tan course = diff. long. × cos mid. lat. (a.) also departure = distance × sin course.
Whence also
Distance × sin course = diff. long. × cos mid. lat. (b.)
If ABC (fig. 29) be the triangle for plane sailing, where AB is the true difference of latitude, AC the distance, BAC the course, and BC the departure; at C make BCD equal to the middle latitude, and produce CD to meet AB produced in D; then CD is evidently the true difference of longitude, and all the problems may be resolved and constructed by the two triangles which have a common side, viz., the departure BC.
Also problems in middle-latitude sailing may be solved by the traverse table; for the relations between middle latitude, difference of longitude, and departure, are the same as those between course, distance, and true difference of latitude, and may therefore be found at once by inspection from the table.
Ex. 1.—Required the compass course and distance from the Island of May, in Lat. 56° 12' N. and Long. 2° 37' W., to the Naze of Norway, in Lat. 67° 50' N., and Long. 7° 37' E.; variation 24 W.
| Latitude Isle of May | 56° 12' N. | |----------------------|-----------| | Latitude Naze of Norway | 67° 50' N. |
Difference of latitude .................................. 1° 38' = 9° 114 2 Middle latitude ........................................... 57° 1
Longitude Isle of May .................................. 2° 37' W. Longitude Naze of Norway ............................... 7° 37' E.
Difference of longitude .................................. 10° 4 = 604° E.
By Construction.—Draw the right line AD (fig. 30) to represent the meridian of the May; with the chord of 60° describe the arc m n, upon which lay off the chord of 32° 59', the complement of the middle latitude from m to n. From D through n draw the line DC equal to 604°, the difference of longitude; and from C draw CB perpendicular to AD; make BA equal to 98°, the difference of the latitudes, and join AC; which, applied to the scale, will measure 343 miles, the distance sought; and the angle A being measured by means of the line of chords, will be found equal to 73° 24', the required course. By Calculation.—To find the course.
L cos mid. latitude $57^\circ 1' = 973591$ Log. diff. longitude $604$ miles $= 278104$ Log. true diff. latitude $98 = 199123$ L tan course $73^\circ 24' = 1032572$ Or course $N. 73^\circ 24' E.$
To find the distance.
L sec course $73^\circ 24' = 1054411$ Log. diff. latitude $98$ miles $= 199123$ Log. distance $343 = 253334$
The true course is N. $73^\circ 24' E.$ or E.N.E.$E.$ nearly.
True course $6^\circ 3'$ right of N. Variation $2^\circ 2'$ right of N. Or $9^\circ 1'$ right of N., or E.S.E.$E.$ Deviation $0^\circ 2'$ left of S. Compass course $7^\circ 1'$ or E.$S.$
Ex. 2.—A ship from Brest, in Lat. $48^\circ 23' N.$, and Long. $4^\circ 30' W.$, sailed S.W.$W.$, $238$ miles; required the latitude and longitude in.
By Construction.—With the course and distance construct the triangle ABC (fig. 31), and the difference of latitude AB being measured will be found equal to $142$ miles; hence the latitude in is $46^\circ 1' N.$, and the middle latitude $47^\circ 12'$. Now make the angle DCB equal to $47^\circ 12'$; and DC being measured will be $231$, the difference of longitude; hence the longitude in is $5^\circ 11' W.$
By Calculation.—To find the difference of latitude.
L cos course $44$ pts. $= 977503$ Log. distance $238$ miles $= 237558$ Log. true diff. latitude $1418 = 215161$ Lat. Brest $48^\circ 23' N.$ Diff. latitude $2^\circ 22' S.$ Half $1^\circ 11' S.$ Latitude in $46^\circ 1' N.$ Mid. lat. $47^\circ 12' N.$
To find the difference of longitude.
Log. distance $238 = 237558$ L sin course $44$ pts. $= 950483$ Log. cos mid. latitude $47^\circ 12' = 983215$ Log. diff. longitude $2813 = 244926$ Long. Brest $4^\circ 30' W.$ Diff. longitude $4^\circ 41' W.$ Longitude in $9^\circ 11' W.$
Ex. 3.—A ship from St. Antonio, in Lat. $17^\circ 0' N.$, and Long. $24^\circ 25' W.$, sailed N.W.$N.$, till, by observation, her latitude was found to be $28^\circ 34' N.$; required the distance sailed, and longitude come to.
Latitude St Antonio $17^\circ 0' N.$ Latitude by observation $28^\circ 34' N.$ Diff. of latitude $11^\circ 34' S.$ Middle latitude $22^\circ 47' N.$
By Construction.—Construct the triangle ABC (fig. 32), with the given course and difference of latitude, and make the angle BCD equal to the middle latitude. Now the distance AC and difference of longitude DC being measured, will be found equal to $864$ and $558$ respectively.
By Calculation.—To find the distance.
L sec course $31$ pts. $= 1009517$ Log. diff. latitude $694$ miles $= 284136$ Log. distance $864 = 293553$
To find the difference of longitude.
L tan course $31$ pts. $= 987029$ Log. diff. latitude $694$ miles $= 284136$ L cos mid. latitude $22^\circ 47' = 986472$ Log. diff. longitude $5583 = 274684$ Long. St Antonio $24^\circ 25' W.$ Diff. longitude $9^\circ 18' W.$ Longitude in $33^\circ 43' W.$
Ex. 4.—A ship from Lat. $26^\circ 20' N.$, and Long. $45^\circ 30' W.$, sailed N.E.$N.$ till her departure is $216$ miles; required the distance run, and latitude and longitude come to.
By Construction.—With the course and departure construct the triangle ABC (fig. 33) and the distance and difference of latitude behind measured will be found equal to $340$ and $264$ respectively. Hence the latitude in is $30^\circ 53'$, and middle latitude $28^\circ 42'$. Now make the angle BCD equal to the middle latitude, and the difference of longitude DC applied to the scale will measure $246$.
By Calculation.—To find the distance.
L sec course $34$ pts. $= 1019764$ Log. departure $216$ miles $= 233445$ Log. distance $3405 = 253109$
To find the true difference of latitude.
L cot course $34$ pts. $= 1009583$ Log. departure $216$ miles $= 233445$ Log. true diff. latitude $2632 = 242028$ Latitude from $26^\circ 30' N.$ Diff. latitude $4^\circ 23' N.$ Half $2^\circ 12' N.$ Latitude in $30^\circ 53' N.$ Mid. lat. $28^\circ 42' N.$
To find the difference of longitude.
L sec mid. latitude $28^\circ 42' = 1005693$ Log. departure $216$ miles $= 233445$ Log. diff. longitude $2462 = 239138$ Longitude from $45^\circ 30' W.$ Diff. longitude $4^\circ 6' E.$ Longitude in $41^\circ 24' W.$
Ex. 5.—From Cape Sable, in Lat. $43^\circ 24' N.$, and Long. $65^\circ 39' W.$, a ship sailed $246$ miles on a direct course between the S. and E., and was then by observation in Lat. $40^\circ 48' N.$; required the course, and longitude in.
Latitude Cape Sable $43^\circ 24' N.$ Latitude by observation $40^\circ 48' N.$ Diff. of latitude $2^\circ 36' = 156 S.$ Sum $84^\circ 12' N.$ Middle latitude $42^\circ 6' N.$
By Construction.—Make AB (fig. 34) equal to $156$ miles, draw BC perpendicular to AB, and make AC equal to $246$ miles; draw CD, making with CB an angle of $42^\circ 6'$, the middle latitude. Now DC will be found to measure $256$, and the course or angle A will measure $59^\circ 39'$.
By Calculation.—To find the course.
Log. diff. latitude $156 + 10 = 1219312$ Log. dist. $246 = 239093$ L cos course $30^\circ 39' = 980219$
To find the difference of longitude.
Log. dist. $246 = 239093$ L sin course $50^\circ 39' = 988834$ L cos mid. latitude $42^\circ 6' = 987039$ Log. diff. longitude $2564 = 24088$ Ex. 6.—A ship from Cape St Vincent, in Lat. 37° 2' N., Long. 9° 2' W., sails between the S. and W.; the latitude in is 18° 16' N., and departure 838 miles; required the course and distance run, and longitude in.
Latitude Cape St Vincent 37° 2' N. Longitude in 65° 39' W.
Ex. 8.—A ship from Lat. 54° 58' N., Long. 1° 10' W., sailed between the N. and E. till, by observation, she was found to be in Long. 5° 26' E., and has made 220 miles of easting; required the latitude in, course, and distance run.
Longitude from 1° 10' W. Longitude in 5° 26' E.
By Construction.—Make BC (fig. 37) equal to the departure 220, and CD equal to the difference of longitude 396; then the middle latitude BCD being measured, will be found equal to 56° 15'; hence the latitude come to is 67° 34', and difference of latitude 158'. Now make AB equal to 158, and join AC, which, applied to the scale, will measure 271 miles. Also the course BAC, being measured on chords, will be found equal to 54°.
By Calculation.—To find the middle latitude.
Log. diff. of longitude 306 + 10 = 12:50760 Log. departure 220 = 2:34242 L sec mid. latitude 56° 15' = 10:25527
Double middle latitude 112° 30' N. Latitude from 54° 55' N.
Latitude in 57° 34' N. True diff. latitude 2° 38' = 158 miles N.
To find the course.
Log. departure 220 + 10 = 12:34242 Log. diff. latitude 158 = 2:19866 L tan course 54° 19' = 10:14376
To find distance.
L sec course 54° 19' = 10:23410 Log. diff. latitude 158 = 2:19866
Log. distance 270 - 9 = 2:43276
Ex. 7.—A ship from Bordeaux, in Lat. 44° 50' N., and Long. 6° 35' W., sailed between the N. and W.; 374 miles, and made 210 miles of westing; required the course, and the latitude and longitude in.
By Construction.—With the given distance and departure make the triangle ABC (fig. 38). Now the course, being measured on the line of chords, is about 34°, and the difference of latitude on the line of numbers is 309 miles; hence the latitude in is 49° 50' N., and middle latitude 47° 25'. Then make the angle BCD equal to 47° 25', and DC being measured, will be 310 miles, the difference of longitude.
By Calculation.—To find the course.
Log. departure 210 + 10 = 12:32222 Log. distance 374 = 2:57287 L sin course 34° 10' = 9:74935
To find the true difference of latitude.
L cos course 34° 10' = 9:91772 Log. distance 374 = 2:57287
Log. diff. latitude 309 - 4 = 2:49059
Latitude from 44° 50' N. True diff. latitude 5° 9' Half. 2° 35' N.
Latitude in 49° 59' N. Mid. lat. 47° 25' N.
To find the difference of longitude.
L sec mld. latitude 47° 25' = 10:16963 Log. departure 210 = 2:32222
Log. diff. longitude 310 - 3 = 2:49185
Ex. 9.—A ship from a port in N. Lat., sailed S.E. 18° 438 miles, and differed her Long. 7° 28'; required the latitudes from and in.
By Construction.—With the course and distance construct the triangle ABC (fig. 38), and make DC equal to 448, the given difference of longitude. Now the middle latitude BCD will be measured 45° 58', and the difference of latitude AB 324 miles; hence the latitude from is 51° 40', and latitude in 46° 16'.
By Calculation.—To find the true difference of latitude.
Log. course 34 pts. = 9:60797 Log. distance 438 miles = 2:64147
Log. true diff. latitude 324 - 5 = 2:61126
To find middle latitude.
Log. distance 438 ms. = 2:64147 L sin course 34 pts. = 9:82708
Log. diff. longitude 448 = 2:65128
L sec mld. latitude 43° 58' = 9:81727
Mid. latitude 45° 58' N. Half diff. latitude 2° 42' S.
Latitude from 51° 40' N. Latitude in 46° 16' N.
CHAP. VII.—OF OBLIQUE SAILING.
Oblique sailing is the application of oblique-angled plane triangles to the solution of problems at sea. This sailing will be found particularly useful in going along shore, and in surveying coasts and harbours.
Ex. 1.—At 11 A.M., the Giraffe Ness bore W.N.W., and at 2 P.M. it bore N.W. by N.; the course during the interval S. by W., five knots an hour; required the distance of the ship from the Ness at each station.
By Construction.—Describe the circle NESW (fig. 39), and draw the diameters NS, EW at right angles to each other. From the centre C, which represents the first station, draw the W.N.W. line CF; and from the same point draw CH, S. by W., and equal to 15 miles, the distance sailed. From H draw HF in a N.W. by N. direction, and the point F will represent the Giraffe Ness. Then the distances CF, HF will measure 19½ and 26¾ miles respectively.
By Calculation.—In the triangle FCH are given the distance CH 15 miles, the angle FCH equal to 9 points, the interval between the S. by W. and W.N.W. points, and the angle CHF equal to 4 points, being the supplement of the angle contained between the S. by W. and N.W. by N. points. Hence CTH is 3 points; to find the distances CF, HF.
To find the distance CF.
\[ \begin{align*} \text{Log. CH} & = 15 \text{ m.} = 1:17609 \\ \text{L sin CHF} & = 4 \text{ pts.} = 9:84948 \\ \text{L sin CFH} & = 3 \text{ pts.} = -9:74474 \\ \text{Log. CF} & = 19:07 \text{ m.} = 1:28083 \end{align*} \]
To find the distance FH.
\[ \begin{align*} \text{Log. CH} & = 15 \text{ m.} = 1:17609 \\ \text{L sin FCH} & = 9 \text{ pts.} = 9:99157 \\ \text{L sin CFH} & = 3 \text{ pts.} = -9:74474 \\ \text{Log. FH} & = 25:48 \text{ m.} = 1:42292 \end{align*} \]
Ex. 2.—Running up Channel E, by S. compass at the rate of 5 knots an hour. At 11 A.M., the Eddystone Lighthouse bore N. by E.½E., and the Start Point N.E. by E.½E.; and at 4 P.M., the Eddystone bore N.W. by N., and the Start N.½N.; required the distance and bearing of the Start from the Eddystone, the variation being 2½ points W.
By Construction.—Let the point C (fig. 40) represent the first station, from which draw the N. by E.½E. line CA, the N.E. by E.½E. line CB, and the E. by S. line CD, which make equal to 25 miles, the distance run in the elapsed time. Then from D draw the N.W. by N. line DA, intersecting CA in A, which represents the Eddystone; and from the same point draw the N.½E. line DB, cutting CB in B, which therefore represents the Start. Now the distance AB applied to the scale will measure 22½, and the bearing per compass BAF will measure 73°½.
By Calculation—
The angle ACD = ACE + ECD = NCE - NCA + ECD
\[ \begin{align*} \text{BCD} & = NCE - NCB + ECD = 8 - 14 + 1 \text{ pt.} = 7 \text{ pts.} \\ \text{ACB} & = ACD - BCD = 8 - 5 + 1 \text{ pt.} = 4 \text{ pts.} \\ \text{ADC} & = N'DC - N'DA = 7 - 3 \text{ pts.} = 4 \text{ pts.} \\ \text{N'DB} & = 2 \text{ pts.} \\ \text{and CBD} & = N'DC + N'DB = 9 \text{ pts.} \\ \text{Also, CAD} & = 16 \text{ pts.} - ACD - ADG = 16 - 7 + 4 \text{ pts.} = 13 \text{ pts.} \\ \text{and CBD} & = 16 - BCD - CDB = 16 - 3 - 7 = 6 \text{ pts.} \end{align*} \]
To find AC.
\[ \begin{align*} \text{Log. CD} & = 25 \text{ m.} = 1:39794 \\ \text{L sin ADC} & = 4 \text{ pts.} = 9:84948 \\ \text{L sin CAD} & = 41 \text{ pts.} = -9:88679 \\ \text{Log. AC} & = 23:86 \text{ m.} = 1:37763 \end{align*} \]
To find BC.
\[ \begin{align*} \text{Log. CD} & = 25 \text{ m.} = 1:39794 \\ \text{L sin BDC} & = 7 \frac{1}{2} \text{ pts.} = 9:99948 \\ \text{L sin CBD} & = 4 \text{ pts.} = -9:88818 \\ \text{Log. BC} & = 32:30 \text{ m.} = 1:50924 \end{align*} \]
To find BAC.
\[ \begin{align*} \text{BC - AC} & = 8:44 \\ \text{BC + AC} & = 66:16 \\ \text{Log. (BC - AC)} & = 8:44 \text{ m.} = 0:92634 \\ \text{L cot } \frac{1}{2} \text{ACB} & = 2 \text{ pts.} = 10:38278 \\ \text{Log. (BC + AC)} & = 56:16 = 1:74943 \\ \text{L tan } \frac{1}{2} (\text{BAC - ABC}) & = 19°56' = 1:55989 \\ \text{∴ BAC} & = 87°26' - 14°4' \\ \text{ABC} & = 47°34' \end{align*} \]
Hence BAF = BAC - CAF
\[ \begin{align*} \text{BAF} & = 87°26' - 14°4' \\ & = 73°22' - 73°1' \text{ nearly.} \end{align*} \]
To find AB, or the distance.
\[ \begin{align*} \text{Log. BC} & = 32:30 \text{ m.} = 1:50924 \\ \text{L sin ACB} & = 4 \text{ pts.} = 9:84948 \\ \text{L sin CAB} & = 87°26' = 9:99956 \\ \text{Log. AB} & = 22:9 \text{ m.} = 1:36016 \end{align*} \]
Many other examples might be given. These and all other cases which can occur in practice are to be resolved by plane trigonometry, from calculating the triangles which the data of the given case afford.
**Chap. VIII.—Of Windward Sailing.**
Windward sailing is when a ship by reason of a contrary wind is obliged to sail on different tacks in order to gain her intended port; and the object of this sailing is to find the proper course and distance to be run on each tack.
Ex.—The wind at N.W., a ship bound to a port 64 miles to the windward proposes to reach it on three boards,—two on the starboard and one on the larboard tack, and each within 5 points of the wind; required the course and distance of each tack.
By Construction.—Draw the N.W. line CA (fig. 41) equal to 64 miles; from C draw CH W. by S., and from A draw AD parallel thereto and in an opposite direction. Erect AC in E, and draw BED parallel to the N. by E. rhumb, meeting CB, AD in the points B and D. Then CB = AD applied to the scale will measure 36½ miles, and BD = 2CB = 72½ miles.
**Chap. IX.—Of Current Sailing.**
The computations in the preceding chapters have been performed upon the assumption that the water has no motion. This may no doubt answer tolerably well in those places where the ebbs and flowsings are regular, as then the effect of the tide will be nearly counterbalanced. But in places where there is a constant current or setting of the sea towards the same point, an allowance for the change of the ship's place arising therefrom must be made. And the method of resolving these problems in which the effect of a current or heave of the sea is taken into consideration is called current sailing. In a calm, it is evident a ship will be carried in the direction and with the velocity of the current. Hence if a ship sails in the direction of the current, her rate will be augmented by the rate of the current; but if sailing directly against it, the distance made good will be equal to the difference between the ship's rate as given by the log and that of the current. And the absolute motion of the ship will be ahead if her rate exceeds that of the current; but if less, the ship will make sternway. If the ship's course be oblique to the current, the distance made good in a given time will be represented by the third side of a triangle, whereof the distance given by the log, and the drift of the current in the same time, are the other sides; and the true course will be the angle contained between the meridian and the line actually described by the ship.
It is evident from the above observations that we may consider the direction of the current in the light of a separate course; and by multiplying the rate of the current per hour by the number of hours it has been running, and treating this as a distance, we may estimate the ship's real place by any of the rules for compound courses.
Ex. 1.—A ship sailed N.N.E. at the rate of 8 knots an hour during 18 hours, in a current setting N.W. by W., 2½ miles an hour; required the course and distance made good.
By Construction.—Draw the N.N.E. line CA (fig. 42) equal to 18 × 8 = 144 miles; and from A draw AB parallel to the N.W. by W. rhumb, and equal to 18 × 2½ = 45 miles; now BC being joined will be the distance, and NBC the course. The first of these will measure 159 miles, and the second 6° 23'.
By Calculation—
The angle CAB = 9 pts. CA = 144 m. AB = 45 m. CA + AB = 189 m. CA - AB = 99 m.
Log. (CA - AB) = 99 m. = 1-095635 L cot ½ CAB = 42 pts. = 9-914173
Log. CA + CB = 185 m. = 2-276462 L tan ½ (ABC - ACB) = 23° 15' = 9-633346 ½ (ABC + ACB) = 39 22' ∴ ACB = 16° 7' and ABC = 62° 37' NCA = 22° 30' ∴ NBC the course = 6° 23'
Log. AB = 45 m. = 1-653212 L sin CAB = 9 pts. = 9-991574
L sin ACB = 16° 7' = 9-443410
Log. BC = 159 m. = 2-201376
Or,
For first course we have course N.N.E., or 2 points. And distance = 144 True diff. latitude = 133° 0 N. Departure = 65° 1 E.
For second course we have course N.W. by W., or 5 points. And distance = 45 And from traverse table, True diff. latitude = 25 N. Departure = 37° 4 W.
For last course—
True diff. latitude = 158 N. Departure = 17° 7 W.
Log. departure = 17° 7 + 10 = 11-24797 Log. true diff. latitude = 158 = 2-19866 L tan course = N. 6° 23' E. = 9-04931 L sec course = 6° 23' - 10 = 0-0271 Log true diff. latitude = 158 m. = 2-19866 Log. distance = 159 = 2-20137
Ex. 2.—A ship from Lat. 35° 20' N. sailed 24 hours in a
By Construction.—Make CE (fig. 43) equal to 22 miles, the difference of latitude by dead reckoning, and EA = 44 miles, the departure, and join CA; make CD = 38 miles, the difference of latitude by observation. Draw the parallel of latitude DB, and from A draw the N.W. by N. line AB, intersecting DB in B, and AB will be the drift of the current in 24 hours; CB being joined, will be the distance made good, and the angle DCB the true course. Now AB and CB applied to the scale will measure 19-2 and 50-5 respectively, and the angle DCB will be 41°.
By Calculation—
ABF = 3 pts. BF = CD - CE = 16 miles.
To find AB.
Log. BF = 16 m. = 1-20412 L sec ABF = 3 pts. - 10 = 0-68915
Log. AB = 19-2 m. = 1-28427 Or drift of current = 19-2 miles.
To find AF.
Log. BF = 16 m. = 1-20412 L tan ABF = 3 pts. - 10 = 0-68915
Log. AF = 10-7 m. = 1-02901
Hence BD = AE - AF = 44 - 10-7 = 33-4.
To find the course.
Log. BD = 33-4 + 10 = 11-52244 Log. CD = 38 = 1-57978
L tan course = N. 41° 14' E. = 9-94266
To find the distance.
Log. sec course = 41° 14' E. - 10 = 0-12376 Log. CD = 38 = 1-57978
Log. distance = 50-5 m. = 7-0354
By Traverse Table.—Taking the current course first, true difference of latitude 16, and course N.W. by N., we find in the traverse table the corresponding distance 19-3, and departure 10-7.
Again, for second course, we have true difference of latitude 38, and departure 44 - 10-7 = 33-3 E.
| Points | Course | Distance | Diff. of Latitude | Departure | |--------|--------|----------|------------------|-----------| | 3 | N.W. by N. | 10-3 | 16 | 10-7 | | N. 41° E. | | 51 | 38 | 33-3 |
Whence the course and distance are found as above.
Or, from the traverse table to nearest degree and minute, we find in the columns of distance and angle, opposite to difference of latitude 38° 3', and departure 35° 5'—distance 51, and angle 41°.
CHAP. X.—OF THE DAY'S WORK AND SHIP'S JOURNAL.
The most usual application of the principles laid down in the preceding chapters, is to ascertain from the several courses and distances run by a ship in the interval between the noons of two successive days, the ship's place at the noon of the latter day,—i.e., its latitude and longitude; its latitude and longitude being given for the noon of the preceding day. This constitutes a day's work; and the ship's place deduced therefrom is called her place by account or dead reckoning. The day aboard ship, like the astronomical day, commences at noon; and the ship's position is always calculated at every noon. In the Royal Navy, the log is hove once in every hour; but in most trading-vessels only once in every two hours. A record of the knots, and tenths of knots, run every hour or every two hours, the course, the direction of the wind, the leeway, and everything which affects the ship's place, is kept in the journal, which, for this purpose, is usually divided into six or seven columns. The first column on the left hand contains the hours... Day's from noon to noon; the second and third, the knots and tenths of knots sailed every hour, or every two hours; the fourth contains the courses steered; the fifth, the direction of the wind; and the sixth, when there are seven columns, contains the leeway; and the last contains general remarks, including phenomena, variation, &c., &c.
The mode of forming a table showing the deviation of the compass for the several positions of the ship's head, has already been given.
The courses steered, as entered in the log-book, must be corrected for variation, deviation, and leeway. The setting and drift of current, and the heave of the sea, are to be marked down. These are to be corrected for variation only. In the day's work, it is usual to treat a current as an independent course and distance. If the ship does not sail from a place whose latitude and longitude are known (which rarely happens), the bearing of some known place is to be observed, and its distance found, which is usually done by estimation. The ship is then supposed to have taken her departure from this place, in a course exactly opposite to the observed bearing, and to have run the estimated distance on it. If there be any reason to suspect the correctness of the estimated distance, it will be easy to obtain the true distance as follows:—Let the bearing be observed of the place from which the departure is to be taken; and the ship having run a certain distance on a direct course, the bearing of the same place is again to be observed. We shall then have a triangle, all of whose angles are known from the observed bearings, and one of its sides, viz., the distance the ship has sailed. The other two sides, viz., the distance of the ship from the place of departure at each of the observations, can be immediately found, as in problem I on "Oblique Sailing." The distances for each course may be obtained by adding together the hourly distances. The courses being thus corrected, and the distances found, the latitude and longitude may be found by any of the methods explained in chap. iv. As the differences of latitude are not usually great, the traverse table may generally be made use of for finding the latitude in; and having found the middle latitude, the longitude may be obtained by the middle latitude method.
The following example will enable the reader to apply the directions we have just given:
Ex.—September 12, 1857, at noon, a point of land in Lat. 64° 29' S., and Long. 59° 40' E., bore by compass S.E., distant 15 miles (ship's head being E.), afterwards sailed as by the following log-account; find the latitude and longitude in, on September 13, at noon.
| H. | K. | P.O.S. | Course | Wind | Leeway | Remarks | |----|----|-------|-------|------|--------|---------| | 1 | 4 | 7 | W. by N. | N.E. | 2 | F.M. | | 2 | 5 | 8 | | | | | | 3 | 6 | 5 | | | | | | 4 | 7 | 7 | | | | | | 5 | 8 | 4 | | | | | | 6 | 9 | 6 | N.E.E. | N.W. | 1½ | (For deviation see the table, page 13.) | | 10 | 11 | 5 | | | | | | 12 | 13 | 6 | | | | |
| H. | K. | P.O.S. | Course | Wind | Leeway | Remarks | |----|----|-------|-------|------|--------|---------| | 1 | 4 | 7 | A.M. | | | | | 2 | 5 | 8 | | | | | | 3 | 6 | 5 | | | | | | 4 | 7 | 7 | | | | | | 5 | 8 | 4 | | | | | | 6 | 9 | 6 | S. by W. | W. | 2½ | During the last 7 hours a current set the ship N.W. at the rate of two knots an hour. |
Departure course, N.W. being the opposite to S.E. Compass course ........................................ 4 pts. 0 qrs. left of N. Variation .................................................. 1 Deviation ................................................... 0 True course ................................................ 4 Dist ............................................................ 15.
First course, W. by N. Compass course ........................................... 7 pts. 0 qrs. left of N. Variation .................................................. 1 Deviation ................................................... 0 Leeway (wind N.E. on starboard tack) ................. 2 Or true course ............................................ 4 Dist ............................................................ 46½
Second course, N.E. Compass course ........................................... 2 pts. 0 qrs. right of N. Variation .................................................. 1 Deviation ................................................... 0 Leeway (wind N.W. on port tack) ....................... 1 True course ................................................ 3 Dist ............................................................ 34½
Third course, S.E. Compass course ........................................... 4 pts. 0 qrs. left of S. Variation .................................................. 1 Deviation ................................................... 0 Leeway (wind E.N.E. on port tack) ..................... 2 True course ................................................ 3 Dist ............................................................ 31½
Fourth course, S. by W. Compass course ........................................... 1 pt. 0 qrs. right of S. Variation .................................................. 1 Deviation ................................................... 0 Leeway (wind W. starboard tack) ....................... 2 True course ................................................ 3 Dist ............................................................ 20½
Current N.W. Compass course ........................................... 4 pts. 0 qrs. left of N. Variation .................................................. 1 Deviation ................................................... 0 Dist ............................................................ 14.
Enter these in a table as under:
| Points | Course | Distance | DoE.Lat. | Departure | |--------|--------|----------|----------|-----------| | 4½ | N.W.3W.| 15 | 8½ | 12½ | | 4 | S.W.3W.| 46½ | 27½ | 36½ | | 3 | N.E.6N.| 34½ | 19½ | | | 3 | S.E.6S.| 31½ | 25½ | 17½ | | 3½ | S.E.by S.H.| 20½ | 16½ | 13½ | | 5½ | N.W.by W.4N.| 14 | 6½ | 12½ |
| Points | Course | Distance | DoE.Lat. | Departure | |--------|--------|----------|----------|-----------| | 44½ | | | 68½ | 49½ | | 44½ | | | 44½ | 49½ |
True diff. lat. S. 23½', Dep. W. 11½'
Lat. from ..... 64° 20' S. Lat. from ..... 64° 29' S. T.D.Lat. ..... 0 23 8' Half. .......... 0 12 8' Lat.in ..... 64 43 S. Mid.Lat. ....... 64 32 S. Log.departure ....... 11-3 = 1-05308 L sec mid.lat. ....... 64° 32' -10 = 0-36654 Log.diff.long. ....... 20-2 = 1-41962
Long.from ..... 59° 40' E. Diff.long. .......... 0 26 W. Long.in ..... 59 14 E.
In this example the true differences of latitude and departures are taken by inspection from the traverse table.
When a ship is bound for a distant port, the bearing and distance of the port must be found. This may be done by calculation or by a chart. If islands, capes, or headlands intervene, it will be necessary to find the several courses and distances between each successively. The true course between the places must be reduced to the compass course. In hard blowing weather, with a contrary wind and a high sea, it is impossible to gain any advantage by sailing. In such cases, therefore, the object is to avoid as much as possible being driven back. With this intention it is usual to lie to under no more sail than is sufficient to prevent the violent rolling to which the vessel would be otherwise subjected, to the endangering of her masts and straining her timbers, &c. When a ship is brought to, the tiller or wheel is put down over to the leeward, which brings her head round to the wind. The wind having then little power over the sails, the ship loses her way through the water; and the action of the water on the rudder ceasing, her head falls off from the wind, the sail which she has set fills, and gives her fresh way through the water, which acting on the rudder, brings her head again to the wind. Thus the ship has a kind of oscillating motion, coming up to the wind and falling off from it again alternately. The middle point between those upon which she comes up and falls off is taken for her apparent course; and the leeway, variation, and deviation are to be allowed from this to find the true course.
It is generally found that the latitude by account does not agree with that by observation. On considering the imperfections of the common log-line, and the uncertainty with regard to variation, an exact agreement of latitudes cannot be expected. When the difference of longitude is to be found by dead reckoning, and the latitudes by account and observation disagree, several writers on navigation have proposed to apply a conjectural correction to the departure or difference of longitude. Thus, if the course is near the meridian, the error is wholly attributed to the distance, and the departure is to be increased or diminished accordingly; if near the parallel, the course only is supposed to be erroneous; and if the course is towards the middle of the quadrant, the course and distance are both assumed to be in error. This last correction will, according to different authors, place the ship upon opposite sides of her meridian by account. As these corrections, therefore, are no better than guessing, they should be absolutely rejected.
If the latitudes do not agree, the navigator should examine his log-line and half-minute glass, and correct the distance accordingly. He is then to consider if the variation and leeway have been properly ascertained; if not, the courses are to be again corrected, and no other alteration whatever is to be made in them. He is next to observe if the ship's place has been affected by a current or heave of the sea, and to allow for them according to the best of his judgment. By applying these corrections, the latitudes will generally be found to agree tolerably well; and the longitude may be corrected in the same way.
It will be proper for the navigator to determine the longitude of the ship by observation as often as possible, and the reckoning is to be carried forward in the usual manner from the last good observation; yet it will perhaps be very satisfactory to keep a separate account of the longitude by dead reckoning. The modes of finding the latitude and longitude of a ship by observation, and the variation of the compass, will be given in the next book.
**Use of the Plane Chart.**
**Prob. I.**—To find the latitude and longitude of a place on the chart.
*Rule.*—Take the least distance of the given place from the nearest parallel of latitude; this distance applied to the graduated meridian from the extremity of the parallel will give the latitude of the place. In the same way the longitude is found by taking the least distance from the nearest meridian, and applying it to the graduated parallel.
Thus the distance between Bonavista and the parallel of 15° being laid from that parallel on the graduated meridian, will reach to 16° 5', the latitude required.
**Prob. II.**—To find the course and distance between two given places on the chart.
*Rule.*—Lay a ruler over the given places; if a parallel ruler be used, keeping the edge of one ruler passing through the places fixed, move the other until it passes through the centre of one of the compasses on the chart; the point of the compass through which this edge passes will show the course.
Or, generally, let a line on the edge of another ruler be placed so as to be parallel to the first ruler, and to pass through the centre of a compass; it will cut the circumference in a point which will determine the course.
The interval between the places being applied to the scale will give the distance.
Thus the course from Palmas to St Vincent will be found to be about S.S.W. 3° W., and the distance 134° or 795 miles.
**Prob. III.**—The course and distance sailed from a known place being given, to find the ship's place on the chart.
*Rule.*—Lay a ruler over the given place parallel to another ruler laid over one of the compasses, with one edge passing through the centre, and the other the point on the circumference which shows the course, and lay off on it the distance taken from the scale; it will give the point representing the ship's present place.
Thus, supposing a ship has sailed S.W. by W. 160 miles from Cape Palmas; then by proceeding as above, it will be found that she is in Lat. 2° 57' N.
The reader will have no difficulty in solving various other problems by means of this chart, being, in fact, only the construction of the various problems in plane sailing on this chart.
**Use of Mercator's Chart.**
The method of finding the latitude and longitude of a place, and the course or bearing between two given places, is the same as in the plane chart, which see.
**Prob. I.**—To find the distance between two given places on the chart.
**Case 1.**—When the given places are under the same meridian.
*Rule.*—The difference or sum of their latitudes, according as they are on the same or on opposite sides of the equator, will be the distance required.
**Case 2.**—When the given places are under the same parallel.
*Rule.*—If that parallel be the equator, the difference or sum of their longitudes, according as they are on the same or on opposite sides of the first meridian, is the distance; otherwise take the distance between the places, lay it off upwards and downwards from the given parallel, and the intercepted degrees will be the distance between the places.
Or take an equal extent of a few degrees on the meridian on each side of the parallel; and the number of extents and parts of an extent contained between the places, multiplied by the length of an extent, will give the required distance.
**Chap. XI.—OF SEA CHARTS.**
The charts usually employed in the practice of navigation are the *Plane* and *Mercator's charts*. The former of these is adapted to represent a portion of the earth's surface near the equator, where the change in the lengths of corresponding arcs of the parallel is very small; and the other for all portions of the earth's surface. (For a particular description of these, see the articles *Chart* and *Geography*.) We shall here only describe their use. CASE 3.—When the given places differ both in latitude and longitude.
Rule.—Find the difference of latitude between the given places, and take it from the equator or graduated parallel; then lay a ruler over the places, and move one point of the compass opened to the difference of latitude just found along the edge of the ruler till the other just touches a parallel; then the distance from the point of the compass on the ruler to the point of intersection of the ruler and the parallel, applied to the equator, will give the distance between the places in degrees and parts of a degree, which, multiplied by 60, will give it in miles.
Prob. II.—Given the latitude and longitude in ; to find the ship's place by the chart.
Rule.—Lay a ruler over the given latitude, and lay off the given longitude from the first meridian by the edge of the ruler, and the ship's present place will be obtained.
Prob. III.—Given the course sailed from the given place, and the latitude in ; to find the ship's present place on the chart.
Rule.—Lay a ruler over the place sailed from, in the direction of the given course; its intersection with the parallel of latitude in, will give the ship's present place.
Prob. IV.—Given the latitude and longitude of the place left, and the course and distance sailed; to find the ship's present place on the chart.
Rule.—Lay a ruler over the given place, in the direction of the given course, take the distance sailed from the equator, and put one point of the compass opened to this distance at the intersection of the ruler with any parallel, and the other point will reach to a certain place by the edge of the ruler. This point being kept fixed, draw in the other point of the compass until it just touch the above parallel when swept round; apply this extent to the equator, and it will give the difference of latitude. Hence the latitude in is known; and the intersection of the edge of the ruler with the parallel of this latitude will give the ship's present place.
The above problems sufficiently illustrate the use of Mercator's Chart. The reader will have no difficulty in solving other problems by means of it.
BOOK II.
CONTAINING THE METHODS OF FINDING THE LATITUDE AND LONGITUDE OF THE SHIP AT SEA, THE VARIATION OF THE COMPASS, AND TIME OF HIGH WATER.
CHAP. I.—DESCRIPTION AND USE OF INSTRUMENTS USED IN OBSERVATIONS.
SECT. I.—OF HADLEY'S SEXTANT AND QUADRANT.
The principal difference between these instruments is in the extent of the angle which can be observed by them; and in the more elaborate and careful workmanship of the latter of the two. Indeed the quadrant is only available for taking observations which determine the latitude. The distances of the moon from the sun or other heavenly body, which are frequently used for the determination of the longitude, can only be observed by the help of the sextant.
Allowing for these differences, the principle on which the quadrant and sextant are constructed is the same. In the Royal Navy sextants are almost exclusively in use, although quadrants are still employed for the observation of altitudes in many trading vessels. The sextant, therefore, will first be described, and afterwards those points in which the quadrant differs from the sextant will be explained.
The reader is supposed to be aware of the ordinary laws with regard to the propagation and reflection of light, viz., that in the same medium, light is propagated in straight lines, the smallest conceivable quantity of which that can be stopped or propagated alone is called a ray; and that when a ray of light is incident on a plane reflecting surface, it is bent or reflected after incidence in such manner, that the incident and reflected rays and the straight line perpendicular to the mirror at the point of incidence (called the normal to the surface) lie all in one plane; and that the incident and reflected rays make equal angles with the normal or the surface.
Let MO (fig. 44) be an arc of a circle, CO and CM two radii, and CI be a moveable radius carrying a plane mirror, silvered through its whole extent, firmly fixed to it; EFG another mirror, the lower part of which FG only is silvered, while the upper part EF is unsilvered, so that a ray reflected from the lower portion FG in direction FH, and a direct ray PFH through the unsilvered part EF, may be seen together by an eye at K. This mirror is fixed to the radius CM in such a manner that when the moveable radius occupies the position ACO, the two mirrors ACB and EFG are both perpendicular to the plane of the instrument, and parallel to one another.
Let now S and P be two distant objects whose angular distance is required to be found. Let the instrument be placed so that its plane passes through S and P, and that a ray from P, passing through the unsilvered glass EF, may be seen directly by an eye at K; and while in this position let the bar be moved round C, CI carrying the mirror with it until a ray from S, falling on ACB, is reflected in the direction CF, and again reflected by FG in the direction FH; so that to the eye at K the images of the two objects S and P are seen together, or coincide.
Produce SA to meet PFH in H; then SHP is the angle through which the ray SA has been deflected, and is also the angular distance between S and P. Let A'B' be the new position of the mirror AB; then ACA' is the angle through which the mirror has turned, and consequently also the angle through which CI has moved.
Now angle of deflection
\[ \text{SHP} = \text{SCF} - \text{CFH} \]
\[ = 180^\circ - 2 \text{FCB'} - (180^\circ - 2 \text{EFC}) \]
because by law of reflection,
\[ \text{SCA}' = \text{FCB'}; \quad \text{and therefore} \]
\[ \text{SCF} = 180^\circ - \text{SCA}' - \text{FCB'} \]
\[ = 180^\circ - 2 \text{FCB'}, \]
and EFC = GFH; and therefore
\[ \text{CFH} = 180^\circ - \text{EFC} - \text{GFH} \]
\[ = 180^\circ - 2 \text{EFC}; \]
\[ \therefore \text{SHP} = 2 \text{EFC} - 2 \text{FCB'}. \]
But EFC = FCB, because EFG is parallel to ACB;
or SHP = 2 FCB - 2 FCB' = 2 ACA'
= twice the angle through which CI has moved.
Hence if the arc OM be divided into degrees, and each degree marked as two degrees, the reading off of the arc OI will be the angle between the distant objects S and P. An instrument constructed on this principle, whose circular arc or limb is a sixth part of a circle, and therefore capable of measuring angles up to 120°, is called a sextant; if the limb contain only an eighth part of a circle, it is a quadrant, and can only measure angles up to 90°.
The Sextant.
![Diagram of Sextant]
(1.) PLM (fig. 45) is the frame of the sextant. (2.) AA the graduated arc or limb. (3.) N the index, carrying the vernier OQ. (4.) I the index-glass. (5.) F the horizon-glass. (6.) D the coloured or dark glasses between the index-glass and horizon-glass. (7.) E the coloured glasses behind the horizon-glass. (8.) K the tube or collar in which the telescope is inserted.
The frame of the sextant consists of an arc AA, firmly attached to the two rails LP, MP, which are bound together by braces, as shown in the figure, to prevent warping and liability to bend.
The index N is a flat bar of brass, and turns on the centre of the sextant; at the lower end of the index there is an oblong opening; to one side of this opening the vernier scale is attached to subdivide the divisions of the arc; at the end of the index there is a piece of brass which bends under the arc, carrying a spring to make the vernier scale lie close to the divisions. It is furnished with a finger-screw C, by which the index is fixed in any position to the limb of the instrument. There is also an adjusting-screw B attached to the index, capable of moving it with greater accuracy than the hand; this screw does not act until the index is fixed by the finger-screw C. Care must be taken not to force the adjusting-screw when it arrives at either extremity of its adjustment. When any considerable movement is required to be given to the index, the screw C at the back of the sextant must be set free; but where the index is brought nearly to the divisions required, this back screw should be tightened, and then the index gradually moved by the adjusting-screw.
Upon the index, and near its axis of motion, is fixed a plane speculum or mirror of glass I, quicksilvered. It is set in a brass frame, which is firmly fixed by a strong cock to the centre plate of the index, with its face perpendicular to the plane of the instrument. This mirror being fixed to the index, moves along with it, and has its direction changed by the motion thereof. As has already been observed, this glass is to receive the rays from the sun or other object, and reflect them upon the horizon-glass. It is furnished with screws at its back, the object of which is to replace it in a perpendicular position, if by any accident it has been deranged.
To the radius PL is attached a small speculum F, whose surface is parallel to the index-glass when zero on the index coincides with zero on the limb. The under part only of this speculum is silvered, the upper half being left transparent, and the back part of the frame cut away, that nothing may impede the sight through the unsilvered part of the glass. The edge of the foil of this glass is nearly parallel to the plane of the instrument, and ought to be very sharp, and without a flaw. It is set in a brass frame, which turns on axes and pivots which move in an exterior frame; the holes in which the pivots move may be tightened by four screws in the exterior frame. G is a screw by which the horizon-glass may be set perpendicular to the plane of the instrument. Should this screw become loose, or move too easy, it may be easily tightened by turning the capstan-headed screw H which is on one side of the socket through which the stem of the finger-screw passes; this screw G is in some instruments under the glass, in others behind it, and in others at the side.
There are four coloured glasses at D, tinged red and green, each of which is set on a separate frame that turns on a centre. They are used to defend the eye from the brightness of the solar image and the glare of the moon, and may be used separately or together as occasion may require. There are three more such glasses placed behind the horizon-glass at E, to weaken the rays of the sun or moon when viewed directly through the horizon-glass. The paler glass is sometimes used in observing altitudes at sea to take off the strong glare of the horizon.
The sextant is furnished with a plane tube K; and in order to render objects distinct, it has two telescopes—one a Galileo's telescope, representing the objects erect in their natural position; the longer one, an astronomical telescope, shews them inverted. It has a large field of view; and has parallel wires placed in the principal focus, where a true image of the object viewed by it is seen; thus rendering the position of the image more exact and more easy to be read off, and is that which should be used in taking observations at sea when great accuracy is required. A little use will soon accustom the observer to the inverted position, and to manage the instrument with ease. By a telescope the contact of the images is more perfectly distinguished; and by the place of the images in the field of view, it is easy to perceive whether the sextant is held in the proper position for observation. By sliding the tube that contains the eye-glasses in the inside of the other tube, the object is suited to different eyes, and made to appear perfectly distinct and well-defined.
The telescopes are to be screwed into a circular ring at K; this ring rests on two points against an exterior ring, and is held to it by two screws; by turning one of these screws, and tightening the other, the axis of the telescope may be set parallel to the plane of the sextant. The exterior ring is fixed on a triangular brass stem which slides in a socket, and, by means of a screw at the back of the sextant, may be raised or lowered so as to move the centre of the telescope to that part of the horizon-glass which shall be deemed most fit for observation. Tinged glasses are provided to screw on the eye-end of either of the telescopes or the plane tube.
The limb of the sextant is divided from right to left into 120 primary divisions, which are to be considered as degrees; the degree is subdivided in some cases into two equal parts, each of which is 30'; in others into three equal parts, each of which is 20'; and in others again into six equal parts, each of which is 10'. If the zero of the index stand exactly at one of the divisions of the limb, the Observation Instruments.
The vernier contains a space equal to nineteen divisions on the limb, and is divided into twenty equal parts; hence the vernier, the difference between a division on the vernier and a division on the limb is one-twentieth of a division of the limb, or 1', if the interval between divisions on the limb is equal to 20'. Or supposing the limb divided into intervals of 10', and that fifty-nine divisions of the limb correspond to sixty divisions of the vernier; it is then evident that the difference between a division of the instrument and of the vernier is $\frac{1}{60}$th part of 10', i.e., 10". This is the most usual kind of division.
To find the actual reading off in any particular case, we must observe which division of the vernier coincides with a division of the limb; the number denoting this, multiplied by the value of the difference between a division of the limb and of the vernier, will give the additional reading. Suppose, for instance, the nearest division of the limb to the zero of the vernier to be 25° 30', and the eighth division of the vernier to be coincident with a division of the limb, the additional angle will be 80" or 1' 20", and the reading off will be 25° 31' 20".
The adjustments of the sextant are to set the mirrors perpendicular to the plane of the instrument, and parallel to one another when the index is at zero; and to set the axis of the telescope parallel to the plane of the instrument.
Adjustment 1.—To set the index-glass perpendicular to the plane of the sextant.
Set the index towards the middle of the limb, and hold the sextant so that its plane is nearly parallel to the horizon; then look into the index-glass, and if the portion of the limb seen by reflection appears in the same plane with the limb seen directly, the speculum is perpendicular to the plane of the instrument. If they do not appear in the same plane, i.e., if the image be seen above or below the arc itself, its position must be gradually and carefully changed by means of the screws at its back until the error is rectified.
Adjustment 2.—To set the horizon-glass perpendicular to the plane of the instrument.
Place the instrument horizontal, and direct the sight to a distant well-defined object, as the sun, so as to view it directly; then move the index until the image of the object seen by reflection is on the field of view, and move the index backwards and forwards so as to make the image pass over the object. If it pass exactly over the object, the fixed mirror is perpendicular to the plane of the instrument; if not, move the screw G until their exact coincidence takes place.
Adjustment 3.—To set the horizon-glass parallel to the index-glass when the zero of the index or vernier-plate coincides with the zero of the graduations of the limb.
Set O on the index exactly to O on the limb, and fix it in that position by the screw on the under side of it; hold the sextant with its plane vertical, and direct the sight to a well-defined part of the horizon; then if the horizon seen on the silvered part coincides with that seen through the transparent part, the horizon-glass is adjusted; but if the horizons do not coincide, the position of the glass must be altered by moving a screw placed near the fixed reflector, which gives it a motion about an axis perpendicular to the plane of the instrument.
This adjustment is seldom made, as turning the adjusting-screw too often renders this part of the instrument very apt to get out of order. It is usual, therefore, to determine the error in the reading called the Index Error.
To do this, direct the sight to the horizon, and move the index until the reflected horizon coincides with that seen by direct vision; then the difference between O on the limb and O on the vernier-plate will be the index error, which is to be added when O of the vernier is to the right of O on the limb; otherwise subtracted.
A more accurate method than the above is to measure the sun's apparent diameter twice with the index placed alternately on the right and on the left of the zero point of the graduated limb. Half the difference of these two measures will be the index error, which must be added to, or subtracted from, all observations, according as the diameter measured with the index to the left of O is less or greater than the diameter measured with the index to the right of the beginning of the divisions. Care must be taken to measure the sun's horizontal diameter, as the vertical diameter is often affected with refraction. This must be done by keeping the plane of the instrument at right-angles to the vertical diameter of the sun.
For example, on January 2, 1857, the sun's diameter, measured with the index first to the right and secondly to the left of the zero point of division, was 33' and 32' 20" respectively, and the index error obtained by taking the semidifference is -20'.
Adjustment 4.—To set the axis of the telescope parallel to the plane of the instrument.
Turn the eye-end of the telescope until the two wires are parallel to the plane of the instrument; and let two distant objects, or two stars of the first magnitude, be selected, whose distance is not less than 90° or 100°; make the contact of these as perfect as possible at the wire nearest the plane of the instrument; fix the index in this position; move the sextant until the objects are seen at the other wire, and if the same points are in contact, the axis of the telescope is parallel to the plane of the sextant. If, however, the objects are apparently separated, or overlap one another, correct half the error by the screws in the circular part of the supporter, one of which is above, and the other between the telescope and sextant; turn the adjusting-screw at the end of the index till the limbs are in contact; then bring the objects to the wire next the instrument, and if the limbs are in contact, the axis of the telescope is adjusted; if not, proceed as at the other wire, and continue till no error remains. In practice, this adjustment is usually made by means of the sun and moon. The mode of bringing the limbs of the sun and moon into contact will be explained when the use of the sextant is treated of. It is sometimes necessary to know the angular distance between the wires of the telescope; to find which, place the wires perpendicular to the plane of the sextant, hold the instrument vertical, direct the sight to the horizon, and move the sextant in its own plane till the horizon and upper wire coincide; keep the sextant in this position, and move the index till the reflected horizon is covered by the lower wire, and the difference of readings off in these two positions will be the angular distance between the wires. Other and better methods will readily occur to the observer on land.
The Quadrant.
It has been already observed, that this instrument differs from a sextant in the extent of the divided limb and in its rougher manufacture. It is only calculated for observing altitudes. Fig. 46 represents a quadrant of the common construction.
The frame, index, index-glass, and F the fore horizon-glass, are much the same as in the sextant. There is, besides, another horizon-glass G, called the back horizon-glass attached to the same radius as F. Instead of a tube or telescope, the quadrant is furnished with vanes or sights H and I. There are but three coloured glasses, two of which are red and the other green. They are fixed at K, as shown in the figure, when the fore horizon-glass is used. If the back horizon-glass be used, they are transferred to N. The back horizon-glass is silvered at both ends, but has a transparent slit in the middle through which the horizon may be seen. Each of the horizon-glasses is set in a brass frame, to which there is an axis passing through the wood-work, and is fitted to a lever on the under side of the quadrant, by which the glass may be turned a few degrees on its axis, in order to set it parallel or perpendicular, according as it is the fore or back horizon-glass, to the index-glass. The lever has a contrivance to turn it slowly, and a button to fix it.
To set the glasses perpendicular to the plane of the instrument, there are two sunk screws, one before and the other behind each glass; these screws pass through the plate on which the frame is fixed into another plate; so that by loosening one and tightening the other of these screws, the direction of the frame, with its mirror, may be altered, and set perpendicular to the plane of the instrument.
The sight-vanes H and I are perforated pieces of brass, designed to direct the sight parallel to the plane of the quadrant. The vane I has two holes, one exactly at the height of the silvered part of the horizon-glass, the other a little higher, to direct the sight to the middle of the transparent part of the mirror.
The limb is divided into ninety primary divisions, which are considered as degrees, and each degree subdivided into three equal parts, which are therefore of 20' each. The vernier-plate is generally so divided as to enable the observer to read off accurately to minutes.
These consist in setting the mirrors perpendicular to the plane of the instrument, and the fore horizon-glass parallel, and the back horizon-glass perpendicular to the index-glass, when the zero of the index or vernier-plate coincides with zero of the graduations on the limb.
The adjustments for the index-glass and fore horizon-glass are performed nearly in the same way as for the sextant. The index error, however, must be ascertained by bringing the horizon by reflection into the same line with the horizon seen directly. The method by taking the distance of two stars of the first magnitude, or the sun and moon, is inapplicable here.
The back horizon-glass is so seldom used, that for its adjustments and the mode of taking observations with it, the reader is referred to Norie's Navigation, and other works in which this subject is treated.
The altitude of an object may be determined by either instrument, and is the reading off on the limb, with the proper index error applied, when by reflection that object appears to be in contact with the horizon. The distance between the sun and moon, or other heavenly bodies, may be observed by the sextant when the limbs of the bodies whose distance is required appear to be in contact. If the quadrant be used for taking the altitude of the sun, when it is so bright that its image may be seen in the transparent part of the fore horizon-glass, the eye is to be applied to the upper hole in the sight-vane, otherwise to the lower hole; and in this case the quadrant is to be held so that the sun be bisected by the line of separation of the silvered and transparent parts of the glass. The moon is to be kept as nearly as possible in the same position, and the image of the star is to be observed on the silvered part of the glass adjacent to the line of separation of the two parts.
With the quadrant two different methods of taking observations may be employed. In the first, the observer faces the sun, and looks to that part of the horizon which is immediately under the sun, and the observation is therefore called the fore observation. In the other method, the observer's back is towards the sun, and he looks to the part of the horizon opposite to that which is under the sun; and this is consequently called the back observation. It is not to be employed except when the horizon under the sun is obscured, or rendered indistinct by fog or other impediment.
In all cases of taking altitudes, it must be considered that it is necessary to be quite sure that the distance of the sun or other body from the horizon is the least possible, otherwise it would not be the altitude that is observed. Consequently, after the instrument has been placed as nearly as possible in a vertical position, and a contact made, a motion about the line of sight of the sun must be communicated to the instrument, so as to keep the image always in the same part of the silvered mirror, the plane of the instrument being inclined. In this way we keep the angular distance of the sun from the line through the eye by which it is viewed the same, and the sun's image describes a small circle, whose angular radius is this distance. The horizon being fixed and viewed directly, will always occupy the same position. If, then, on giving this vibratory motion to the instrument, the arc described by the sun touches the horizon, the angular distance observed is the altitude. If it should cut the horizon, so that a portion of the sun's image goes below it, the index must be moved back until this arc simply touches the horizon. In the back observation with the quadrant, and in observing with the sextant furnished with the inverting telescope, the images are inverted, and the arc described by the sun's image lies below the horizon, to which line it is convex.
The motion must be given round the axis passing through the observer's eye and the sun. To do this, a motion about the axis of vision must be given to the instrument, and at the same time the observer must turn himself about upon his heel; for the motion about the line of sight of the sun may be resolved into these two motions; and the observer has no means of giving the requisite motion directly by one movement. When the sun is near the horizon, the line from the eye to the sun will not be far removed from the axis of vision, and the principal motion of the instrument will be performed on this axis; while that part of the motion made about the vertical axis will be small. On the contrary, if the sun be near the zenith, the line from the eye to the sun is nearly vertical and perpendicular to the axis of vision; hence the motion about the vertical axis is the greatest, and that about the axis of vision very trifling. In intermediate positions of the sun the motions of the instrument about these two axes will be more equally divided. When the distance between the moon and sun, a planet or a star, is to be observed, the sextant must be so held that its plane may pass through the eye of the observer and both objects; and the reflected image of the brighter of the two is to be brought into contact with the other seen directly. To effect this, therefore, it is evident that when the brighter object is to the right of the other, the face of the sextant... must be held upwards, and if to the left, downwards. When the face of the sextant is held upwards, the instrument should be supported with the right hand, and the index moved with the left hand. But when the face of the sextant is from the observer, it should be held with the left hand, and the motion of the index regulated with the right hand. Sometimes a sitting posture will be found convenient for the observer, particularly when the reflected object is to the right of the direct one. In this case the instrument is supported by the right hand; the elbow may rest on the right knee; the right leg at the same time resting on the left knee. If the sextant be provided with a hall and socket, and a staff, one of whose ends is attached thereto, and the other rests in a belt fastened round the observer's body, the greater part of the weight of the instrument will be supported by his body. In all cases where the sextant is used, when the contact is nearly made, the index should be fixed by the under screw, and the remaining small motion given by the adjusting screw.
Error may arise from two kinds of causes: one inherent in the construction of the instrument—as defect of parallelism or perfect planeness in the fore and back surfaces of the mirrors, as also of the coloured glasses, and of the true circular form of the arc, and true centring, for which no remedy can be provided; and the other arising from the bending and elasticity of the index or moveable radius.
The parallelism of the two surfaces of the mirrors may be tested by viewing through them obliquely a distant distinct object. If the image is perfect and well defined, the surfaces are parallel; otherwise not.
To ascertain whether the surfaces of the mirrors are plane, observe the angle between two distant objects which are nearly of the same altitude, the image of the left-hand object being brought into contact with the right-hand object viewed directly; then move the instrument in its own plane so as to bring the image of the right-hand object into contact with the left-hand object viewed directly. If they continue in contact, the surfaces are plane; otherwise not.
To test the form of the dark glasses, measure the sun's diameter to the right and to the left of the zero point with different combinations of the glasses. If the sum of the diameters so measured be nearly equal to four times the semidiameter given in the Nautical Almanac, the form of the glasses is satisfactory. For the true centring of the arc, and its truly circular form and correct graduation, the navigator must trust entirely to the skill of the maker.
By reason of the bending and elasticity of the index, and the resistance it meets with in turning round the centre, its extremity, on being pushed round the arc, will sensibly advance before the index-glass begins to move, and may be seen to recoil when the force acting on it is removed. Mr Hadley, in order to remedy this defect, which he seems to have apprehended, gave special directions that the index be made broad at the end next the centre, and the centre or axis itself have as easy a motion as is consistent with steadiness; that is, an entire freedom from looseness or shake, as the workmen term it. By strictly complying with these directions, the error in question may indeed be greatly diminished, so as to be nearly insensible, when the index is made strong, and the proper medium between the two extremes of a shake at the centre on the one hand, and too much stiffness there on the other, is nicely hit; but it cannot be entirely corrected, for to more or less of bending the index will always be subject, and some degree of resistance will remain at the centre, unless the friction there could be totally removed, which is impossible.
Of the reality of the error to which he is liable from this cause, the observer, if he is provided with an instrument furnished with an adjusting screw for the index, may thus satisfy himself:—After finishing the observation, lay the instrument on a table, and note the angle; then cautiously loosen the screw which fastens the index, and it will immediately, if the instrument is not remarkably well constructed, be seen to start from its former situation more or less according to the perfection of the joint and strength of the index. This starting, which is due to the index recoiling after being released from the confined state it was in during the observation, will sometimes amount to several minutes; and its direction will be opposite to that in which the index was moved by the screw at the time of finishing the observation. But how far it affects the truth of the observation depends on the manner in which the index was moved in setting it to O, for adjusting the instrument, or in finishing the observations necessary for finding the index error.
The easiest and best rule to avoid these errors seems to be this:—In all observations made by Hadley's quadrant or sextant, let the observer take notice constantly to finish his observations by moving the index in the same direction which was used in setting it to O for adjusting, or in the observations necessary for finding the index error. If this rule is observed, the error arising from the spring of the index will be obliterated. For as the index was bent the same way, and in the same degree, in adjusting as in observing, the truth of the observations will not be affected by this bending.
To Observe the Sun's Altitude at Sea.
Turn down one of the dark glasses before the horizon-glass (if the instrument be the quadrant, the fore horizon-glass is to be used) according to the sun's brightness; direct the sight to that part of the horizon which is under the sun, and move the index until the coloured image of the sun appears in the horizon-glass. Then give the instrument a slow vibratory motion about the axis of vision, as already described; move the index until the upper or lower limb of the sun is nearly in contact with the horizon at the lowest or highest part of the arc (according as the image is seen erect or inverted) described by this motion; and complete the contact by the tangent-screw, if the sextant be used—if not, by moving the index. The reading off of the limb will be the altitude of the sun.
To Observe the Moon's Altitude at Sea.
Turn down the green glass, and observe the moon in the silvered part of the horizon-glass, the eye being directed towards the horizon; move the index gradually, and proceed as already described in the case of the sun, until the enlightened limb is in contact with the horizon at the lowest or highest point of the arc described by the vibratory motion. The reading off will be the altitude of the moon's observed limb. If the lower limb be observed, the moon's semidiameter must be added; and if the upper limb be observed, it must be subtracted from the observed altitude, in order to obtain the altitude of the moon's centre. If the observation is made in the day-time, the coloured glass is not to be used;
To Observe the Altitude of a Star or Planet.
Put the index to zero; then direct the sight to the star so as to see it through the unsilvered part of the horizon-glass; turn the instrument a little to the left, and the image of the star will be seen in the silvered part of the glass. Now move the index, and the image will be seen to descend; continue to move the index gradually, until the star is in contact with the horizon at the lowest point of the arc described by the vibratory motion; in the case of the sextant, clamping the index when the contact is nearly made, and completing it with the adjusting or tangent screw. To find the Altitude of the Sun on Shore with an Artificial Horizon.
A flat dish, containing a small quantity of mercury, is generally used for this purpose. The surface of the mercury is horizontal, and is a good reflector of the sun's rays. Let the observer so stand that he may receive on his eye rays from the sun which have been reflected from the surface of the mercury. He will then, according to the principles of optics, see the reflected image of the sun as much below the surface of the mercury as the sun is above it. If, then, he looks at the image through the unsilvered part of the horizon-glass, instead of at the horizon, and brings the image of the sun reflected at the index-glass and horizon-glass into contact with this, it is evident that the angle observed will be double of the sun's altitude. The details of this process are the same as have been already explained. Having read off the angle when the contact has been made, it must be corrected for the index error; and the result, divided by 2, will be the apparent altitude of that limb of the sun which has been observed.
To Observe the Distance between the Moon and any Celestial Object with the Sextant.
1. Between the Sun and Moon.—Put the telescope in its place, and the wires parallel to the plane of the instrument; and if the sun is very bright, raise the plate before the silvered part of the speculum; direct the telescope to the transparent part of the horizon-glass, or to the line of separation of the silvered and transparent parts, according to the brightness of the sun; and turn down one of the coloured glasses. Then hold the sextant so that its plane produced may pass through the sun and moon, having its face upwards or downwards, according as the sun is to the right or left of the moon; direct the sight through the telescope to the moon, and move the index till the limb of the sun is nearly in contact with the illuminated limb of the moon; now clamp the index, and, by a gentle motion of the instrument, make the image of the sun move alternately to one side and the other of the moon; and when in that position, where the limbs are nearest each other, make the contact of the limbs perfect by the tangent-screw; this being effected, read off the degrees and parts of a degree shown by the index on the limb, using the magnifying-glass; and thus the angular distance between the nearest limbs of the sun and moon is obtained.
2. Distance between the Moon and a Planet or Star.—Direct the middle of the field of the telescope to the line of separation of the silvered and transparent parts of the horizon-glass; if the moon is very bright, turn down the lightest-coloured glass, and hold the sextant so that its plane may be parallel to that passing through the eye of the observer and both objects; its face being upwards if the moon is to the right of the star, but downwards if it be to the left. Now direct the sight through the telescope to the star, and move the index till the moon appears by the reflection to be nearly in contact with the star; clamp the index, and turn the adjusting or tangent screw till the coincidence of the star and the enlightened limb of the moon is perfect; and the reading off of the limb at the index will be the observed distance between the moon's enlightened limb and the star.
The contact of the limbs must always be observed in the middle, between the parallel wires.
It is sometimes difficult for those not much accustomed to observations of this kind, to find the reflected image in the horizon-glass; it will perhaps, in this case, be found more convenient to look directly to the object, and, by moving the index, to make its image coincide with that seen directly.
Sect. II.—Of the Corrections to be Applied to Observed Altitudes and Distances.
1. Parallax.
In order that the place of a heavenly body may be fixed in space, it is necessary to suppose that all observations are taken from one point. This point is the centre of the earth. Consequently the places of the sun, moon, and planets, whose distances from the earth are measurable, as observed, must be reduced to what they would be if seen from the centre. The correction for this object is called parallax. The fixed stars are at so great a distance from the earth that they have no sensible parallax, and this correction is not to be applied to them.
Let C (fig. 47) be the centre of the earth, P the place of the observer on its surface, and Z his zenith; and let S be a heavenly body whose position is observed. Then ZPS is the observed zenith distance, or complement of the observed altitude, and ZCS the true zenith distance, —i.e., the zenith distance as observed at the earth's centre.
Then clearly the angle ZPS is greater than the angle ZCS by the angle PSC, which is also in the plane of the vertical circle through S. It is evident from the figure that a heavenly body is depressed by parallax; and the observed altitude is less than the true altitude by a certain amount depending on the altitude, which is called the correction of parallax. The correction of parallax, therefore, must always be applied with the positive sign. Its amount may be easily found; for, let \( \angle ZPS = z' \), and ZCS = z, and PSC the parallax = p; then, in triangle PSC,
\[ \sin PSC = \frac{PC}{SC} \sin SPC = \frac{PC}{SC} \sin SPZ. \]
Let PC = r, the radius of the earth, and SC = R, the distance of the heavenly body from C; also PSC is always very small, and \( \sin PSC = PSC = p \) very nearly.
Hence \( p = \frac{r}{R} \sin z' \).
r and R being invariable, p is greatest when \( z' = 90^\circ \), or when S is in the horizon. Let P be the value of p in this position; then
\[ P = \frac{r}{R}, \]
and \( p = P \sin z' = P \cos a' \),
if \( a' \) be the observed altitude.
This will illustrate the principle on which tables of correction for parallax for different heavenly bodies, as sun, moon, &c., for different angles of altitude, are calculated.
2. Refraction.
Refraction is a correction to be applied in consequence of the rays from every heavenly body being bent or refracted as they pass through the successive layers of the earth's atmosphere, in consequence of which they describe curvilinear paths, having their convexity turned towards the zenith of the observer. The tangent to this curve at the eye of the observer is the direction in which he sees the object, and is evidently, from what has been said, bent towards the zenith. Hence the effect of refraction is to raise the heavenly bodies in the heavens above their true places; and the correction must therefore be applied to the observed altitudes of all bodies with a negative sign.
The law which this correction follows is very complex; it is very great when the body is near the horizon, and vanishes when it is in the zenith. The best position of heavenly bodies for observation, so far as this correction is concerned, is near the zenith.
A table of this correction for every 5° of altitude is calculated, and is to be found in all nautical tables.
3. Correction for Semidiameter.
When the sun or moon, or a planet, is observed, the altitude of one of the limbs is the observed altitude, the altitude of the centre must be obtained by adding or subtracting the semidiameter of the observed body according as the lower or upper limb is observed. The semidiameters of the sun and moon are continually changing; and tables are given in the Nautical Almanac of their values at noon of every day at Greenwich, and in the case of the moon at midnight also. Their values at any intermediate hour may be calculated from these, as will be more fully shown hereafter.
This correction is not to be applied when a fixed star is the object observed.
4. Correction for Dip.
The altitude is supposed to be observed from the true horizon—i.e., from a horizontal plane through the place of the observer. This, however, is never the case, for the observer's eye must then be on the earth's surface. It is, in fact, always elevated above it, and the apparent horizon is depressed in consequence below the true horizon.
The nature of this correction will be easily seen from fig. 48. B the place of observation at a distance \(AB = d\) feet above the surface, \(BH'\) touching the surface at \(H'\); then \(BH'\) is the plane of the sensible horizon; \(S\) a heavenly body to be observed. Draw \(BA\) parallel to \(AH\), the true horizon of \(A\), and let \(BH'\) and \(AH\) intersect in \(J\). Then, since \(AB\) is very small compared with \(BS\) and \(AS\), the \(\angle ABS\) may be considered as equal to \(\angle HAS\). Also let \(R =\) earth's radius, \(CA\) or \(CH'\). Hence observed altitude of \(S\)
\[H'BS = ABS + ABH'\]
\[= hBS + AJB = ABS + ACH'.\]
Or true altitude = observed altitude - \(ACH'\).
Now tan \(ACH' = \frac{BH'}{CH'} = \sqrt{\frac{(2R+d)}{d}} \frac{d}{R}\)
\[= \sqrt{\frac{2d}{R}} \text{nearly, because } \frac{d^2}{R^2} \text{ may be neglected in comparison of } \frac{d}{R}.\]
Also \(ACH' = \tan ACH' \text{nearly;} \quad \therefore \quad ACH' = \sqrt{\frac{2d}{R}}\)
Now \(R\) in feet \(= 6120 \times 3958\).
Hence \(ACH'\) in seconds
\[= 57.29577 \times 60 \times 60 \times \sqrt{\frac{2d}{6120 \times 3958}}.\]
Hence, putting \(d = 1, 2, 3, \ldots\), feet respectively, we can find the dip when the eye is 1, 2, 3, etc., feet respectively above the surface.
Whence tables for the dip at different elevations may be calculated.
5. Index Error.
This error has already been explained.
It will be observed that all observed altitudes are affected with refraction and dip, of which the first is always subtractive, and the second is subtractive in all observations, except when the back horizon-glass of the quadrant is made use of, when it is additive.
Parallax and semidiameter affect only those bodies whose distance from the earth is not very great, and which have a sensible diameter, as the sun and moon; but no fixed stars. Parallax is always additive; and the semidiameter is to be added or subtracted according as the lower or upper limb of the heavenly body is observed.
**Chap. II.—Preliminary Problems in Nautical Astronomy, and Use of Nautical Almanac and Tables.**
**SECT. I.—OF TIME.**
A day is the interval between two successive transits of a heavenly body over the meridian, and derives its name from the body whose motion is observed; and of whatever denomination it be, it is divided into 24 hours, each hour into 60 minutes, and each minute into 60 seconds. The interval between two successive transits of the sun is a solar day; of the moon, a lunar day; of a fixed star, a sidereal day. The earth's revolution about her axis is performed always in the same time; hence if all the heavenly bodies retained the same position with regard to one another, all days of whatever name would be of the same length. The sun (or, more strictly speaking, the earth), the moon, and planets are always varying their position with respect to one another and the fixed stars; and they move with velocities not only different from each other, but variable in different parts of their own orbits. The length of the day, therefore, determined by these bodies is variable. In order to obtain a definite and uniform measure of time, the mean solar day, which is the average of all the apparent solar days in the year, is employed. An imaginary body, called the mean sun, is supposed to describe the equator uniformly with the true sun's average or mean daily motion, and the interval between two successive transits of this imaginary sun is a mean solar day.
Clocks and chronometers are adapted to mean solar time; so that a complete revolution through twenty-four hours of the hour-hand of one of these instruments would exactly correspond with the revolution of the earth about her axis with regard to the mean sun. Time reckoned in mean solar days and parts of mean solar days is called mean solar time.
The true sun sometimes passes the meridian before and sometimes after the mean sun; the difference in time of the transits of the true and mean suns is called the equation of time, and is sometimes to be added and sometimes to be subtracted from the one of these times to obtain the other, as is pointed out in the Nautical Almanac.
As the earth revolves about her axis from E. to W., different meridians successively come under the mean sun; and since, after the lapse of twenty-four mean solar hours, Nautical the same meridian again comes under the sun; it follows that for every 15° of longitude between the places, there is a difference of 1h of mean time; and that mean noon is 1h earlier for every 15° of E. longitude, and 1h later for every 15° of W. longitude.
A sidereal day is the interval between two successive transits over the same meridian of the vernal equinox, or the first point of Aries. This is not strictly a uniform measure because of the change of the position of this point of Aries in consequence of precession and nutation. Time, therefore, so reckoned, ought strictly to be called apparent sidereal time; and mean sidereal time to be reckoned from the transit not of the true but of the mean equinoctial point. The smallness of the fluctuations to which a clock regulated to apparent sidereal time, compared with one regulated to mean sidereal time, is subject, amounting at the utmost to 2s in 19 years, has prevented the practical inconvenience of this being felt; no clock being sufficiently perfect to go for so long a period without requiring frequently to be re- adjusted.
The sun's apparent revolution in his orbit round the earth takes place in 365.242218 mean solar days; the mean sun, therefore, describes an angle of 59° 8'33" in a mean solar day; hence, in the interval between two suc- cessive transits of the mean sun over the same meridian, the earth revolves through 360° 59° 8'33"; but in a sidereal day the earth revolves through 360°. Whence we obtain the proportion—
A sidereal day : a mean solar day :: 360° : 360° 59° 8'33".
Whence also it is evident that, if the same interval of absolute time be expressed in mean solar and also in sidereal time, it will be greater in the latter denomination than in the former.
Tables are calculated for the acceleration of sidereal on mean solar time, and, conversely, for the retardation of mean solar on sidereal time, which are useful for convert- ing intervals of sidereal into mean solar time, and con- versely. These are to be found at page 516 of the Nau- tical Almanac.
Astronomical mean time always begins at noon, and goes on to the succeeding noon through 24 hours; so that a clock which shows astronomical mean time, set for any place (as Greenwich), shows 0h 0m 0s when the mean sun is on the meridian, and shows 24 hours, or 0° 0' 0", when the mean sun is again on the meridian.
A clock set to show sidereal time shows 0h 0m 0s when the mean equinoctial point is on the meridian, and shows 24 hours, or 0° 0' 0", again when this point is next on the meridian.
Sidereal Time, in the Nautical Almanac, is the sidereal time at mean noon for the meridian of Greenwich; or, in other words, the hour angle of the equinoctial point when the mean sun is on this meridian; and is very useful in all cases where mean time is to be deduced from observations of heavenly bodies.
The civil day is reckoned from midnight to midnight, and is divided into 12 hours, from midnight to noon, called A.M.; and into 12 hours, from noon to midnight, called P.M. Evidently time P.M., and astronomical mean time up to midnight, are the same. But for time A.M., add 12 hours, and date the day one back.
Thus, June 3, 7h 5m 27s P.M. is, June 3, 7h 5m 27s astronomical time; but, June 3, 5h 12m 37s A.M. is, June 2, 17h 12m 37s A.M.
Conversely, if the astronomical time is given, and the corresponding civil time is required, if under 12 hours, the given time is the same time P.M. for the same date; if above 12 hours, subtract 12 hours from it, and date the day one forward, and the result is civil time A.M.
Thus, September 18, 7h 10m 15s is, September 18, 7h 10m 15s P.M.; but September 18, 22h 13m 45s astro- nautical is, September 19, 10h 13m 45s A.M.
Prob. 1.—To convert degrees, or parts of the equator, into time.
Rule.—Divide the degrees by 15; the result gives hours. Multiply the remaining degrees, if any, by 4; the result is minutes. Divide the number of minutes by 15 for minutes, and then multiply the remaining minutes by 4 for seconds. Divide the seconds by 15 for seconds, and decimals of seconds, if necessary.
Ex. 1.—Reduce 26° 46' to time. Divide 26 by 15, the quotient is 1, with remainder 11. Hence, by the rule—
\[ \begin{align*} 26° &= 1h \ 44m \\ 45' &= 0 \ 3 \\ \text{or } 26° 45' &= 1 \ 47. \end{align*} \]
Ex. 2.—Reduce 139° 48' 18" to time. \[ \begin{align*} 139° &= 9h \ 16m \ 0s \\ 48' &= 0 \ 3 \ 12 \\ 18" &= 0 \ 0 \ 1 \ 2 \\ \text{or } 139° 48' 18" &= 9 \ 19 \ 13 \ 2. \end{align*} \]
Prob. 2.—To convert time into degrees, minutes, and seconds.
Rule.—Multiply the given time by 10, and add to the result half of the product. The result will be the corre- sponding degrees, minutes, and seconds.
Ex.—Convert 3h 4m 28s into degrees, etc.
\[ \begin{align*} 3h \ 4m \ 28s &= 10 \\ 30 \ 44 \ 40 \\ \text{One half } &= 15 \ 22 \ 20 \\ 46 \ 7 \ 0 \\ \text{Answer} &= 46° 7'. \end{align*} \]
Prob. 3.—Given the time under any known meridian, to find the corresponding time at Greenwich.
Rule.—Let the given time be reckoned from the pre- ceding noon to turn it into astronomical time; convert the longitude of the known meridian into time, which add to the given time if the place be W. of Greenwich, and sub- tract from the given time if it be E.; and, if necessary, increase the time by 24 hours, reckoning the day one back. If the resulting time exceed 24 hours, put the date one day forward, and subtract 24 hours.
Ex. 1.—It is 6h 15m P.M., June 23, in a ship whose longitude is 76° 45' W.
\[ \begin{align*} 76° 45' &= 5h 7m \\ \text{Given time, June 23} &= 6h 15m \\ \text{Longitude} &= 6 \ 7 W. \\ \text{Time at Greenwich} &= 11 \ 22 \ 23 \text{ June 23}. \end{align*} \]
Ex. 2.—The time at a ship in longitude 139° 48' 15" E. is, July 24, 5h 25m 20s P.M.; required the Greenwich time.
\[ \begin{align*} 139° &= 9h \ 16m \ 0s \\ 48' &= 0 \ 3 \ 12 \\ 15" &= 0 \ 0 \ 1 \\ \text{or } 139° 48' 15" &= 9 \ 19 \ 13. \end{align*} \]
This being greater than the given time, we next add 24h to the latter, and we have—
\[ \begin{align*} 29h \ 25m \ 20s &= 9 \ 19 \ 13 \\ \text{Time at Greenwich} &= 20 \ 6 \ 7 \text{ July 23}. \\ \text{Or civil time} &= 8 \ 6 \ 7 \text{ A.M. July 24}. \end{align*} \]
Ex. 3.—The time at a ship in longitude 175° 45' W. is, June 28, 4h 18m 12s A.M.; what is the Greenwich time?
\[ \begin{align*} 175° &= 11h \ 49m \\ 45' &= 0 \ 3 \\ \text{Longitude} &= 11 \ 43 W. \\ \text{Time at ship is, June 27} &= 16h \ 18m \ 12s \\ \text{Longitude} &= 11 \ 43 W. \\ \text{Time} &= 22 \ 1 \ 12 \text{ June 27}. \\ \text{Or} &= 4 \ 1 \ 12 \text{ June 28}. \end{align*} \] Prob. 4.—To find the Greenwich time by a chronometer whose error on Greenwich mean time is known.
Rule.—Add or subtract the error of the chronometer to the time shown by it, according as it is slow or fast; the result is Greenwich time. Sometimes 12 hours must be added to this result, and the day reckoned one back. The best way to obviate this error, is to obtain Greenwich time approximately, by Problem 3, by help of the ship's mean time and longitude, by account; if the difference between the two (Greenwich) dates, so found in these two ways, is nearly 12 hours, then the date by chronometer must be increased by 12 hours, and the day reckoned one back if necessary, so as to make the two dates agree in the day and hour nearly.
Ex. 1.—June 15, 1857, at 7h 45m P.M. mean time nearly, in longitude 45° W., a chronometer showed 10h 53m 12s, being 5° 15' fast; required the Greenwich time.
By Chronometer. By Rule 3. June 15, Chronometer 10h 53m 12s Ship, June 15... 7h 45m Error on Greenwich | 0 3 15 Longitude...... 3° 0 W. Greenwich time....... 10 49 57 Greenwich time, 10 45
Ex. 2.—October 12, 1857, at 11h 45m P.M., in longitude 110° 35' W., a chronometer showed 7h 0m 40s, being 9° 5' slow; required the Greenwich date.
By Chronometer. By Rule 3. Oct. 12................. 7h 0m 40s Ship, Oct. 12... 11h 45m 0s Error (slow)........... 0 9 5 Longitude........... 7° 22 20 7 9 45 Greenwich, Oct. 12... 19 7 20
Greenwich, Oct. 12, 19 9 45
Prob. 5.—Given the Greenwich date, to find the date under a given meridian.
Rule.—The Greenwich date being reduced to the preceding noon, reduce the longitude to time, and subtract it from Greenwich date, increased by 24 hours, and put one day back if necessary, if the longitude be W.; and add if the longitude be E.
Ex. 1.—An eclipse of the moon commenced at Greenwich, December 25, 1852, at 22h 14m 36s; at what time did it begin at a place in longitude 175° E.?
Greenwich time, Dec. 25............. 22h 14m 36s Longitude......................... 11 40 E. (1.) December 26................. 9 54
Ex. 2.—On the 6th of December 1857, at 9° at Greenwich, the distance of the moon from the sun was 106° 44' 34"; at what time will the distance be the same in longitude 168° 43' 43" W.?
Long. = 11h 13m 28s 86 W. Greenwich, Dec. 5... 33h 0m 0s Add 24 to Greenwich time. Longitude........... 11 13 28s 86 W. Date required, Dec. 5............. 21 46 57 14.
Sect. II.—Of the Sun's Declination, Right Ascension, Equation of Time, and Semidiameter and Sidereal Time.
All of these quantities are given in the Nautical Almanac for the noon of every day. They all vary, and consequently their value at any time intermediate between two successive noons is different from that at either noon. For ordinary purposes, it is sufficient to suppose that the change of these quantities is proportional to the time from the preceding noon. Hence, if the difference of the value between the preceding and succeeding noon is taken from the Nautical Almanac, a simple proportion will give the quantity to be applied to the value at the preceding noon, which must be added if the quantity is increasing, and subtracted if it is decreasing. As the values of the quantities are given for Greenwich noon, it is necessary to reduce the ship's time to Greenwich time, in order to obtain their proper value.
This process may be materially shortened by the use of proportional logarithms, which are given in nautical tables, and to which the reader is referred for an explanation. For example, to take out the sun's declination at a given time in a given place, get a Greenwich date, as already explained; take out from the Nautical Almanac the declination at two successive noons between which the date lies; take their difference; take out from the tables the logarithm of the Greenwich date of the sun; add to it the proportional logarithm for the change in 24 hours—the result is the proportional logarithm of the change of declination for the given time. This will be found from the table, and added or subtracted, according as the declination is increasing or decreasing. Sometimes the declination at two successive noons will be of different names; the difference must then be found by adding them.
The very same mode applies to finding the equation of time, and also the sun's apparent right ascension.
The sun's semidiameter may always be taken to be that corresponding to the nearest noon.
The right ascension of the mean sun, called in the Nautical Almanac the sidereal time, may be found for any other meridian than Greenwich in the same way. Since, however, the motion of the mean sun is uniform throughout the year, the change in any given number of hours, minutes, and seconds is the same for all days, and can be found at once from the "Table of Time Equivalents" given in the Nautical Almanac.
Opposite to each of these quantities in the Nautical Almanac is given the difference for one hour, which enables us to find the quantity for any other hour than noon, by multiplying this hourly difference by the number of hours in the date from noon, and adding to it the proportional parts for the additional minutes and seconds, and applying this difference to the value for the preceding noon, with a positive or negative sign, according as the quantity is increasing or decreasing.
The following examples will illustrate these directions:
Ex. 1.—October 11, 1857, in Long. 36° 45' E., at 7h 45m A.M.; required the sun's declination, right ascension, equation of time, and sidereal time, i.e., the right ascension of the mean sun when crossing the meridian of observer.
October 11, 7h 45m A.M. civil ls, October 10... 19h 45m Longitude......................... 2° 27 E. Greenwich date, Oct. 10............. 17 18
To find Declination.—(1.) By proportional logarithms, from Nautical Almanac.
Declination, Oct. 10............. 6° 44' 0° 9 8. Diff. for 24h................. 0 22 42 6 Greenwich date, logarithm of C = -14217 Proportional logarithm for 22 42' 6 = -89925 Prop. log. for 19h 22m = 1° 4142 Declination at noon, October 10... 6° 44' 0° 9 8. Increase in 17h 18m............. 0 16 22 Declination required........... 7° 0 22 9 8.
(2.) By hourly differences.
Difference for 1h, October 10, 58° 77' from Nautical Almanac.
17° 39739 5677 865-09
965°09' = 16° 5'09" difference for 17°. 15° is 1/4 14°19' 3° is 1/2 2°33' 16° 22°11' Declination, October 10... 6° 44' 0° Declination required... 7° 0° 23°01'
To find right ascension. Right ascension, Greenwich, October 10... 13° 3° 8°50' "... " "... 11° 13° 6°50'02" Greenwich date logarithm of 0° = -14217 Prop. log. for 3° 41' = 1-68903 Prop. log. for 2° 39' = 1-83120 Sun's right ascension at noon, October 10... 13° 3° 8°50' Increase in 17° 18° = 0° 2°39' Sun's right ascension required... 13° 5° 47°80'
Or, from Nautical Almanac— Difference for 1°... 9°216 17 64512 9216 156672 15° is 1° 2°304 3° is 1° 461 15° 43° or 2° 39° nearly. Sun's right ascension, October 10, moon... 13° 3° 8°50' Increase for 17° 18° = 0° 2°39' Right ascension required... 13° 5° 47°
To find equation of time. Equation of time, October 10... 12° 59°81' "... " "... 11° 13° 15°15' 0° 15°34' Greenwich date log. of 0° = -14217 Prop. log. for 15°3° = 2-84873 Prop. log. for 11° = 2-29090 Equation of time, October 10... 12° 59°81' Increase in 17° 18° = 0° 11' 13° 10°81'
Or, from Nautical Almanac— Increase for 1°... 0°639 17 4473 639 10°883 15° is 1° 129 3° is 1° 25 11°07° or 11° nearly, which, added to the equation of time at mean noon of October 10, gives 13° 10°81° for equation of time required.
Right ascension mean sun, October 10... 13° 16° 8°61' "... " "... 11° 13° 20° 5°17' Difference... 0° 3°56°56' Greenwich date of 0° = -14217 Prop. log. for 3° 56° = 1-66051 Prop. log. for 2° 50° = 1-80288 October 10... 13° 16° 8°61' Increase... 0° 2°50' Mean sun's r. ascen. or sidereal time, 13° 18° 58°61'
Or, by table of time equivalents, Mean sun's right ascension, October 10... 13° 16° 8°61' Correction for 17°... 0° 2°47' Correction for 18°... 0° 0°3' Mean sun's right ascension required... 13° 18° 58°61'
Ex. 2.—To find the sun's declination at a place in Long. 97° 45' W., at 3° 46° P.M., on 20th March 1857.—Greenwich date, March 20, 10° 17'. Sun's declination, March 20... 0° 3°35°58' "... " "... 21° 0° 20° 6°1 N. Difference... 0° 23°41°6
Greenwich date logarithm 0° = -36908 Prop. log. for 23° 41' = -38983 Prop. log. for 10° 9° = 1-24891 Declination... 6° 33°5 N.
Or, by hourly difference, Difference for 1°, March 20... 5°9°23' 10 Difference for 10°... 59°230 15° is 1° 14°80 2° is 2°5 1°97 60°07 Correction... 10° 9° And the declination, subtracting from this... 3° 35°5 Gives, as before... 6° 33°5 N.
SECT. III.—OF TAKING OUT MOON'S DECLINATION, RIGHT ASCENSION, SEMIDIAMETER, AND HORIZONTAL PARALLAX.
The moon's declination and right ascension are given for every hour in the Nautical Almanac; and the difference for 10° at the beginning of every hour is recorded. From these data, the moon's declination and right ascension at any hour can be readily obtained; for by multiplying the difference for 10° by the number of intervals of 10° in the given time since the last hour, and taking parts for the additional minutes and seconds, and adding the result if the declination be increasing, and subtracting if decreasing, to the declination at the last preceding hour, we shall find the declination. The difference must always be added to the right ascension. Or they may be found by a table of logistic logarithms; thus, add together the logistic logarithms of the minutes and seconds in the Greenwich date, and the proportional logarithms of the difference; the result will be the proportional logarithm of the correction to be applied.
The moon's semidiameter and horizontal parallax are given in the Nautical Almanac for every 12°,—viz., from mean noon to mean midnight, and mean midnight to mean noon, of every day at Greenwich. Hence we may proceed as for the sun's right ascension and declination, except that we must always take out of the table the declination or right ascension for the mean noon or mean midnight, according as the time lies between noon and midnight or midnight and noon; and if the Greenwich date exceed 12°, we must subtract 12° from it, and reckon the difference from the preceding midnight.
Ex. 1.—August 10, 1857, in Long. 84°30' E., at 7° 44° 15° P.M.; required the moon's right ascension and declination. Ship, August 10... 7° 44° 15° Longitude... 5° 38° 0° E. Greenwich date, August 10, 2° 6° 15° Moon's r. ascen., Aug. 10, 2°, 1° 8° 43°49' Decl., 9° 22° 48°7 N. "... " "... 3° 1° 10° 50°91' "... 9° 38° 55°1 N. Difference... 0° 2°742 Log. log. 6° 15° = -98227 Log. log. 6° 15° = -78927 P. log. for... 2° 7° = 1-92962 P. log. for... 16° 6° = 1-04845 P. log. for... 0° 13° = 2-91189 P. log. for... 1° 41° = 2-03072 Moon's r. ascen., Aug. 10, 2°, 1° 8° 43°49' Decl., 12° 22° 48°7 Correction... 0° 0°13° Cor. ... 0° 1°41° Moon's right ascen. required, 1° 8° 56°49' Decl. req., 9° 24° 29°7 Or for right ascension—Or for declination— Difference for 1°... 2°742 Difference for 10°... = 161°05 6° is 1°... 12°742 5° is 1°... 80°52 15° is 1°... 530 1° is 1°... 10°10 13°272 15° is 1°... 4°02 Or correction is 1°, as before. Or correction is 1° 41°, as before.
Ex. 2.—September 12, 1857, in Long. 149°18' W., at 5° 36° P.M.; required the moon's semidiameter and horizontal parallax. Ship, September 12... 5° 36° 0° Longitude... 9° 57° 12° W. Greenwich, September 12... 15° 33° 12° Nautical Astronomy
| Date | Moon's Semi-diameter | Moon's Horizontal Parallax | |---------------|----------------------|---------------------------| | Sept. 12, midnight | 15° 33' 2" | Sept. 12, midnight | 58° 10' | | Sept. 13, noon | 15° 49' 3" | Sept. 13, noon | 57° 59' |
Difference: 0° 3' 7" Difference: 0° 14' 1"
And both are decreasing.
Greenwich date log (3° 35') = -32895 Greenwich date log (3° 35') = -32895 Prop. log. for 3° 7' = 3° 46522 Prop. log. for 1° 1' = 2° 88490 Prop. log. for 1° 1' = 3° 99117 Prop. log. for 4° 2' = 3° 41315
Moon's semi-diameter at 15° 33' 12" = 15° 52' 1" Horiz. parallax at 15° 33' 12" = 58° 5' 8"
These problems may also be solved by simply taking the proportional parts. Thus, for the horizontal parallax—
Difference for 12° = 14° 1'
3° is 1° 3° 5' 30° is 1° 58° 5'
Or the correction is 4°.
And for semidiameter—
Difference for 12° = 3° 7'
3° is 1° 2° 92 30° is 1° 15° 15
Or the correction is 1° 07'.
Sect. IV.—Of Lunar Distances
Prob.—Having given a lunar distance, to find its Greenwich date.
Rule.—Lunar distances are found for every third hour of Greenwich mean time, and are recorded in the Nautical Almanac, as also the proportional logarithms of the differences of the distances at intervals of every three hours. To find, then, the Greenwich date corresponding to a given distance, find in the table the nearest distance preceding in order of time the given distance, and take the difference between it and the given distance; from the proportional logarithm of this difference subtract the proportional logarithm of the said nearest distance; the remainder will be the proportional logarithm of the correction to be applied to the Greenwich date answering to the nearest distance in order to obtain the approximate Greenwich time.
The distances are here supposed to increase uniformly, which is not the case. It is therefore necessary to apply a correction to the time above obtained, if great accuracy is required. This may be found by help of a Table showing the Corrections required on account of Second Differences, in the Nautical Almanac, as follows:—(1) Find the approximate interval by the preceding rule. (2) Take the difference between the proportional logarithms which stand opposite to the distances which include the given distance. (3.) Look down the column in the table at the head of which is this difference of proportional logarithms, and along the column at the left hand side of which stands the approximate interval; the number so found is the correction, which is to be added to the approximate time if the proportional logarithms are decreasing, and subtracted if they are increasing.
Ex.—Required the time at Greenwich at which the reduced distance of the moon from Jupiter will be 42° 55' 18" on the 23rd December 1857.
From the table, it appears this will be between moon and 3°.
Distance at noon...... 43° 58' 19" Prop. log...... -2° 500 Reduced distance...... 42° 55' 18" Prop. log...... -2° 553 Difference........... 1° 3' 1" Prop. log...... -1° 088 Approximate interval. 1° 54' 24" Prop. log...... -1° 088
The difference of proportional logarithms at moon and 3° is 12°; and on looking in the table for second differences down the column headed ('), and along the column on the left of which is 2°, the nearest to the approximate interval, we find 3° the correction, which is to be added, as the proportional logarithms are decreasing. Hence Astronomy the Greenwich time is 1° 54' 27".
Sect. V.—Of Mean and Apparent Solar Time, and Sidereal Time.
Prob. 1.—Given mean solar time at any given place, to find apparent solar time; and, conversely, given apparent, to find mean solar time.
Find the time at Greenwich, and correct the equation of time for this date. The equation of time so corrected applied to the given time, with the proper sign (as shown in the Nautical Almanac), will give the time required.
Ex. 1.—June 16, 1857, mean solar time 8° 15' P.M., in longitude 25° W.; to find the apparent time.
Ship, June 16 .................................................. 8° 15' Longitude .................................................................. 1° 40 W.
Greenwich, June 16 .................................................. 9° 55'
Equation of time (p. 11, Nautical Almanac) subtractive—
June 16 ................................................................. 19° 43' 17 ................................................................. 32° 34'
Difference ............................................................ 12° 91'
Greenwich date, log. .................................................. -3° 8835 Propor. log. for 12° 9' .................................................. -2° 92283 Propor. log. for 5° 3' .................................................. -3° 0668 Equation of time ....................................................... -0° 0' 24° 73' Ship, June 16 .......................................................... 8° 15' 0'
Apparent time, ship .................................................. 8° 14' 35' 27'
Ex. 2.—Oct. 15, 1857, at 6° 51' P.M., apparent time, in Long. 113° 45' E.; find mean time.
Ship, Oct. 15 .......................................................... 6° 51' Longitude .................................................................. 7° 35 E.
Greenwich, Oct. 14 ..................................................... 23' 16'
Equation of time (p. 1, Nautical Almanac)—
Oct. 14 ................................................................. -13° 57' 22' 15 ................................................................. 14° 11' 12'
Difference ............................................................ 0° 13' 20'
Difference for 1° ...................................................... -5° 50 -23 -1650 -1100
Difference for 23° .................................................... -12° 250 15° is 1° .............................................................. -1° 37 1° is 1° .............................................................. -0° 09
Or 12° 8 nearly.
Hence mean time at ship, 6° 50' 47° 2' P.M., Oct. 15.
Prob. 2.—Given solar time, to find sidereal time.
Rule.—If the given time be mean solar, find the Greenwich date; correct the right ascension of mean sun for this date, and add to the mean time at ship; the result is sidereal time.
If the given time is apparent solar, convert it into mean solar time, as in Problem 1, and proceed as before.
Ex.—Nov. 18, 1857, at 9° 35' 30' A.M., mean time, in Long. 15° 7' W.; required the sidereal time.
Ship, Nov. 17 .......................................................... 21° 35' 30' Longitude .................................................................. 1° 12' 28 W.
Greenwich, Nov. 17th .................................................. 22° 47' 58' Right ascension of mean sun, Nov. 17 ................. 15° 45' 67' 72' Correction for 22° .................................................... 0° 3' 38' 84' " ................................................................. 0° 0' 7' 72' " ................................................................. 0° 0' 0' 16'
Right ascension of mean sun ........................................ 15° 49' 42' 44' Ship mean time ......................................................... 21° 35' 30' Sidereal time .......................................................... 13° 25' 12' 44'
Rejecting 24° in the result. By help of the last problem, we may find what bright stars in the catalogue of the Nautical Almanac will be on the meridian of a given place next after any time, mean or apparent.
For having found the sidereal time at a ship, it is evident that the star whose right ascension is next greater, is the first that will pass the meridian in question.
It is sometimes required to find all the bright stars that will pass a given meridian between two specified hours. We must in that case find the sidereal time for both hours, and all the bright stars in the catalogue whose right ascensions lie between these limits will pass the given meridian between the specified hours.
Ex. 1.—To find what bright star will pass the meridian in Long. 18° 7' W., on November 18, 1857; next after 9h 25m 30s A.M. (See Example to Problem 2.)
Looking in the catalogue of bright stars in the Nautical Almanac, we find that the bright star whose right ascension is the next greater than 13h 25m 12s 44s, is β Corvi, which is the star required.
Ex. 2.—What bright stars in the Nautical Almanac passed the meridian of a place in Long. 64° E., between the hours of 6 and 9, on March 10, 1857?
| Ship, March 10 | 6h 0m 0s | |---------------|----------| | Longitude | 4 16 0 E. | | Greenwich, March 10 | 1 44 0 |
(1.) Right ascension of mean sun—
Greenwich, March 10 | 23h 12m 25s 64s | Correction for 1h 0m | 0 0 9 3655 | " 0 44 | 0 0 7 2281 |
| Ship, March 10 | 6h 0m 0s | |---------------|----------| | Sidereal time | 12h 42m 72s 46s |
(2.) Right ascension of mean sun—
Greenwich, March 10 | 23h 12m 25s 64s | Correction for 4h 0m | 0 0 39 4259 | " 0 44 | 0 0 7 2281 |
| Ship, March 10 | 6h 0m 0s | |---------------|----------| | Sidereal time | 13h 12m 29s 40s |
On looking at the catalogue, we find that the stars whose right ascensions lie between these limits are from β Tauri to ι Argus, which are the stars required.
Prob. 3.—Given sidereal, to find mean solar time.
Rule.—Take out of the Nautical Almanac the right ascension of the mean sun for noon of the given day. Subtract this from the sidereal time, increased, if necessary, by 24 hours; the remainder is mean time nearly. By help of the table which gives the relation of mean solar and sidereal time already referred to, correct this approximate time, and the result will be the mean time required.
Ex.—July 21, 1857, when a sidereal clock showed 6h 45m 35s, find the mean time.
| Sidereal time | 6h 45m 35s | |---------------|------------| | Right ascension mean sun at mean noon | 7 58 47 66 | | Mean time nearly | 22 44 47 34 | | Correction for 22s | 3s 40m 69 | " 0 48s | 0 0 7 88 | " 0 47s | 0 0 13 0 54 60 |
Mean time | 22 44 52 74 |
By means of the above we can find the error of a clock or chronometer by comparing the time shown by it with that of a sidereal clock in an observatory. Thus, Greenwich, November 4, 1857, a sidereal clock showed 18h 15m 30s, and a chronometer showed 3h 28m 15s, the sidereal clock being 3m 13s slow; required the error of the chronometer.
| Sidereal clock | 18h 15m 30s | |----------------|-------------| | Error (slow) | 3m 13s |
| Sidereal time | 18 18 43 5 | |---------------|-------------| | R. ascen. of mean sun at mean noon | 14 54 42 49 | | Mean time nearly | 3 24 1 01 | | Correction for 2s | 29 37 | " 24s | 3 0 33 51 | " 1s | 0 0 48 5 |
Mean time | 3 23 27 59 | Chronometer | 3 28 15 0 | Error (fast) | 0 4 48 5 |
Probl. 4.—To find at what time any heavenly body will pass the meridian of a given place on a given day.
Rule.—From the right ascension of the given star taken out of the Nautical Almanac, increased, if necessary, by 24 hours, subtract the right ascension of the mean sun at mean noon (or sidereal time, as it is called in the Nautical Almanac). This gives the mean time at the ship approximately. Apply the longitude to this, and thus get a Greenwich date, and by this date correct the right ascension of the mean sun, and subtract the quantity so found from the star's right ascension; this will be the time required.
Ex.—At what time will β Corvi pass the meridian of a place in Long. 62° 20' E., on July 24, 1857?
| Right ascension β Corvi, July 24 | 12h 26m 53s 88 | | Right ascension of mean sun at mean noon | 8 8 37 33 | | Sidereal time ship, approximate | 4 18 16 55 | | Longitude | 4 9 20 0 E. | | Greenwich date, July 24 | 0h 8m 58 55 | | Right ascension of mean sun at mean noon | 8 8 37 33 | | Correction for 8° | 0 0 1 31 | " 56° | 0 0 0 16 | | Right ascension of mean sun | 8 8 39 79 | | β Corvi | 12 26 53 88 | | Time of star's passing the meridian | 4 18 15 09 |
SECT. VI.—OF THE MOON'S AUGMENTATION—CORRECTION OF SEMIDIAMETER ON ACCOUNT OF REFRACTION—THE MOON'S MERIDIAN PASSAGE, &c., &c.
1. Of the Moon's Augmentation.—The moon's semidiameter given in the Nautical Almanac is calculated as if seen from the earth's centre. Moreover, on account of the comparatively moderate distance of the moon from the earth, the moon's semidiameter subtends an angle at the earth's surface which sensibly varies with the altitude, being greatest when the moon is in the zenith, and least when in the horizon of the observer. The correction to be applied to the semidiameter from this cause is called the augmentation. It is computed for every degree of altitude, and is to be found in all nautical tables.
2. Contraction of Moon's Semidiameter on account of Refraction.—The correction for refraction varies very rapidly near the horizon, and consequently the moon's lower limb is sensibly more raised than the upper; and the moon's apparent form is that of an ellipse instead of a circle. Hence, in taking a lunar distance when the moon is near the horizon, the moon's semidiameter to be added to the observed distance, as taken from the tables, is greater than it ought to be; and a correction must be applied in consequence of the moon's diameter being an oblique diameter of the elliptic face, and not the horizontal or greatest diameter. This correction is always subtractive, and is given in nautical tables.
3. Of the Moon's Meridian Passage.—In west longitude the moon crosses the meridian later with regard to the sun than at Greenwich, and in east longitude earlier; because the moon's distance from the sun in right ascension is constantly changing. Hence to the Greenwich meridian passage of the moon must be applied a correction for every other place, depending on the longitude. All nautical tables contain this correction.
The corrections to be applied to the observed altitudes and distances of the heavenly bodies have been now sufficiently explained, and the reader will, with a little experience, easily understand how the several corrections are to be taken out of the tables. Index error, dip, and refraction are common to all observed altitudes, and must be applied in the order in which they are here set down. If the body observed be a star, this is all that is required. If a planet, the parallax for the given altitude, taken from the proper table, must be added. If the observed body be the sun, after the dip apply the sun's semidiameter, and then correct for refraction and parallax. These two corrections are frequently given together, under the name of correction in altitude, which is always subtractive, because the refraction is always greater than the parallax. If the observed body be the moon, the semidiameter and horizontal parallax cannot be considered as invariable for 24 hours; they must therefore be corrected for the proper Greenwich date, as already pointed out. To the semidiameter thus corrected must be added the augmentation, from the proper table. The rest of the process is exactly the same as for the sun.
Ex. 1.—Feb. 1, 1857, at 2° 40' p.m., in Lat. 53° 20' N., and Long. 13° 20' E., the observed altitude of the sun's lower limb was 13° 35', the index error was +1° 20', and the height of the eye above the sea 18 feet; to find the sun's true altitude.
| Observed altitude | 13° 35' 0" | |------------------|------------| | Index correction | + 0° 1° 20' | | Dip | - 0° 3° 33' | | Sun's apparent diameter | 13° 32' 47" | | Apparent altitude | 13° 49' 12" | | Correction in altitude | 0° 3° 45' | | Sun's true altitude | 13° 45' 17" |
Ex. 2.—August 24, 1857, in Long. 18° 20' W., the observed meridian altitude of the moon's lower limb was 54° 20' 20", the index correction was +2° 20', and height of the eye above the sea 20 feet; required the moon's true altitude.
| August 24. Time at ship | 0h 0m 0s | |-------------------------|----------| | Longitude | 1° 13' 20" W. | | Greenwich date, August 24 | 1° 13' 20" | | Moon's semidiameter | 14° 54' 4" | | Moon's horizontal parallax, 24th, Noon | 54° 34' 7" | | Midnight | 54° 25' 9" | | 0° 2' | 0° 8' 8" | | Greenwich date, log. moon | 26401 | | Prop. log. for 2° 4' | 3.63321 | | Prop. log. for 5° 8' | 3.68894 | | 4° 6' 22" | 4° 08' 25" |
Part 0° 3' 9" Hence moon's semidiameter = 14° 54' 1" Moon's horizontal parallax = 54° 33' 8" Augmentation = 0° 11' 3" Hence semidiameter = 15° 54' 4"
Observed altitude = 54° 20' 20" Index correction = +0° 2° 20' Dip = -0° 4° 24' Semidiameter = 0° 15' 54" Apparent altitude = 54° 43' 21' 4" Correction in altitude = 0° 30' 30" True altitude = 55° 14' 10" 4"
**CHAP. III.—ON FINDING THE LATITUDE.**
**SECT. I.—OF FINDING THE LATITUDE BY MERIDIAN ALTITUDES.**
1. **By a single Meridian Altitude above the Pole.**
Subtract the true meridian altitude (corrected from observed meridian altitude) from 90°; the result is the zenith distance—which mark N. or S., according as the zenith is north or south of the observed heavenly body: find also the declination, which must be marked N. or S. If the zenith distance and the declination have the same name, their sum will be the latitude, with the name of either. If they be of different names, take their difference, which will be the latitude, with the name of the greater.
The process will differ in detail, according to the nature of the body observed.
(1.) If it be a fixed star, the altitude must be corrected only for index correction, dip, and refraction; and the declination is given at once by the tables.
(2.) If it be a planet, the declination must be corrected by getting a Greenwich date, applying the proportional increase or decrease for this date. If great accuracy is required, the semidiameter and horizontal parallax must also be found, and the altitude corrected for these.
(3.) If it be the sun, the Greenwich date must be got, and the declination corrected for this; the altitude must be corrected for index, dip, semidiameter, refraction, and parallax.
(4.) If it be the moon, a Greenwich date must be got. If it be only known that the observed altitude is a meridian altitude, the meridian passage at Greenwich must be corrected for the longitude, and the Greenwich date obtained by applying the longitude in time as already shown. Th moon's semidiameter, horizontal parallax, and declination, must be corrected for the Greenwich date, and the true altitude and declination thus found. When the true altitude and declination are found, the remainder of the process is according to the rule given above, and is the same for all.
Ex. 1.—May 20, 1857, the observed meridian altitude of Castor was 49° 20' 30" (zenith N. of star), the index correction was -3° 40", and the height of the eye above the sea was 18 feet; required the latitude.
| Observed altitude | 49° 20' 30" | |-------------------|-------------| | Index error | - 3° 40" | | Dip | - 0° 4° 11" | | Correction for refraction | 0° 5° 50" | | True altitude | 49° 11' 49" | | Zenith distance | 90° 0° 0" | | From Nautical Almanac, declin. of Castor | 32° 12' 17" N. | | a° Geminorum on May 20 | 73° 0° 12" N. |
Ex. 2.—October 10, 1857, the observed meridian altitude of a Ophiuchi was 54° 20' 30" (zenith, N. of the star), the index correction was -3° 20", and the height of the eye above the sea was 16 feet; required the latitude.
| Observed altitude | 54° 20' 30" | |-------------------|-------------| | Index correction | - 3° 20" | | Dip | - 0° 3° 56" | | Correction in altitude | 0° 4° 42" | | True altitude | 54° 12' 32" | | Zenith distance | 90° 0° 0" | | Declination of a Ophiuchi | 35° 47' 28" N. | | Latitude | 48° 27' 30" 4" N. |
Ex. 3.—January 6, in Long. 59° 30' E., the observed meridian altitude of the sun's lower limb was 60° 20' 20" (zenith N. of the sun), the index correction was +4° 10", and the height of the eye above the sea was 10 feet; required the latitude.
| January 6, ship | 0h 0m 0s | |----------------|----------| | Longitude | 3° 58' 0" E. | | Greenwich, January 5 | 2° 20" | | Sun's declination, Jan. 5 | 22° 35' 43" 6" S. | | Diff. | 0° 7° 77" | | -0° 7846 | 1° 40' 30" | | 1° 48' 14" Part | 0° 6° 6" | | True declination | 22° 29' 37" 8" S. | | Sun's semidiameter | 16° 18' 2" | Finding the Latitude.
Observed altitude ........................................... 59° 20' 36" Index .................................................................. +0 4 10 Dip ...................................................................... -0 3 7 Semidiameter ..................................................... 60 21 33 Correction in altitude ........................................... -0 0 28 Zenith distance .................................................... 29 22 36 8 N. Declination .......................................................... 22 29 37 6 S. Latitude ............................................................... 6 52 59 2 N.
Ex. 1.—April 4, in Long. 55° 20' E., the observed meridian altitude of the moon's lower limb was 58° 40' 10" (zenith 8° of the moon), the index correction was +3° 20", and height of the eye above the sea was 9 feet; required the latitude.
Moon's meridian passage, April 4 ......................... 9h 0m 5 Corr. ................................................................. 0 47 1 Meridian passage (ship) ........................................ 8 53 5 = 9h 53m 30s Longitude ............................................................ 3 41 20 E. Ship, April 4 ....................................................... 5 12 10
Moon's Horizontal Parallax Declination. Ap. 4, Noon.15° 14'9 Noon.55° 49'7 5°.17° 25' 42" 6 N. Mid.15 10'4 Mid.55 33'4 6°.17 13 19 4 Diff. 0 4'5 Diff. 0 16'3 0 12 23 2 -36318 -36318 -69298 330921 282124 118243 374339 318442 185411
Part. ........ 0 2 Part 0 7:1 Part. 0 0 2:1 Semi. ....... 15 12'9 H.P.55 42'6 Decl.17 23 11 6 N. Augment. .... 0 12'9 Semidiam. .. 15 25'8
Observed altitude .............................................. 58° 40' 10" Index .................................................................. +0 3 20 Dip ...................................................................... -0 2 57 Semidiameter ..................................................... 58 40 33 Correction .......................................................... 0 15 28 58 55 48 8 0 27 49 0 0 21 59 23 58 8 90 0 0 39 36 1:2 S. Declination .......................................................... 17 23 11 6 N. Latitude ............................................................... 13 12 49 6 S.
2. By a single Observed Altitude below the Pole.
Rule.—Correct the altitude, and to it add 90°, and from this sum subtract the declination; the remainder is the latitude.
Ex. 1.—June 8, 1857, the observed meridian altitude of β Chamaeleontis, under the S. pole, was 29° 55' 40", the index correction was -2° 40", and the height of the eye above the sea was 15 feet; required the latitude.
Observed altitude .............................................. 29° 50' 40" Index correction .................................................. -0 2 40 Dip ...................................................................... -0 3 49 Correction in altitude ........................................... -0 1 41 True altitude ....................................................... 29 50 11 90 0 0 Declination of β Chamaeleontis .............................. 119 48 30 S. Latitude ............................................................... 41 17 0 S.
Ex. 2.—Nov. 20, 1857, the observed meridian altitude of α Draconis under the N. pole, was 31° 40' 20", the index correction was -2° 20", and height of the eye above the sea, 10 feet; required the latitude.
Observed altitude .............................................. 31° 40' 20" Index .................................................................. +0 1 20 Dip ...................................................................... -0 4 24 Correction in altitude ........................................... -0 1 34 True altitude ....................................................... 31 35 42 90 0 0 Declination of α Draconis, Nov. 10 ......................... 121 35 42 N. Latitude ............................................................... 39 45 33 N.
In the following example, the altitude is observed by means of an artificial horizon.
June 8, 1857, in Long. 1° 6' W., the observed altitude of the sun's lower limb (in quicksilver horizon) was 121° 9' 4", index correction 1° 20"; required the latitude.
Ship, June 8 ....................................................... 0h 0m 0s Longitude ............................................................ 0 4 24 W. Greenwich, June 8 ............................................... 0 4 24 Sun's declination, June 8 ....................................... 22° 52' 43" N. 9 ................................................................. 22 57 50 9 2:55630 .......................................................... 0 0 7 8 1:54487 .......................................................... 0 0 0 9 4:09117 .......................................................... 0 0 0 9 Sun's declination ................................................ 22 52 43 N.
Semidiameter ..................................................... 15' 47" 3
Observed altitude .............................................. 121° 9' 40" Index .................................................................. -0 1 20 2:121 8 20 60 34 19 Semidiameter ..................................................... 0 15 47 3 60 40 57 3 Correction in altitude ........................................... -0 0 28 True altitude ....................................................... 60 49 29 3 90 0 0 Zenith distance .................................................... 29 10 30 7 N. Declination .......................................................... 22 52 43 2 N. Latitude ............................................................... 52 3 14 6 N.
SECT. II.—BY OBSERVED ALTITUDES OUT OF THE MERIDIAN.
1. By Observations of the Altitude of the Pole-Star (a Polaris).
The tables for this purpose are given in the Nautical Almanac, pp. 513 to 516.
Rule.—From the observed altitude, corrected for the error of the instrument, refraction, and dip of the horizon, subtract 2'.
Reduce the mean time of observation at the place to the corresponding sidereal time.
With the sidereal time found, take out the first correction with its proper sign. If the sign be +, the correction must be added to the reduced altitude; but if it be -, it must be subtracted; in either case the result will give an approximate latitude.
With the altitude and sidereal time of observation, take out the second correction; and with the day of the month and the same sidereal time, take out the third correction. These two corrections, added to the approximate latitude, will give the latitude of the place.
Ex.—January 20, 1857, at 9h 40m p.m. (mean time nearly), in Long. 30° 20' E., the observed altitude of Polaris was 31° 40' 20", the index correction was -2° 20", and height of the eye above the sea, 10 feet; required the latitude.
Ship, Jan. 20 ....................................................... 9h 40m 0s Longitude ............................................................ 2 1 20 E. Greenwich, Jan. 20 ............................................... 7 38 40 3. By Observed Altitudes of a Heavenly Body very near the Meridian.
The sun is the body most usually observed for the purpose. The apparent time from noon or hour angle is supposed to be known; as also the latitude by account, which is supposed to differ from the true latitude only by a small quantity.
Rule.—Add together the tabular logs. of cosines of the latitude by account, and of the declination of the observed body. To these add log. rising, if a table of log. rising be used; or, $6\cdot301030 + \log.$ haversine hour angle, if a table of haversines be used; or twice the log. sine of half the hour angle with the same constant log., if a table of log. sines only be used.
In all cases cast out the tens from the result, and the remainder is the logarithm of a natural number, which, added to the natural sine of the true altitude, will give the natural cosine of the meridian zenith distance. The declination applied to this—as already pointed out in preceding problems—will give the latitude. The hour angle in this rule must be found by applying to the mean time at the ship the equation of time corrected for the Greenwich date.
If the latitude thus found differ much from the latitude by account, the work must be repeated, using the latitude found instead of latitude by account.
Ex.—October 11, 1857, in Long. 114° E., and Lat. by account 46° 10' N., the chronometer showed 4h 38m 10s, being slow on Greenwich 14s. 10s. The altitude of the sun's lower limb was observed to be 32° 25' 20", the index error was -2° 20', and the height of the eye above the sea 16 feet; required the latitude.
| Chronometer | 4h 38m 10s | | Error (slow) | 0 14 10 | | Equation of time | October 10... + 0h 12m 59s | | 4 52 29 | 11... 0 13 15s | | 12 0 0 | 0 0 15s |
Greenw. Oct. 10... 16 32 20
Sun's declination,
October 10... 6° 44' 05" 9
= 11... 6 43 5
-15318 0 21 42 6
-28142
1°7166 0 14 59
Declination S... 6 58 59 9
Semi-diameter ... 0 16 4
Lat. by acc... 46 16 4
Declination ... 6 58 57 9
Log... 0 41 30 61
Equation of time 0 13 10 61
Chron. showed... 4 38 10
Error (slow)... 14 10
Greenw. m.t. 12+ 4 52 20
Longitude... 7 36 0 E.
Equa. of time... 0 13 10 61
Hour angle... 0 41 30 61
Log. rising... 19 837225
Nat. number by table of log. rising, 1124.
4. By two Observed Altitudes of the Sun, and the Time between; having given also the Latitude by Account (Douce's Method).
Rule.—To find the hour angle corresponding to the greater meridian altitude—i.e., the hour from apparent noon, at which the greater altitude is observed.
To the log. secant of the latitude by account, add the log. secant of the sun's declination (the mean between the declinations at the first and second observation); this, rejecting 20, is called the logarithm ratio. To this add the log. of the difference of the natural sines of the two altitudes, and the logarithm of the half-elapsed time from the proper column (taken from nautical tables, which are furnished with special tables for this purpose).
Find this sum in the column of middle times, and take out the time answering thereto; the difference between this and the half-elapsed time will be the time from noon, when the greater altitude was observed.
Or from any table:
Find the logarithm ratio as before.
Subtract each of the true altitudes from 90°, to get the true zenith distances, and take half their sum and half their difference; and to the logarithm ratio add the sum of the log. of the sines of this semi-sum and semi-difference, and from this sum subtract the log. sine of the half-elapsed time; the remainder is the log. sine of middle time, which find from a table of log. sines, and the difference of the middle time and half-elapsed time is the time from noon, at which the greater altitude is observed.
We have now an observed altitude and the time from noon or hour angle corresponding; the remainder of the solution is therefore the same as in the last case, except that the log. ratio, being the sum of the logarithms of the secants, may be used instead of the sum of the logarithms of the cosines, with the negative sign. If the latitude so found differs much from the latitude by account, the work must be repeated, using the computed latitude for the latitude by account.
This rule is only approximate, and must be used under the following restrictions:—1. The observations must be taken between nine in the forenoon and three in the afternoon. 2. If both the observations be made in the forenoon, or both in the afternoon, the interval must not be less than the distance of the time of observation of the greatest altitude from noon. 3. If one observation be in the forenoon and the other in the afternoon, the interval must not exceed 4h; and in all instances the nearer to noon the greater altitude is observed the better. 4. If the sun's meridian zenith distance be less than the latitude, the limitations are still more con- Finding the Latitude.
If the latitude be double the meridian zenith distance, the observations must be taken between half-past nine in the morning and half-past two in the afternoon; and the interval must not exceed 3½ hours. The observations must be taken still nearer to noon if the latitude exceed the zenith distance in a greater proportion.
As the ship is generally in motion between the two observations, it is necessary to apply to the first altitude a correction for its run in the interval, so as to obtain what the altitude would have been if observed at the same time as the second.
If θ be the angle between the course of the ship and the bearing of the sun, the correction for run is \( y = \pm m \cos \theta \), where \( m \) is the number of minutes in the angle subtended by the ship's run,—i.e., number of miles run,—and \( \theta \) the angle between the bearing of the sun and the course; and the sign + or − is to be used according as the ship is moving towards or from the heavenly body: in the former case \( \theta \) is less than 90°, and in the latter greater. It is evident, from the form of the expression, that this correction may be found in a traverse table by looking out the difference between the course and bearing, or what it wants of 8 points for a course, and the ship's run as a distance; the corresponding difference of latitude, multiplied by 60, will then be the number of seconds in the correction for run, which is to be added or subtracted according as the angle is less or greater than 8 points or 90°. Thus, the bearing of the sun was S.E., the run was S.S.E. 18 miles; required the correction for run. Here the angle between the bearing and the course is 2 points, and the distance 18 miles; on entering a traverse table with these, we find the true difference of latitude 16°6′. Hence the correction for run is +16°12′.
Ex.—April 23, 1857, in Lat. 44°20′ N., Long. 35°50′ E., the following double altitude of the sun was observed:
| Mean Time, nearly. | Chronometer. Obs. Alt. Sun's L.L. True Bearing. | |-------------------|-----------------------------------------------| | 10h 40m A.M. | 8° 30' 20" 20° 53° 5' 45" S.E. by S. | | 2 40 P.M. | 0 29' 40" 41 0 30' S.W. by W. |
The run of the ship in the interval was S.S.E. 25 miles, the index correction was −4°40′, and the height of the eye above the sea 15 feet; required the true latitude at the last observation.
The angle between the true bearing at first observation and ship's course is 1 point; the true difference of latitude, corresponding to distance 25 and course 1 point in traverse table, is 24°5′. Hence correction for run to be applied to the greater altitude is +24°30′.
First observation—Second observation—
| Ship, April 19, 22h 40m 0s | Ship, April 20, 2h 40m 0s | |----------------------------|---------------------------| | Longitude... 2 23 20 E. | Longitude... 2 23 20 E. | | Greenwich, Ap.19, 16 40 | Greenwich, Ap.20, 16 40 | | Semi-diam. Sun's Declin. | Semi-diam. Sun's Declin. | | April 19, 11°15'16"7 | April 20, 11°35'53"8 | | 15°56'9 | 11°56'53"8 | | Zenith dist. | Zenith dist. | | 90°00 | 90°00 | | Semi-sum zenith dist. | Semi-sum zenith dist. | | 2°85 15'57"5 | 2°85 15'57"5 | | Log. sine | Log. sine | | 9-830813 | 9-830813 |
Or, if a table of logarithmic sines, &c., only be used, we have—
| First obs. alt... 53°5'45" | Second obs. alt. 41°0'30" | |-----------------------------|----------------------------| | Index... −9 4 40 | Index... −9 4 40 | | Dip... 53 1 5 | Dip... 40 55 50 | | −9 3 49 | | | 52 57 16 | | | Semidiameter... +0 15 55"9 | Semidiameter... 0 15 55"6 | | 53 13 12"9 | | | Cor. in alt... −0 0 38 | Cor. in alt... 0 1 0 | | 53 12 34"9 | | | True alt... 53 37 49 | | | True alt... 41 6 57"6 | |
Similarly, by taking this approximate latitude, we shall get the true latitude, 44°38′ nearly, as before.
1 See Mackay's Treatises on Longitude and Navigation, &c.; Requisite Tables, 3d edition; Mendoza Rico's Tables; Norte's and Riddle's Treatises on Navigation, &c. Finding the Latitude.
5. By any Two Altitudes of the same or different Heavenly Bodies, and the Polar Angle between them.
In this case the declinations of the heavenly body or bodies at the two observations are supposed to be known.
If the same body be observed at different times, the polar angle is the elapsed time measured sidereally, if necessary.
If different bodies be observed at the same time, this angle is the difference of their right ascensions.
If different bodies be observed, but not at the same time, to the right ascension of the first observed body, add the elapsed time measured sidereally, and the difference between this sum and the right ascension of the second observed body is the polar angle required.
The polar angle being given, the following rule will give the latitude:
(1.) If the sun be the body observed: from the declination find the polar distance by subtracting it from 90°, if of the same name with the latitude; and adding it, if of an opposite name.
Add together log. sine polar distance at the greater bearing, log. sine polar distance at lesser bearing, and log. haversine of polar angle; reject 10 from the index; the result is the haversine of an arc, which look out, and call arc (1).
(2.) If two stars be observed: find the polar distances; add together log. sine polar distance at greater bearing, log. sine polar distance at lesser bearing, and log. haversine of the polar angle; the result is the haversine of an arc, which find from the table, and add its versed sine to the versed sine of the difference of the polar distances; the result is the versed sine of an arc, which is arc (1), as before.
Find the difference of the polar distances, and take the difference and the sum of this quantity and of arc (1). Add together log. cosecants of the arc (1) and of polar distance at greater bearing, and the halves of the log. haversines of the two arcs just obtained; the sum, rejecting 10 in the index, is the log. haversine of an arc, which take from tables, and call arc (2).
Again, take the difference of arc (1) and the zenith distance at greater bearing, and take the sum and difference of this arc and the zenith distance at less bearing. Add together log. cosecants of arc (1) and of zenith distance at greater bearing, and the halves of the log. haversines of the arcs just found; and the result, rejecting 10 in the index, is the log. haversine of an arc, which call arc (3).
Arc (4) is the difference between arc (2) and arc (3); or the sum if the arc joining the heavenly bodies at the two observations passes between the zenith and the pole.
Add together log. sine polar distance at greater bearing, log. sine zenith distance at greater bearing, and log. haversine arc (4); the sum, rejecting 10 in the index, is the log. haversine of an arc, which call arc (5).
Add together versed sine of the difference of the zenith distance and polar distance at greater bearing, and the versed sine of arc (5). The result is the versed sine of the colatitude. Find this from the tables, subtract it from 90°; and the result is the latitude.
This method of finding the latitude, which is at once the most general and accurate, is due to the Rev. Dr Inman, to whose excellent work on Navigation the reader is referred for further information respecting it.
Ex. 1.—March 23, 1857, in Lat. 54° 47' N., and Long. 96° E., the following double altitude of the sun was observed:
Mean time, nearly. Chronometer. Obs. Alt. L. L. Bearing. 9h 20m A.M. 10h 26m 26° 0' 40" S.E. 1h 45 m P.M. 2h 39m 84° 6' 20" S.W. by S.
The run of the ship in the interval was S.S.E. 18 miles, the index correction was +2° 10', and the height of the eye above the sea was 18 feet; required the latitude at the last observation.
Ship, March 24. 21° 30m Ship, March 25. 1h 45m Longitude. 6° 32' E. Longitude. 6° 32' E. Greenwich, Mar. 24, 14 58 Greenwich, Mar. 24, 19 13 Ex. 2.—Nov. 30, 1857, in Lat. by account 30° N., the following altitudes of stars were taken at the same time, required the true latitude.
True alt. Orionis. Bearing. True alt. Hydra. Bearing. 42° 45' 15". S.W. 39° 35' 15". S.S.E.
From Nautical Almanac we find—
Orionis, R.A. 5h 7m 4s Declination, S. 8° 21' 57" N.P.D. at G.B. 98 21 57 Declination S. 8° 21' 57"
Hydra, R.A. 9h 20' 36" Declination S. 8° 21' 57" Polar angle ... 4° 12' 52" N.P.D. at L.B. 98 21 57 P.D. at G.B. ... 98 21 57 L sine ... 9-993333 L.B. ... 98 21 57 L sine ... 9-993333 Diff. ... 0° 19' 17" Haversine polar angle ... 9-438871 Haversine arc ... 9-429932
Arc ... 62° 29' 58". Nat. ver. sine ... 62° 29' 58" = 0537993 Nat. ver. sine ... 0° 19' 17" = 0000015 Nat. ver. sine arc (1) ... 0° 19' 17" = 0538256 Arc (1) ... 62° 30' 1".
To find arc (2).
Arc (2) ... 94° 45' 8".
To find arc (3).
True alt. at G.B. ... 42° 45' 15" Log. cosec ... 052069 Z.D. at G.B. ... 98 21 57 Diff. ... 35 51 56 P.D. at L.B. ... 98 21 57 Sum ... 133 54 26 Diff. ... 62 10 34 Haversine arc (2) ... 9733537
To find arc (4).
Arc 2 ... 94° 45' 8" Arc 3 ... 60° 11' 18" Arc 4 ... 34 33 50
To find arc (5).
P.D. at G.B. 88° 21' 57" L sine ... 9-993334 Z.D. at G.B. 47 14 45 Diff. ... 51 7 12 Haversine arc (4) ... 8-945729 Log. haversine arc (5) ... 8-806941 Arc (5) ... 29° 20' 4".
Nat. ver. sine ... 61° 7' 12" = 0372263 Nat. ver. sine, colatitude ... 0500534 Colat ... 60° 2' 7" Lat. N. ... 29° 57' 53"
CHAP. IV.—ON FINDING THE LONGITUDE BY OBSERVATION.
SECT. I.—INTRODUCTION.
The observations necessary to determine the longitude by this method are the altitudes of the sun or other heavenly body, or the distance between the sun and moon, the moon and a planet, or the moon and a fixed star near the ecliptic, together with the altitude of each. The planets used in the Nautical Almanac for this purpose are the following:—Venus, Mars, Jupiter, and Saturn. The stars are, α Arietis, Aldebaran, Pollux, Regulus, Spica Virginis, Antares, α Aquilae, Fomalhaut, and a Pegasi; and the distances of the moon's centre from the sun, and from one or more of these planets and stars, are contained in the xiii. to xviii. pages of the month, at the beginning of every third hour mean time by the meridian of Greenwich. The distance between the moon and one of these objects is observed with a sextant; and the altitudes of the objects are taken as usual with a sextant or a Hadley's quadrant.
In the practice of this method it will be found convenient to be provided with three assistants. Two of these are to take the altitudes of the sun and moon, or moon and star, at the same time that the principal observer is taking the distance between the objects; and the third assistant is to observe the time, and write down the observations. In order to obtain accuracy, it will be necessary to observe several distances, and the corresponding altitudes, the intervals of time between them being as short as possible; and the sum of each divided by the number will give the mean distance and mean altitudes; from which the time of observation at Greenwich is to be computed by the rules to be explained.
If the sun or star from which the moon's distance is observed, be at a proper distance from the meridian, the time at the ship may be inferred from the altitude observed at the same time with the distance. In this case the chronometer is not necessary; but if that object be near the meridian, the chronometer is absolutely necessary, in order to connect the observations for ascertaining the mean time at the ship and at Greenwich with each other.
An observer without any assistants may very easily take all the observations, by first taking the altitudes of the objects, then the distance, and again their altitudes, and reduce the altitudes to the time of observation of the distance; or, by a single observation of the distance, the time being known from which the altitudes of the bodies may be computed, the longitude may be determined.
A set of observations of the distance between the moon and a star or planet, and their altitudes, may be taken with accuracy during the time of the evening or morning twilight; and the observer, though not much acquainted with the stars, will not find it difficult to distinguish the star from which the moon's distance is to be observed. For the time of observation nearly, and the ship's longitude by account being known, the estimated time at Greenwich may be found; and by entering the Nautical Almanac with the reduced time, the distance between the moon and given star will be found nearly. Now set the index of the sextant to this distance, and hold the plane of the instrument so as to be nearly at right angles to the line joining the moon's cusps, direct the sight to the moon, and, by giving the sextant a slow vibratory motion, the axis of which being that of vision, the star, which is usually one of the brightest in that part of the heavens, will be seen in the transparent part of the horizon-glass.
SECT. II.—TO FIND MEAN TIME BY A SINGLE OBSERVATION OF THE SUN OR MOON, OR OTHER HEAVENLY BODY.
Prob. 1.—Given the latitude of a place, the altitude, and declination of the sun, to find the true time and the error of the chronometer.
Rule.—If the latitude and declination are of different signs, take their sum; otherwise take their difference. From the natural cosine of this sum or difference, take the natural sine of the altitude (corrected); find the logarithm of the remainder, and to it add the log. secants of the lati- Finding tude and declination; the sum will be the log. rising of the horary distance of the sun from the meridian, and hence the apparent time will be known; by applying to it the equation of time, the mean time may be found, and the error of the chronometer.
Or, if a table of log. haversines be used, take the difference or sum of the latitude and declination as before. Under this difference place the zenith distance of the sun, and take their sum and difference.
Take the sum of half of the log. haversines of these quantities and of the log. secants of the latitude and declination; the result, rejecting the tens, is the log. haversine of the hour angle.
Or, if a table of sines only be used, add together one-half of the log. sines of half the arcs above found, of the log. secants of latitude and declination; the result is the log. sine of one-half the hour angle.
Ex.—March 25, 1837, at 3h 30m p.m. nearly, in Lat. 53° N., and Long. 32° W., when a chronometer showed 5h 46m 20s, the observed altitude of the sun's lower limb was 23° 12' 10", and the index correction was -4° 20', and the height of the eye above the sea 20 feet; required the true time, and error of chronometer.
Ship, March 25, 3h 30m 0s Longitude, 2° 21 20 Greenwich, March 25, 5h 51 20
Sun's declination, March 25, 1° 54' 32" N. Sun's semidiameter, 16° 36' 7" 20° 21 8 Equation of time +6° 32' 20 Diff., 0 23 33-1 Diff. for 1°, 0 767 -5 -61334 -88328 Diff. for 30° is 383 1° 49' 56" 0 5 44 Sun's dec., 2° 0 16' 9
Equation, 5° 58' 703 = 4° 49'
Sun's obs. altitude, 23° 12' 10" Index correction, -0 4 20 Dip, 23 7 50 Semidiameter, +0 16 37 Correction in altitude, -0 2 13 True altitude, 23 17 16' 7 Zenith distance, 66 42 43
Lat. N., 53° 0' 0" Log. sec., 220337 Declination, 2° 0 16' 9 Log. sec., 000266 Diff., 50 59 43-1 Nat. cosine, 0623360 Nat.sin, 23° 17' 16', 0234031
Hour angle, 3h 29m 24s Equation of time, +0 5 59 Ship's mean time, 3h 35 23 Longitude, 2° 21 20 Greenwich mean time, 5h 56 43 Chronometer showed, 5h 46 20
Or error of chronometer is 10m 23s slow on Greenwich.
Or— Lat., 53° 0' 0" N. Log. sec., 220337 Declination, 2° 0 16' 9 Log. sec., 000266 Zenith dist., 66 42 43 Sum, 117 42 26-3 haversine, 4932397 Diff., 15 43 0-1 Log. haversine hour angle, 9° 28' 945 Hour angle, 3h 29m 24s, as before.
By taking another observation at the interval of a few days, the daily rate of the chronometer may be found.
Pron. 2.—Given the latitude of a place, and the altitude of a known fixed star, to find the mean time of observation and error of chronometer.
The right ascension of the mean sun or sidereal time must be corrected for the Greenwich date of observation. The star's hour angle must then be found as in the last problem. To the hour angle thus found add the star's right ascension, and from the sum (increased, if necessary, by twenty-four hours) subtract the right ascension of the mean sun; the remainder is the mean time at the place at the instant of observation.
Ex.—January 16, 1837, at 8h p.m. (mean time, nearly), in Lat. 49° 57' N., and Long. 32° 10' W., when a chronometer showed 10h 23m, the observed altitude of Regulus E. of meridian was 8° 20' 30", the index correction was -5° 20', and the height of the eye above the sea, 20 feet; required the mean time, and error of chronometer on Greenwich mean time.
Ship, Jan. 16, 8h 0m 0s Longitude, 2° 8 40 W. Greenwich, Jan 16, 10h 8 40 Observed altitude, 8° 20' 30" Index correction, -5 20 Dip, 8 15 10 Refraction, 8 10 46 True altitude, 8 4 19 True zenith distance, 81 55 41 Star's right ascension, 10h 0m 46s Star's declination, 19° 43' 28" Right ascension of mean sun, 19h 43' 28" Correction for 10h, 0 1 38-6 " 8m, 0 0 1-3 " 40', 0 0 0-1 True right ascension of mean sun, 19h 45' 8"
Latitude, 49° 57' 0" N., Log. sec., 188493 Declination, 12° 39' 50" N., Log. sec., 010696 Diff., 36 57 10 Zenith dist., 81 55 41 Sum, 118 62 51 Half, 59 26 25-5 Log. sin., 9-935054 Difference, 44 58 31 Half, 22 29 15 Log. sin., 9-582011
Hence half hour angle, 3h 4m 49s Star's hour angle, 17 50 22 " right ascension, 10 0 46 Sun's right ascension, 27 51 8 Ship's mean time, 8 6 0 Longitude, 2° 8 40 W. Greenwich mean time, 10h 14 40 Chronometer, 10h 23 30 Error (fast on Greenwich), 0 8 50
SECT. III.—TO FIND THE LONGITUDE BY MEANS OF A CHRONOMETER.
In order to find the longitude at sea by means of a chronometer, its daily rate in mean solar or sidereal time must be established by observations made at some particular place, and its error ascertained for the meridian of that or of any other known place.
An observatory is the most proper and convenient place for this purpose, as there the rate and error may be both determined with the utmost accuracy by equal altitudes, or transits over the meridian of the sun or stars. But if an observatory is not adjacent, the rate and error of the chronometer may be found by altitudes taken daily for several Finding days from the horizon of the sea, or by the method of reflection from an artificial horizon.
If by these observations the daily rate is found to be nearly the same—that is, if the chronometer gains or loses nearly the same portion of absolute time daily—it may be depended on for finding the longitude; but if its rate is unequal, it must be rejected, as the longitude inferred from it cannot be expected to be accurate.
It would be proper to have two chronometers, and that they should be wound up at different stated times of the day, so that if one should be found stoppeth through neglect in winding up or otherwise, it may be set by the other, observing to apply the former interval of time between them, and the change in their rates of going in that interval.
PRON.—To find the longitude of a ship at sea by a chronometer.
Let several altitudes of the sun or a fixed star or planet be observed, and find the true mean altitude; with which, and the ship's latitude, and declination of observed body, compute the mean time of observation as in sect. ii.
To the mean of the times of observation, as shown by the chronometer, apply the error and accumulated rate. Hence the mean time under the meridian of the place where the error and rate were established will be known; to which apply the difference of longitude in time between that place and Greenwich, and the mean time of observation under the meridian of Greenwich will be obtained.
The difference between the time at the place of observation and that at Greenwich will be the longitude of the ship in time; and it is east or west according as the time is later or earlier than the Greenwich time.
Ex. I.—May 29, 1857, at 3 p.m. (mean time nearly), in Lat. 30° 20' S., and Long. by account 155° 10' E., when a chronometer showed 4° 40' 50", the observed altitude of the sun's L.L. was 22° 16', the index correction was -7° 10', and the height of the eye above the sea was 20 feet; required the longitude.
On May 20 the chronometer was fast on Greenwich mean time 4° 50', and its daily rate was 4° 5 losing.
Ship, May 29 ........................................... 3° 0' 0" Longitude .................................................. 10 20 40 E. Greenwich, May 29 ..................................... 16 39 30 Chronometer daily rate .................................. 2° 5 losing
Accumulated rate ........................................ 24° 1666
Chronometer showed .................................... 4° 40' 50"
Original error (fast) ...................................... 0 4 50
Greenwich mean time .................................... 16 35 24 17
Sun's declination, Equation of time, subtractive— May 29, N. .............................................. 21° 32' 43" 9 30 .......................................................... 21° 48' 46" 1
0 9 22
Sun's declination, Equation of time, subtractive— May 29, N. .............................................. 21° 32' 43" 9 30 .......................................................... 21° 48' 46" 1
0 9 22
Sun's semidiameter, 15° 48" 5.
Observed altitude ........................................ 22° 10' 0"
Index correction ......................................... 0 7 10
Dip .......................................................... 0 4 24
Semidiameter ............................................... 0 15 48 5
Correction in altitude ..................................... 0 2 14
Longitude .................................................. 155° 10' 42" 45 E.
Ex. 2.—August 20, 1857, at 08 30' A.M. (mean time nearly), in Lat. 50° 20' N., and Long. 142° 15' E., when a chronometer showed 2° 41' 13", the observed altitude of a Aquila, west of meridian, was 36° 59' 50", the index correction was +6° 20', and height of the eye above the sea 20 feet; required the longitude.
On August 4, at noon, the chronometer was slow on Greenwich mean time 17° 50', and its daily rate was 4° 5 losing.
Ship, August 19 ........................................... 12° 30' 0" Longitude .................................................. 9 29 0 E. Greenwich August 19 ..................................... 3 1 0 Interval ..................................................... 154 3 Daily rate .................................................. 4 5
Accumulated rate ........................................ 1 8
Chronometer showed .................................... 2 41 13
Original error .............................................. 0 17 50
Greenwich mean time .................................... 3 0 11
Observed altitude ........................................ 36° 59' 50"
Index .......................................................... 0 6 20
Dip .......................................................... 0 4 24
Refraction .................................................. 0 1 46
True altitude ............................................... 37 0 29
True zenith distance ....................................... 52 59 31
Right ascension of mean sun (sidereal time)— Aug. 19 .................................................. 9° 51' 7" 80 Part for 3° .................................................. 0 0 29 57 11° .......................................................... 0 0 0 0 03
Star's right ascension ..................................... 19h 43m 51s 33 Star's declination .......................................... 8° 29' 43" N.
Latitude .................................................... 50° 20' 0" N. Log. secant .................................................. -194962 Declination .................................................. 8° 29' 43" N. Log. secant .................................................. -004791
Zenith dist .................................................. 52 59 31
Sum .......................................................... 94 49 48 Half .......................................................... 47 24 54
Difference .................................................. 11 9 14
Half .......................................................... 5 34 37
Log. sine .................................................... 8 857594
Log. sine half-hour angle .................................. 9 527193
Half-hour angle ............................................ 11° 18' 41" 6
Hour angle .................................................. 2 37 23 2
Star's right ascension ..................................... 19 43 51 33
Right ascension of mean sun ......................... 9 51 37 4
Ship, mean time ........................................... 12 29 37 13
Longitude .................................................. 142° 21' 31" E. Finding Sect. IV.—To find the longitude by means of a lunar distance.
Prob. 1.—Given the apparent distance between the moon and sun, or a fixed star or planet, and the apparent and true altitudes of these bodies, to find the true distance, or, as it is called, to clear the distance.
Rule 1. (Borda's method).—Add together the logarithmic cosines of the true altitudes, the logarithmic secants of the apparent altitudes, the logarithmic cosines of one-half the sum of the apparent altitudes and apparent distance, and of this last arc, less the apparent distance. From one-half of this sum subtract the logarithmic cosine of one-half of the true altitudes; the result, rejecting 10 in the index, is the logarithmic sine of an arc. Find the logarithmic cosine of this arc, and add to it the logarithmic cosines of one-half of the true altitudes; the result, rejecting 10 in the index, is the logarithmic sine of one-half of the true distance required. Or,
Rule 2.—Find the auxiliary angle A (which is given in the nautical tables of Inman, Norie, Riddle, and others).
Take the versed sine of the difference of the true altitudes. To it add the versed sines of the sum and difference of the auxiliary angle A and the apparent distance; under the difference of the apparent altitudes place the auxiliary angle A, and take their sum and difference; add together the versed sines of these two arcs, and subtract from the former quantity; the result is the versed sine of the true distance.
Ex.—Required the true distance of the moon from the sun, having given—
App. alt. sun ... 34° 21' 32" True alt. sun ... 34° 20' 14" App. alt. moon ... 57 11 22 True alt. moon ... 57 40 11 App. distance of centres ... 35° 47' 24"
By Rule 1—
True alt. moon ... 57° 40' 11" L cos ... 9-782227 True alt. sun ... 34 20 14 L cos ... 9-916839 App. alt. moon ... 57 11 25 L sec ... -266131 App. alt. sun ... 34 21 32 L sec ... -083267
App. distance ... 35 47 24
2127 20 21 63 40 10 L cos ... 9-946941 35 47 24 27 52 45 L cos ... 9-946353
Sum of true alt ... 92° 0' 25" Half ... 46 0 12 L cos ... 9-841746 L sin ... 9-952133
Arc ... 63° 35' 10" L cos 63° 35' 10" ... 9-648216 L cos 46 0 12 ... 9-841746 L sin half true dist ... 9-489062
Half true distance ... 17° 59' 37"
True distance ... 35 59 14
Rule 2.—Auxiliary angle A (taken from the tables) being 60° 23' 16".
Moon's true alt ... 57° 40' 11" Sun's true alt ... 34 20 14 Diff ... 23 19 57 Vert ... 0081777 A ... 60 25 16 App. distance ... 35 47 24
Sum ... 96 12 40 Vert ... 1108192 Diff ... 24 37 52 Vert ... 0090990
App. alt. sun ... 34 21 32 App. alt. moon ... 57 11 25 Diff ... 22 49 53 A ... 60 25 16
Sum ... 83 15 9 Vert ... 0882506 Diff ... 37 35 23 Vert ... 0207601
True dist. 35° 59' 18" Nat. vert ... 0190862
Prob. 2.—To find the longitude by a lunar observation. Get a Greenwich date, and to this date take out the moon's horizontal parallax and semidiameter. Increase the semidiameter for the augmentation corresponding to the moon's altitude.
Find the apparent and true altitudes of the centre of the moon and of the sun or star which is observed, and the apparent central distance. From these elements compute the true distance as in the last problem, and find the mean time at Greenwich corresponding to it, as shown in chap. ii., sect. iv., p. 38.
If the sun or star be not too near the meridian at the time of observation, compute the mean time at the ship from its altitude; if it be too near, compute the mean time from the moon's altitude. The difference between the mean times of observation at the ship and Greenwich will be the longitude of the ship in time, which is E or W, according as the time at the ship is later or earlier than the Greenwich mean time.
Ex. 1.—January 3, 1857, at 2° 30' P.M., in Lat. 49° 20' N., and Long. 15° 40' E., the following lunar was taken—
Obs. alt. Obs. alt. Obs. dist. N.L. Sun's alt. L. Moon's alt. Sun's dist. N.L. 24° 40' 10" 21° 28' 40" 20° 28' 10" Index error... +0 1 20 -0 1 10 0 1 20
Height of eye above the sea, 18 feet; required the longitude.
Ship, Jan. 3, 2° 30' 0" Longitude ... 1 240 E. Greenwich, Jan. 3, 1 27 20
Sun's declination Equations of Jan. 3, 8° 29' 45" 38° 6" +4° 54' 03" " 4, 22 42 24 6" 5 21 33
1:21885 0 6 14 1:21885 0 27 33 1:46055 0 0 26 2:59726 2:67940 0 0 26 3:81611
Moon's semi. 3, moon 16° 9' 2" Moon hor. par. 69° 8' 9" mild. 16 10 7" 69 14 2
-01782 0 1 5 -01782 0 5 3 3:85733 3:30915
4:77515 0 0 1 4:22697 0 0 7
Aug. 0 6
16 15 3
Sun's Alt. Moon's Alt. Lunar Dist. Obs. alt. 24° 40' 10" 21° 28' 40" 20° 28' 10" Ind. cor.+0 1 20 -0 1 10 +0 1 20 Dip ... 24 41 30 21 35 30 20 29 30 Semi ... 0 16 18 0 16 15 App. dis ... 91 2 3 Cor. insalt. 0 1 57 -0 52 23
True alt. 24 51 40 Tr. alt. 22 40 5
65 8 20 true zenith distance.
Aux. angle A ... 60° 11' 46"
To find ship's mean time.
Sun's decl. ... 22° 48' 15" S. Sec. ... 0 03348 Latitude ... 49 20 0 N. Sec. ... 0 18291 Sun's zen. dist ... 65 8 20
Sum ... 137 16 36 Half ... 68 38 18 Sin ... 0 93909 Diff ... 6 59 50 Half ... 3 29 58 Sin ... 0 785504
2:1876022
W 488011 Finding the Longitude.
Half-hour angle... 1° 11' 39" 6
Hour angle... 2 23 19
Equation of time... 4 53" 7
Ship mean time... 2 28 12
To find Greenwich mean time.
Sun's true alt... 24° 51' 40"
Moon's true alt... 23 40 5
Diff... 2 11 35 Vers... 0000726
Apparent dist... 91 2 3
Auxiliary angle... 60 11 48
Sum... 151 13 51 Vers... 1878447
Difference... 30 50 15 Vers... 0141338
Sun's appar. alt... 24 53 37
Moon's do... 21 47 34
Difference... 3 6 3
Auxiliary angle... 60 11 48
Sum... 63 17 51 Vers... 0550161
Difference... 57 5 45 Vers... 0455931
(1.) True distance... 90° 38' 44" 1011269
(2.) Distance at noon... 89 52 28 054
(3.) Distance at 3 P.M... 91 30 14 215
Diff. (1.) and (2.), 6° 46' 16" Prop. log... -59060
" (2.) and (3.), 1° 37' 46" " -26568
Interval... 1° 25' 11" -32492
Greenwich mean time... 1 25 11
Ship mean time... 2 28 13
Longitude... 1° 3 2
Or... 15° 45' 30" E.
Ex. 2.—February 28, 1857, at 7° 40' P.M., in Lat. 55°, and Long. 57° 29' W., the following star lunar was taken:
Obo. Alt. Pollex E. of Mer. Obs. Alt. Moon Lower Limb. Obs. Dist. Further Limb.
60° 10' 0" 46° 10' 0" 70° 40' 20"
Index error -0 2 10 +0 1 20 -0 3 10
The height of the eye above the sea was 20 feet; required the longitude.
Ship, Feb. 28... 7° 40' 0a
Longitude... 3 49 20 W.
Greenwich, Feb. 28... 11 29 20
Right ascension of mean sun,
Feb. 28... 22° 33' 6" 09 Star's R. A... 7° 36' 35"
Part for 11°... 0 1 48 42 Declination... 28 22 10 N.
" 23°... 0 0 47 6
" 20°... 0 0 05
22 34 53 32
Moon's semi. 28, noon... 16° 23' 2 Hor. par. noon... 59° 59' 8
mid... 16 20 0 n mid... 59 48 3
-0 1911 -0 1911
3 52827 2 97273
3 54738 Part 0 31 2 99184 Part 0 11 0
16 20 1 Aug. 0 12 5
16 32 6
Star's Altitude, Obs. alt. 60° 10' 0" Moon's Altitude, Apparent Distance.
Inx. cor.-0 2 10
60 7 50 46 11 20 70 37 10
Dip... -0 4 24 -0 4 24 Semi... -0 16 32 6
App. alt. 60° 3 26 Refr... 0 0 34 Semi... 0 16 32 6
True alt. 60° 2 52 App. alt. 46° 23 28 6
Cor. alt. 0 39 48
T. Z. dis. 29 57 8 True alt. 47° 3 49 6
Vol. XVI.
Auxiliary angle A... 60° 23' 33"
Variation of the Compass.
To find ship's mean time.
Star's dec... 23° 22' 10 N. Sec... 0 055555
Latitude... 55 0 0 N. Sec... 0 241409
Diff... 26 37 50
Star's zen. dis... 29 57 8
Sum... 56 34 58 Half havers... 4 675736
Diff... 3 19 18 Half havers... 3 462142
Hav. hour angle 8° 434852
Hour angle (E. M.)... 22° 44' 1"
Star's R. A... 7 36 35
R. A. mean sum... 22 34 53
Ship mean time... 7 45 43
To find Greenwich mean time.
Star's true altitude... 60° 2' 52"
Moon's... 47 3 49 6
Diff... 12 69 2' 4 Vers... 0025564
App. distance... 70 20 37
A... 60 23 59
Sum... 130 44 35 Vers... 1652539
Diff... 9 56 28 Vers... 0014991
Star's app. altitude... 60 3 25
Moon's... 46 23 28 6
A... 13 39 57 4
Sum... 74 3 50 4 Vers... 0725202
Diff... 40 44 1' 6 Vers... 0314605
Vers. true distance... 1040075
True distance... 69° 42' 13"
Distance at 9h... 71 16 1
" midnight... 60 27 42
First diff... 1° 33' 48" Prop. log... -28307
Second... 1 48 19 Prop. log... -22058
-0 6149
Time from 9h... 2h 36' 14"
9 0 0
Greenwich mean time... 11 36 14
Ship mean time... 7 45 43
Longitude... 57° 37' 45" W.
Chap. V.—OF THE VARIATION OF THE COMPASS.
The variation of the compass is the deviation of the points of the mariner's compass from the corresponding points of the horizon, and is denominated east or west variation, according as the north point of the compass is to the east or west of the true north point of the horizon.
(A particular account of the variation, and of the several instruments used for determining it from observation, may be seen under the article MAGNETISM, where the method of communicating magnetism to compass-needles is also fully described.
Besides the variation, there is also the deviation of the compass arising from the local attraction of the iron on board ship, of which we have already given an account in the former part of this article. This deviation is always taken into account in ships of the Royal Navy, but not always in ships belonging to the mercantile navy. In the latter case the variation is the whole difference between the observed bearing of the sun and the compass bearing; in the former, allowance must be made for deviation. We shall take deviation into account, as it is easy to omit it when it is not required.
To correct the variation for deviation, it will be sufficient to place under the variation, when determined by observation with its proper name, the deviation with a name opposite to its true name. Add these when the names are alike, and subtract the less from the greater if the names are different; and the remainder, with the name of the greater, is the true variation.
Prob. 1.—Given the latitude of a place, and the sun's declination, and the sun's magnetic amplitude, to find the deviation. [Obs.—The amplitude is the distance from the east point at which it rises, or from the west point at which it sets.]
Rule.—To the log. secant of latitude add log. sine of sun's declination; the sum, rejecting 10 from the index, will be the log. sine amplitude, which is cast if the body is rising, and west if it is setting. The variation is the difference between the true and magnetic amplitudes if these be of the same name, and their sum if of different names. Also the variation is east if the true bearing is to the right of the compass bearing, west if the true bearing is to the left of the compass.
Ex.—May 18, 1857, about 5° 25' A.M., in Lat. 51° 3' N., Long. 143° W.; the sun rose by compass E. 6° 40' S., the ship's head being E.; required the variation.
| Ship, May 17 | 17h 25m 0s | |--------------|-------------| | Longitude | 9° 32' 0 W. | | | 26° 57' 0 | | | 24° 0 |
Greenwich, May 18
Sun's declin., 18°...10° 36' 18" N.
| Latitude | 51° 3' N. Sec. | |--------------|----------------| | Sin amplitude| 9-728212 |
True bearing...E. 32° 20' 0" N.
Compass...E. 6° 40' 0" S.
Deviation...8° 50' 0" W.
Variation...47° 50' 0" W.
It may be remarked, that the sun's amplitude ought to be observed at the instant the altitude of its lower limb is equal to the sum of fifteen minutes and the dip of the horizon. Thus, if an observer be elevated 18 feet above the level of the sea, the amplitude should be taken at the instant the altitude of the sun's lower limb is 19° 11'.
Prob. 2.—Given the magnetic azimuth, the altitude and declination of the sun, together with the latitude of the place of observation, to find the variation of the compass.
Rule.—Find the polar distance by adding 90° to the declination if the altitude and declination have unlike names, and subtracting the declination from 90° if they have the same name.
Take the difference of the latitude and altitude, and obtain the sum and difference of this quantity and the polar distance.
To the log. secants of the altitude and declination add the halves of the log. haversines of the last two arcs; the result, rejecting 10 in the index, is the haversine of the true bearing, which take from the table. Or if a table not containing haversines is used, to the log. secants as before add the log. sines of half the arcs obtained as before; one-half of the result is the log. sine of half the true bearing. Double the arc taken out of the table is the true bearing.
Mark the true bearing N. or S., according as the latitude is N. or S., and E. or W., according as the observed body is E. or W. of the meridian. The variation then can be found as before.
Ex.—On May 6, 1857, about 8° 10' 0" A.M., mean time, in Lat. 51° 10' N., Long. 140° W., the sun bore by compass S. 65° 25' E., and the observed altitude of the sun's lower limb at the same time was 28° 30' 10", the index correction was +1° 20", and the height of the eye above the sea 15 feet, the ship's head being N.W.; required the variation of the compass.
| Ship, May 4 | 20h 10m 0s | |-------------|-------------| | Longitude | 9° 20' 0 |
Greenwich, May 5
Sun's semidiameter...0° 15' 53"
Sun's declination, May 5...16° 19' 13" 2 N.
| 0-63985 | 16° 35' 8 | |---------|-----------| | 1-02996 | | | 1-66681 | |
Sun's altitude—
| Obs. altitude | 28° 30' 10" | |---------------|-------------| | Index correction | +0 1 20 | | Dip | -0 3 49 | | Semidiameter | 28 27 41 | | Correction in altitude | -0 1 38 | | True altitude | 28 41 56 |
Latitude...51° 10' 0" Sec...20°2833
Altitude...28° 41' 56" Sec...0°4923
Diff...22° 28' 4
Polar dist...73° 36' 54
Sum...96° 4' 58 Half havers...4°671355
Diff...51° 8' 50 Half havers...4°635150
9°760101
True bearing...N. 99° 37' 22" E.
Magnetic bearing...N. 114° 35' 0" E.
Deviation...4° 50' 0" E.
Variation...19° 47' 38" E.
Chap. VI.—OF THE TIDES.
The theory of the tides has already been explained under the article Astronomy, and will again be further illustrated under that of Tides. In this place, therefore, it remains only to explain the method of calculating the time of high water at a given place.
As the tides depend upon the joint actions of the sun and moon, and therefore upon the distance of these objects from the earth and from each other; and as, in the method generally employed to find the time of high-water, whether by the mean time of new moon, or by the ephemeris, or tables deduced therefrom, the moon is supposed to be the sole agent, and to have a uniform motion in the periphery of a circle whose centre is that of the earth; it is hence obvious that this method cannot be accurate, and by observation the error is sometimes found to exceed two hours. This method is therefore rejected, and another given, in which the error will seldom exceed a few minutes, unless the tides are greatly influenced by the winds. ### Table I.—For Determining the Time of High Water.
| Moon's Transit | 60° | 59° | 58° | 57° | 56° | 55° | 54° | |---------------|-----|-----|-----|-----|-----|-----|-----| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
### Table II.—For Finding the Height of the Tide.
| Time of Transit | Moon's Hor. Par. 60° | Moon's Hor. Par. 57° | Moon's Hor. Par. 54° | |-----------------|----------------------|----------------------|----------------------| | | Multipliers | Multipliers | Multipliers | | h. m. | h. m. | h. m. | h. m. | | 0 0 | 12 0 | 0-935a + 0-149a | 0-883a + 0-117b | | 0 40 | 12 40 | 1-04a + 0-038a | 0-970a + 0-0305 | | 1 20 | 13 20 | 1-132a + 0-009a | 1-000a + 0-0005 | | 2 0 | 14 0 | 1-04a + 0-038a | 0-970a + 0-0305 | | 2 40 | 14 40 | 0-935a + 0-149a | 0-883a + 0-117b | | 3 20 | 15 20 | 0-833a + 0-319a | 0-750a + 0-250b | | 4 0 | 16 0 | 0-668a + 0-527a | 0-557a + 0-413b | | 4 40 | 16 40 | 0-460a + 0-749a | 0-413a + 0-5875 | | 5 20 | 17 20 | 0-284a + 0-938a | 0-250a + 0-750b | | 6 0 | 18 0 | 0-133a + 0-883b | 0-117a + 0-883b | | 6 40 | 18 40 | 0-634a + 1-288a | 0-500a + 1-000a | | 7 20 | 19 20 | 0-600a + 1-277a | 0-500a + 1-000a | | 8 0 | 20 0 | 0-634a + 1-288a | 0-500a + 1-000a | | 8 40 | 20 40 | 0-133a + 1-177a | 0-117a + 0-883b | | 9 20 | 21 20 | 0-284a + 0-938a | 0-250a + 0-750b | | 10 0 | 22 0 | 0-460a + 0-749a | 0-413a + 0-5875 | | 10 40 | 22 40 | 0-688a + 0-527a | 0-557a + 0-413b | | 11 20 | 23 20 | 0-853a + 0-319b | 0-750a + 0-250b | | 12 0 | 24 0 | 0-955a + 0-149a | 0-883a + 0-117b | | | | | | | | | | | | | | | | | | | | | | | | | | TO FIND THE TIME OF HIGH-WATER.
Rule.—Let the approximate time of high-water be found, by taking the corrections for the moon's horizontal parallax for the nearest noon or midnight from Table I. Again, to this time and the given longitude take from the Nautical Almanac the moon's horizontal parallax. Also to the time of the moon's transit over the meridian of Greenwich apply the variation answering to the longitude and daily variation between the given and preceding day if the longitude is E. Subtract this from the transit over the meridian of Greenwich, and the remainder will be the time of transit over the meridian of the given place. But if the longitude be W., the correction answering to the longitude and daily variation of transit between the given and following day must be added to the time of transit over the meridian of Greenwich, to obtain the time of transit over the meridian of the given place.
To the time of high-water, if new and full moon at the given place, add the reduced time of transit over the meridian of the same place, and to the sum apply the equation from the table answering to the time of transit and horizontal parallax formerly found; the result will be the true mean time of high-water required. The apparent time may be found by applying the equation of time, with its proper sign.
Ex. 1.—Required the time of high-water at Leith on Wednesday the 16th of May 1837, in Long. 3° 11' W.
By the rule, the time of high-water will be about six o'clock in the evening. In this case, the moon's horizontal parallax will be 54° 16', and the time of transit 4° 50' mean time, or 4° 54' apparent time by applying the equation of time 3° 50' by addition.
Apparent time of transit of upper meridian... 4° 54' Equation from the table to horizontal parallax 54° 16', and transit 4° 54', subtract... -1° 18' Remainder... 3° 38' Time of high-water at new and full moon... +2° 20' Apparent time of high-water... 5° 58' Equation of time... -9° 4' Mean time of high-water... 5° 52'
If the sum exceed 12° 25', subtract this number from it; if it exceed 24° 50', subtract as before, and the remainder will be the time of high-water in the afternoon of the given day nearly. The time of high-water of the tide preceding may be found nearly by subtracting 25' from it, and the succeeding tide by adding 25' to it. In cases of great accuracy, however, a computation should be made for each tide in a manner similar to that above.
Ex. 2.—Required the time of high-water at Aberdeen on the 21st of June 1837, in Long. 2° 6' W.
As before, the time of high-water will readily be found to be about three o'clock.
Here the horizontal parallax of the moon will be 69° 30', and the mean time of transit on the given day 15h 32m. But as this transit exceeds 12h, it will be necessary to take the time of transit over the under meridian, or, what comes to the same thing, half the sum of the transits on the given and preceding days, or, \( \frac{1}{2} (14h 2m) \).
Correction from the table... -0° 44' Remainder... 14° 18' High-water at new and full moon... +1° 10' Sum exceeding 12h... 15° 28' By rule, subtract... 12° 25' Apparent time of high water... 3° 3' Equation of time... +0° 1' Mean time... 3° 4'
Ex. 3.—Required the depth at Aberdeen at the same time, the rise of spring tides being 19 feet, denoted by \( \alpha \) in Table II., part 1, and that of the neap 14 feet, by \( \beta \).
Now, by Table II., part 1, to transit 15h 2m, and horizontal parallax 60° 30', will be obtained \( 0.917 \times 19 + 0.242 \times 14 = 20.8 \) feet.
Ex. 4.—Required the height of the tide at 3° 15' after high-water.
By part 2, 20.8 x 0.5... = 10.4 feet.
In this manner, the time and rise of the tide may be readily obtained nearly, unless both are much influenced by the strength and direction of the wind.
In the preceding pages we have not considered it our province to supply the reader with the tables which have been made use of in the solution of the several problems—e.g., traverse tables, tables of meridional parts, corrections for dip, parallax, refraction, and log. haversines, log. rising, middle time, half-elaps'd time, &c. For these, and for further information, we refer him to the works on Navigation already mentioned. Some of the rules we have given will be found to differ in some respects from those given in these works. These discrepancies, however, arise entirely from the slightly different form in which the formulae on which the rules are based are made to appear, but not at all on any difference of principle. We have generally retained the forms of the formulae which are best known to astronomical students. For example, we have retained Borda's method of clearing the distance, in lieu of the slight deviations from it which are frequently employed, not merely as being equally correct with the latter, but as possessing some historical interest. In clearing the distance by natural versed sines, it is not difficult to put the formulae in such a form that the sum only of the versed sines may be taken. We have retained the formula in which two of the versed sines appear with a negative sign, as being that with which the readers of Hymers Astronomy are already acquainted. Whenever the reader is familiar with astronomical formulae, we recommend him to study the rules given in these pages with the appropriate formulae before him; these he will find in any work on astronomy treated mathematically. In all cases, however, it is believed that, by a careful study of the rules and directions contained in this article, the reader will find himself possessed of all the information which will enable him to navigate a ship.
(J. W.—Y.)
INLAND.
The mariner's compass, it is believed, was first used at the commencement of the fourteenth century. Its introduction gave a new character to commercial enterprise, as it afforded the means of promoting to an almost unlimited degree the progress of maritime discovery. Since that early period the pursuit of Navigation has not only been the grand object to which the labours of Columbus and of all subsequent explorers of the world have been directed; but the researches of the philosopher, the astronomer, the geographer, the mechanician, and the engineer, have all been instrumental in bringing to maturity and perfection the various branches which constitute the system of Navigation as it now exists.
That system, though made up of many subsidiary parts, may conveniently and naturally be divided into two great departments. One of these is treated of in a separate article under the head of NAVIGATION; the other, which forms the subject of the present treatise, is termed Inland Navigation. It may shortly be defined as that branch of navigation which extends from the sea to the land, and affords the means of transport throughout the interior of a country. To form a correct estimate of the importance of this subject, it must be viewed in connection with the entire system of which it forms a part. For, how can we fully enjoy the benefits of those mighty results of science and of art, by which sailing vessels of all classes are now enabled to transport their cargoes from shore to shore, with comparative ease and safety, and gigantic steamers to traverse the ocean with certainty and despatch, if we do not, in addition to exhibiting the beacon light to welcome the approach of ships to our coasts, afford the means of withdrawing them from the ocean billows into sheltered havens, where their lading may be discharged, and cargoes of our country's produce may be shipped for foreign lands. It should be borne in mind, that it is only when the mariner approaches his destined port that the dangers caused by rocks, shoals, sand-banks, tides, and currents, beset his course; and the means of securing shelter for his vessel, and of opening up a passage into the interior of the country, may be held as embraced under the extensive subject of Inland Navigation.
The article HARBOURS fully discusses the construction of piers and breakwaters; and our present treatise will therefore be confined to Canal and River Navigation.
Under these general heads we propose to give a brief account of canals, as applied to the purpose of transport by means of boats, and also on the larger scale, as affording to sea-borne vessels a sheltered and direct route to their destined ports. Our notice of rivers will embrace the navigation of their upper or landward streams, and also the varied means employed in opening up and rendering navigable their seaward or tidal compartments, which will necessarily lead us to consider the conservation of estuaries and the formation of bars. Viewed even in this restricted light, it will be found that inland navigation forms an extensive and intricate department of hydraulic engineering.
It is proper in the outset to state, that it is not our intention to explain the nature or principles of the varied class of works which the engineer finds it necessary to adopt in carrying out such operations as those to which we have alluded. At the present time, when so much is written on all branches of engineering, such a course would be uncalled for, and would indeed extend the present treatise greatly beyond the limits to which it must necessarily be restricted. For information as to such details, we must therefore direct the reader to the different books to which we shall have to refer, as containing full information on the subjects of which they treat. Our aim is rather to present the reader with a general résumé of the state of our knowledge respecting the practice of engineering, as applicable to inland navigation in all its branches, and to confine such detailed remarks as we may have to offer, to those parts of the subjects only, which are not fully treated in works already published; and here we must express our regret, that although we have many treatises expounding the principles of engineering, nevertheless the engineers of the present day have given comparatively few accounts of the effects that have followed the application of these principles in particular cases. In drawing up the following pages, the writer has found great difficulty in obtaining authentic information on applied engineering; and this must be his excuse for having in some of the sections been obliged to apply, it may be thought too largely, to his own experience for illustrations of his subject.
SECT. I.—CANALS.
It must be obvious to all, that railways, from which we Canals have of late years derived such inestimable advantages, have now in a very great measure superseded, and certainly for the future must prevent, the general extension of canals. The great objections to relying on canals as the medium of regular and uninterrupted internal communication in this country, are the difficulty of obtaining a sufficient supply of water to prevent stoppages during dry seasons, the interruption to which they are exposed from ice during winter, and above all, in these days of express railway trains and electric telegraphs, the very limited speed at which the boats which navigate them can be propelled. Sir John Rennie, in speaking of the successful attempts made to introduce swift boats on canals, and the great improvement that was thereby effected in canal transport, says—"All this, however, came too late; for although it would have been readily acknowledged at an earlier period, and might perhaps for a while have retarded the railway system, yet when once the latter was established, its superiority became manifest, and its progress irresistible." These truly are considerations which make canals, when compared to railways as a means of transport and communication, appear so very disadvantageous, that it may at first sight be considered as uncalled-for to describe, even briefly, a class of works which, in the present day, may be regarded by some readers as almost entirely superseded. But although this remark may perhaps be justly considered applicable to those canals which effect a purely inland communication from town to town, it does not, in any degree, apply to that larger class of works called ship canals, which afford to sea-borne vessels an inland course, and enable them to avoid the dangers of a lengthened coasting voyage—an object of the highest importance to navigation, and one which it is obvious cannot be superseded by a railway. But, independently of this reservation on behalf of these peculiar works, it appears to us that the simple consideration of the great antiquity of navigable canals, their wide-spread introduction throughout the world, the important place which they have so long occupied in the commercial history of every country, and above all, the noble specimens which they afford of hydraulic engineering, should lead us naturally and imperatively to give some notice of their origin and subsequent progress; and this we shall do as briefly as possible, not so much from any feeling that the subject is superseded, or is unimportant, but because it will
1 Transactions of Inst. of Civil Engineers, vol. v., p. 78. be found fully and ably treated in the works to which we shall have occasion to refer.
From the writings of Herodotus, Aristotle, Pliny, and other ancient historians, we learn that canals existed in Egypt before the Christian era; and there is reason to believe that, at the same early period, artificial inland navigation also existed in China. Almost nothing, however, save their existence, has been recorded with reference to these very early works; but soon after the commencement of the Christian era, canals were introduced, and gradually extended throughout Europe, particularly in Greece, Italy, Spain, Russia, Sweden, Holland, and France.
In speaking, however, of the earliest of these works, it is not to be supposed that they resembled the modern canals as now constructed in our own and other countries. Early as inland navigation was introduced, it was not until the invention of canal-locks, by which boats could be transferred from one level to another, that the system was rendered generally applicable and useful; and it has been truly remarked, "that to us, living in an age of steam-engines and daguerreotypes, it might appear strange that an invention so simple in itself as the canal-lock, and founded on properties of fluids little recondite, should have escaped the acuteness of Egypt, Greece, and Rome." Not only, however, had the invention escaped the notice of the ancients, but what is more striking, the several gradations made towards the attainment of that simple but valuable improvement appear to have been so gradual that, like many discoveries of importance, great doubts exist as to the person and even the nation, by whom canal-locks were first introduced. One class of writers attributes the discovery to the Dutch; and Messrs Telford and Nimmo, who are understood to have written the article on Inland Navigation in Brewster's Edinburgh Encyclopaedia, adopt the conclusion that locks were used in Holland nearly a century before their application in Italy; while, on the other hand, the invention has been strongly and not unreasonably claimed for engineers of the Italian school, and, in particular, for Leonardo da Vinci, the celebrated engineer and painter. Without, however, entering into a discussion of this question, which it is now probably impossible to solve, we may safely state, that during the fourteenth century the introduction of locks, whether of Dutch or Italian origin, gave a new character to inland navigation, and laid the basis of its rapid and successful extension. And here it may be proper to remark, that the early canals of China and Egypt, although destitute of locks, do not appear to have been on that account formed on a uniformly level line unadapted to varying heights. It is very doubtful, indeed, if the use of locks has even yet been introduced into China, intersected as it is by many canals of great antiquity and extent; and in order to pass boats from one level to another, the Chinese have, from a very early period, employed stop-gates and inclined planes of rude construction. Nevertheless, the invention of locks was, as already noticed, a most important step in the history of canals; and that mode of surmounting elevations may be said to be almost universally adopted throughout Europe and America. Inclined planes and perpendicular lifts have, it is true, been employed in these countries, as will be noticed hereafter; but the instances of their application are undoubtedly rare.
But in proceeding to illustrate the progress of canals, we may, without tracing their gradual introduction from country to country, remark at once, that we find the French, at the end of the seventeenth century, in the reign of Louis XIV., forming the Languedoc Canal, designed by Riquet, between the Bay of Biscay and the Mediterranean, a gigantic work which was finished in 1681. It is 148 miles in length, and the summit level is 600 feet above the sea; while the works on its line embrace upwards of one hundred locks and about fifty aqueducts,—the whole forming an undertaking which is a lasting monument to the skill and enterprise of its projectors; and with this work as a model, it seems strange that Britain should not, till nearly a century after its execution, have been engaged in vigorously following so laudable an example. This seems the more extraordinary, as the Romans in early times had executed works in England, which, whatever might have been their original use, whether for the purposes of navigation or drainage, were ultimately, and that even at an early period, converted into navigable canals. Of these works, we particularly specify the Caer-Ancient Dyke and Foss Dyke cuts in Lincolnshire, which are by Roman consent admitted to have been of Roman origin. The former extends from Peterborough to the River Witham, near the city of Lincoln, a distance of about 40 miles; and the latter extends from Lincoln to the River Trent, near Torksey, a distance of 11 miles. The Caer Dyke exists now only in name; but the Foss Dyke is at this moment an efficient and flourishing navigation; and having been lately professionally engaged in its improvement, the writer had occasion to inquire somewhat minutely into its history. Regarding this oldest British canal, Camden, in his Britannia, states that the Foss Dyke was a cut originally made by the Romans, and that it was deepened in the year 1121 by Henry I.; but to what extent it was deepened does not appear. In 1762 it was reported on by Smeaton and Grundy, who found the navigable depth at that time to be 2 feet 8 inches, and recommended several works for its improvement, which appear, however, not to have been executed. In 1782 Smeaton was again employed, and deepened the navigation to 3 feet 6 inches; but it does not appear that its width was increased; and from that period it remained in a very imperfect state till 1840, when Messrs Stevenson of Edinburgh were employed to design works for assimilating the Foss Dyke, both as regarded the breadth and depth of the navigable channel, to the Rivers Witham and Trent, with which it communicated. Upon examination, the depth of the Foss was found to be 3 feet 10 inches, and its breadth in many places was insufficient for the passage of boats, for the convenience of which occasional passing places had been provided; and it was resolved to increase its dimensions, and otherwise repair the whole work. Accordingly, the canal was widened to the minimum breadth of 45 feet, and deepened to the extent of 6 feet throughout; alterations which were accomplished without stopping the traffic. The entrance-lock was renewed, and a pumping-engine was erected for supplying water from the River Trent during dry seasons; and that ancient canal, which is quoted by Telford and Nimmo as "the oldest artificial canal in Britain," is now in a state of perfect efficiency, forming an important connecting link between the Trent and Witham navigations.
Notwithstanding the existence of this early work, however, and of some others in the country, particularly the water Sankey Brook navigation, opened in 1760, it is generally admitted that the formation of the Bridgewater Canal in Lancashire, the act for which was obtained in 1759, was the commencement of British canal navigation; and that Francis, Duke of Bridgewater, and Brindley the engineer, who were its projectors, were the first to give a practical impulse to a class of works which, under the guidance mainly of Smeaton, Watt, Jessop, Nimmo, Rennie, and Tel-
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1 Fulton On Canal Navigation, London, 1796; Vallancey's Treatise on Inland Navigation, Dublin, 1763; Tatham's Political Economy of Inland Navigation, London, 1789; "Inland Navigation," Brewster's Edinburgh Encyclopedia. 2 Quarterly Review, No. cxlvii., p. 231; Treatise on Navigable Canals, by Paul Friuli. 3 The Imperial canal of China is about 1000 miles in length. 4 Smeaton's Reports, vol. i., p. 55, London, 1786. Navigation, Inland.
Canals have been very generally adopted throughout the country, and has undoubtedly been of vast importance in promoting its commercial prosperity. It is believed that the canals which have been constructed in Britain exceed in the aggregate 4713 miles, and the system has been extensively carried out both in Europe and America.
The introduction of canals adapted for the passage of boats was soon followed by a larger class of works, suited for the accommodation of sea-borne vessels. Thus the Forth and Clyde Canal, projected by Smeaton in 1766, and the Crinan Canal, executed by Rennie, are examples of navigations to enable sea-borne vessels of small size to pass from opposite coasts of the country, and escape long, and it may be hazardous, sea voyages. But these works are completely surpassed by others which have been formed on a scale of much greater magnitude, to admit vessels of heavy burden and large draught of water. Of these we may mention the Great North Holland Canal, designed and constructed by M. Blanken. That canal, which extends from Amsterdam to the Helder, a distance of 51 miles, was finished in 1825. It is about 125 feet in breadth at the water surface, 31 feet at the bottom, and no less than 20 feet in depth of water; and what is most worthy of notice, and is indeed a characteristic of all Dutch engineering works, the greater part of it is protected from the German Ocean by embankments faced with wicker-work, the surface of the water in the canal being below the level of the sea at high water. At the time the writer inspected this work the sea was several feet higher than the surface of the water in the canal, and the vessels were actually being locked down from the ocean into the fertile plains of Holland. The object of this canal is to enable vessels trading from Amsterdam to avoid the islands and sand-banks of the dangerous Zuider Zee, the passage through which in former times often occupied as many weeks as the transit through the canal now occupies hours.
But our own country furnishes us with a similar work of great magnitude and boldness; we allude to the Caledonian Canal, originally projected by Watt and Jessop, and ultimately executed by Telford, which forms an inland navigation, composed partly of natural lakes, and partly of artificial canal, extending from Inverness to Fort-William, a distance of 60 miles. The artificial part of it is 120 feet in width at the top-water level, 50 feet at the bottom, and affords 20 feet of maximum depth. By means of this inland communication vessels are enabled to avoid the dangers of the Pentland Firth, and also in some measure the intricate navigation of the Western Islands; and while the Dutch in their great canal had to encounter the difficulties occasioned by the proverbial lowness of their country, Telford, in constructing the Caledonian Canal, had to deal with the ruggedness of a succession of Highland glens, and to surmount the summit-level of Loch Oich, which is about 80 feet above the level of the sea. Accordingly, in addition to many heavy works which occur in its course, there is at one point on the Caledonian Canal a succession of eight locks, by means of which a vessel of nearly the largest class of merchantmen can be raised or lowered through a height of 60 perpendicular feet. The locks, which are in close succession, rise one above another like a series of gigantic steps; and this unique and extensive marine ladder has not inappropriately been termed "Neptune's Staircase."
It must be obvious that, in successfully carrying out works of such a nature, and on so gigantic a scale, no ordinary amount of engineering skill is requisite. Vast Difficulties reservoirs must in some cases be formed for storing the incoming water necessary to supply during dry seasons the loss occurring by leakage, and evaporation. Feeders must be made to lead this water to the canal; hills must be pierced by tunnels; valleys must be crossed on lofty embankments, or spanned by spacious aqueducts; and above all, the whole must be conceived and laid out with scrupulous regard to the all-important object of securing the works against injury from an overflow of water during floods, and a consequent inundation of the surrounding country. Moreover, the necessity of laying out the canal in level stretches, and surmounting elevations by means of locks or inclined planes, occurring at intervals, often occasions much difficulty, and greatly restricts the resources of the engineer. Taking, then, all these circumstances into consideration, and bearing in mind that canals were the pioneers of railways, we think it may safely be affirmed that the canal engineers of former days had much more serious physical difficulties to contend with than are experienced in carrying out the railways in modern times; if we except such works as the Britannia Bridge, the high-level bridge of Newcastle, the Boxhill Tunnel, and some other kindred works. But, indeed, their mechanical difficulties were also greater; for the introduction of steam, and its wide-spread application to all engineering operations, affords facilities to the engineers of modern times which Smeaton at the Eddystone, Stevenson at the Bell Rock, and Rennie and Telford in their early navigation works, did not enjoy. We therefore gladly embrace this opportunity of acknowledging the distinguished merits of the engineers who practised at the end of the former and the commencement of the present century.
We have already said that we cannot in this treatise General enter into details as to the construction of the various works principles adopted in executing canal navigations; and we shall here close our short historical notice of these works by submitting the following digest of the general principles which should guide the engineer in selecting the route and designing the construction of a line of canal:
1. The first object to which attention ought to be directed is the supply of water, on which the efficiency of a canal mainly depends. If there be no natural lake in the district available for storage, the engineer must select such situations as are suitable for the construction of artificial reservoirs. The conditions to be attended to in selecting the positions for these works are, that they command a sufficient area of drainage to supply the necessary amount of water; that their offsets are at such an elevation as to admit of water being conveyed to the summit-level of the canal; and that the embankments for retaining the water be erected on sites affording a favourable foundation, and so situated with reference to the valley above them that they shall, with the minimum height and breadth of embankment, dam up the maximum amount of water. It is further necessary to consider whether the subsoil of the valley forming the reservoirs is throughout of so retentive a nature as to prevent leakage; and it is also essential to provide, by means of waste weirs, for the discharge of floods. The Caledonian Canal is, in this respect, very favourably situated; no artificial reservoir having been required. Nearly the whole supply is
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1 History of Inland Navigation, particularly those of the Duke of Bridgewater, London, 1788. Hughes' Memoir of Brindley, Weale's Quarterly Papers, London, 1843. 2 It was here that Bakker, a burgomaster of Amsterdam in 1688, introduced his "camel" for floating large vessels over the shoals of the Pampus, by means of which, according to Sir John Leslie, an Indianman which drew 15 feet water had its draught reduced to 11 feet. 3 The connection of the Atlantic and Pacific oceans by means of a navigable canal has long been under consideration, and the question has of late years assumed a more practical aspect. (For a review of the various schemes which have been proposed for carrying out this desirable object, the reader is referred to communications by Mr Joseph Glynn and Lieut.-Col. Lloyd, in the Trans. of the Society of Civil Engineers, vol. IX., p. 58, and vol. VI., p. 399.) Canals derived from Lochs Ness, Oich, and Lochy, which, indeed, constitute the greater part of the navigation; they afford ample depth of water, and though on different levels, they extend in an almost continuous line through the country. In other cases, such as the Union, Forth and Clyde, Crinan, Birmingham, and other canals, it has been found necessary to construct large artificial reservoirs, from which the water is led in feeders to points convenient for forming a junction with the canal. The water in these reservoirs, whether artificial or natural, is stored up in winter, and let off as required during the droughts of summer. In situations where the canal communicates with the sea or a tidal river, and where the natural supply is small, as in the case of the Foss Dyke, the water may be raised by pumping-engines.
2. In determining the direction of a canal, it is of importance to consider the levels of the country through which it passes, and to lay out the work in a succession of level reaches, so as to overcome elevations in canals at those places where it can be most advantageously effected. This arrangement not only leads to a saving of attendance and expense in working the canal, but is also more convenient as presenting fewer stoppages to the traffic. The means of overcoming the difference of level between the various level reaches must depend very much on circumstances. With few exceptions, the change of level is effected by means of locks, which generally have a lift of from 8 to 10 feet, though in some cases it is somewhat greater. The dimensions of the locks ought to be regulated by the traffic; but they should, in order to save water, be as near as possible the size of the craft to be passed through them. The smallest class of canals have locks about 8 feet in breadth, and from 70 to 80 feet long; those on the Forth and Clyde are 20 feet in breadth, and 74 feet long; on the Caledonian Canal they are 40 feet broad, and 180 feet long; and on the great Holland ship-canal they are 51 feet broad, and 297 feet long. The water is gradually admitted into and flows from each lock by sluices formed in the gates. Sir William Cubitt, in carrying out the improvements of the Severn navigation, introduced the water through a culvert parallel to the side wall of the lock, and opening in the centre by means of a tunnel, which admits 16,000 cubic feet of water to flow into or out of the lock in 1½ minute; and in little more than that time loaded vessels can be passed through. Inclined planes and perpendicular lifts, which have the great advantage of saving water, have also been adopted in a few cases. In 1837 the writer inspected the Morris Canal in the United States, constructed by Mr Douglas of New York, on the line of which there are no fewer than 23 inclined planes, having gradients of about 1 in 10, and an average lift of 58 feet each. The boats weighed, when loaded, 50 tons, and after being grounded on a carriage, were raised by water-power up the inclines with great ease and expedition. The length of the Morris Canal, which connects the Rivers Hudson and Delaware, and is a most interesting work, is 101 miles, and the whole rise and fall is 1557 feet, of which 223 are overcome by locks, and the remaining 1334 by inclined planes. But inclined planes were used on the Ketling Canal in Shropshire in 1789, and afterwards on the Duke of Bridgewater's Canal. Mr Green introduced on the Great Western Canal a perpendicular lift of 46 feet; and more recently Mr Leslie, of Edinburgh, and Mr Bateman, constructed an inclined plane on the Monkland Canal, wrought by two high-pressure steam-engines of 25 horse-power each. The height from surface to surface is 96 feet, and the gradient is 1 in 10. The boats are not wholly grounded on the carriage, but are transported in a caisson of boiler-plate, containing 2 feet of water. The maximum weight raised is from 70 to 80 tons, and the whole transit is accomplished in about 10 minutes. For the five years previous to the end of 1856, the average number of boats that passed over the incline each year was 7500. Sir William Cubitt has also introduced three inclined planes, having gradients of 1 in 8, on the Chard Canal, Somersetshire. One of these inclines overcomes a rise of 86 feet, and they are said to act very satisfactorily.
3. An essential adjunct to a canal is a sufficient number of waste weirs to admit of the discharge of the surplus water which accumulates during floods, and which may, if not provided with an exit, rise to such a height as to overflow the tow-path, and cause a breach in the banks, producing stoppage of the traffic and damage to the adjoining lands. In determining the number and positions of these waste weirs, the engineer must be guided entirely by the nature of the country through which the canal passes. Whenever an opportunity occurs of discharging surplus water into a stream crossed by the canal, a waste weir may safely be introduced; but, independently of this natural facility, the engineer must consider from what quarters, and at what points, the greatest influx of water may be apprehended, and must at such places not only form waste weirs of sufficient size to void the surplus, but prepare artificial courses for their discharge into the nearest streams. These waste weirs are overflows placed at the top water-level of the canal, so that in the event of a flood occurring, the water flows over them, and thus relieves the banks. The want of sufficient escape for flood-water has occasioned overflows of canal banks which were attended with very serious injury to the works, and lengthened suspension of the traffic; and attention to this particular part of canal construction is of essential importance.
4. Another very necessary precaution is the introduction of stop-gates at short intervals of a few miles, for the purpose of dividing the canal into isolated reaches, so that in the event of a breach occurring, the stop-gates may be shut, and the discharge of water confined to the small reach intercepted between them, instead of extending throughout the whole line of canal. In large works these stop-gates may be most advantageously formed in the same manner as the upper gates of locks, two pairs of gates being made to shut in opposite directions. In small works they may be made of thick planks, which are slipped into grooves formed at those narrow parts of the canal which occur under road bridges, or at contractions made with grooves at intermediate points to receive them. Self-acting stop-gates have been tried, but their success has not been such as to lead to their general introduction. Stop-gates are further found to be very useful in cases of repairs, as they admit of the water being run off from a short reach, when the repairs can be made, and the water restored, with comparatively little interruption to the traffic. Their value in obviating serious accidents was well exemplified on one occasion in the experience of the writer, when the water during a heavy flood flowed over the towing-path at the end of an aqueduct adjoining a high embankment, and the uncontrolled current carried away the embankment, and the soil on which it rested, to the depth of 80 feet, as measured from the top water-level. The stop-gates were, on the occasion referred to, promptly applied, and the discharge confined to a short reach of a few miles, otherwise the injury (which was, even in its modified form, very considerable) would have been enormous.
5. For the purpose of draining off the water to admit of repairs after the stop-gates have been closed, it is necessary to introduce, at convenient situations, a series of exits called offlets, consisting of pipes placed at the level of the bottom of the canal, and fitted with sluices which can be opened and shut when required. These offlets are generally formed at aqueducts or bridges crossing rivers where the contents of the canal can be run off directly into the bed of the stream, the stop-gates on either side being closed so as to isolate the part of the canal from which the water is withdrawn.
(6.) In executing the work, provision must be made for the proper drainage of the tow-paths, especially in cuttings. The drainage of the tow-paths should be carried to a sky drain at the bottom of the cutting, and at intervals passed below the tow-path into the canal. The preservation of the banks at the water-line is also a matter of importance. "Pitching" with stones and "facing" with brushwood are employed, and, in the writer's experience, the latter, if well executed, forms an economical and effectual protection.
(7.) In forming the alveus or bed of the canal, care must be taken, particularly on embankments, and also in cuttings, if the soil is porous, to provide against leakage by the application of puddle. And here it is proper to remark, that an all-important matter, as affecting the construction of the works, is the possibility of getting clay in the district, or such other soil as may be worked into puddle, on the good quality of which the stability of the reservoir embankments, and the imperviousness of the beds and banks of the canal, mainly depend.
These are the only points of general application in the construction of canals to which we can advantageously direct attention in the present communication. In carrying them into practice, the engineer must be guided partly by the valuable details to be found in the works to which reference is made in this article, but mainly by that experience which can be gained only by the study of works in actual operation.
We do not propose to extend our remarks to the means of conducting traffic on canals and rivers, and have to refer the reader for information on that subject to observations and works on Traction and Steam Navigation. On the former subject the reader may consult the observations, by Mr Walker and Mr George Rennie, in the Transactions of the Royal Society and of the Institution of Civil Engineers; and especially the very valuable researches on Hydrodynamics, by Mr Scott Russell, in the Edinburgh Philosophical Transactions. On the latter he is referred to the articles Steam-Engine, and Steam Navigation.
SECT. II.—RIVERS, THEIR PHYSICAL CHARACTERISTICS.
From what has been said, it will be seen that a canal may be described as a work by which water is diverted from its natural course, and made to occupy a channel prepared for its reception, extending through the country for the transport of boats and vessels. Canal navigation is thus entirely artificial in its character. In this respect it differs from river navigation, which may be described as the art of using, for the purposes of inland communication, rivers flowing in their natural courses, and of applying means to render them subservient to the purposes of navigation in cases where the depth is limited, or where rapid currents exist. Our consideration of rivers must therefore necessarily comprehend a general sketch of their physical characteristics, and the laws of their motion, as a necessary introduction to the more practical part of the subject, embracing the engineering works required for their improvement, with which we have chiefly to deal in this treatise.
As introductory, therefore, to the remarks which are to follow, it seems desirable to premise, as described by the writer in a communication to the Royal Society of Edinburgh, that in all rivers affected by tidal influence, two physical boundaries, more or less apparent, are invariably found to exist, caused by the influx of the tidal wave through firths or bays, and the modification it receives in its passage up the river, gradually rising inclination or slope of a river's bed. These boundaries again produce three compartments. The seaward, or lowest of these, the writer termed the "sea proper;" the next, or intermediate one, into which the sea ascends, and from which it again withdraws itself, was termed the "tidal compartment of the river;" and the highest, or that which is above the influence of the sea, the "river proper." Their relative extent in different situations is influenced not only by the circumstances under which the great tidal wave of the ocean enters the river, but by the size of its stream, the configuration and the slope of its bed, and, in short, by every natural or artificial obstruction which is presented to the free flow of the tidal currents along its channel.
These three compartments possess very different physical characteristics. The presence of unimpaired tidal phenomena in the lowest, the modified flow of the tide, produced by the inclination of the river's bed in the intermediate, and the absence of all tidal influence in the highest compartment, may be shortly stated as the phenomena by which these spaces are to be recognised. The tides in the "sea proper" compartment of an estuary, for example (although the place of observation be several miles embayed from what in strictness could be called the "sea" or "ocean"), will be found to resemble those of the adjoining sea with which it communicates,—1st, in the identity of the levels of low water; 2nd, in the shortness of the time which elapses between the cessation of ebbing and the commencement of flowing, or, in other words, the absence of any protracted period of low water, during which the surface appears to remain stationary at the same level; 3rd, in the symmetrical form traced by the passage of the tidal wave; and 4th, in the range of tide, so far as that is not influenced by the formation of the shores in narrow seas or channels. In ascending into the intermediate compartment, however, the level of the low water is no longer the same; the range of tide, excepting in peculiar cases, becomes less, and is gradually decreased as the bed of the river rises, and at length a point is reached where its influence is not perceptible. In this intermediate section the phenomena of ebbing and flowing are still found to take place, but the times of ebb and flow do not remain constant, that of ebb gradually gaining the ascendancy; the duration of low water being gradually protracted as we proceed upwards, until the influence of tide is unknown. This forms the boundary line of the upper compartment, the characteristic of which is the total absence of ebbing and flowing; the river at all times pursuing its downward course in an uninterrupted stream, and at an unvarying level, except in so far as may result from the changes due to land floods.
In the investigation of these different characteristics, the variable nature of the elements to be dealt with must be kept in view. The river, for example, is liable to be affected by floods, and the state of the tides by winds and other causes; and therefore a great degree of precision in defining these spaces cannot in all cases be expected, nor indeed is it necessary for the purpose of the present inquiry. But it is satisfactory to know that the termination of the low-water level at the separation of the seaward and intermediate spaces, as laid down by marine surveyors, simply from ob-
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1 Researches on Hydrodynamics, from Transactions of Royal Society of Edinburgh, 1837, by J. S. Russell; *On the Resistance of Fluids to Bodies passing through them* (Philosophical Transactions, 1829), by James Walker, C.E.
2 Proceedings of the Royal Society of Edinburgh, vol. ii., p. 26. observation of the tidal phenomena, has in several situations been found to agree exactly with the position of that boundary as determined by engineers by means of accurate levelling, combined with careful tidal observations.
But an example in actual practice will best illustrate what is meant, and for this purpose we shall refer to the investigation of the tidal phenomena, made by the writer in 1842, of the Firth of Dornoch and Kyles of Sutherland in Cromartyshire. By referring to the small chart of the Dornoch Firth in Plate I., the reader will be better able to follow the illustrations to be given. The harbour of Portmahomac, marked A on the chart, about 3 miles from Tarbetness Lighthouse, was selected as the place at which to observe the ocean or sea wave. The second station at which it was found convenient to institute observations was within the Firth at Meikleberry, marked B, about 3 miles above the town of Tain, and 11 miles distant from Portmahomac. The third station was at Bonar Quarry, marked C, situated on the north shore of the Firth, and 8 miles inland from Meikleberry; and the fourth station was at Bonar Bridge, marked D, one mile from the Bonar Quarry. Beyond Bonar Bridge the observations were also extended as far as the junction of the Rivers Oykell and Cassley, marked E, a distance of 12½ miles; so that the whole distance embraced in the investigation was 33½ miles. Graduated tide-gauges were fixed at Portmahomac, Meikleberry, Bonar Quarry, and Bonar Bridge; and by means of two distinct sets of observations, the levels of these gauges, in relation to each other, were accurately determined, so that all the tidal observations made at them could be reduced to the same datum line. The result of the observations was, that the low-water of each tide is, practically speaking, on the same level at Portmahomac, Meikleberry, and Bonar Quarry. We use the word practically, because the level of the sea is more or less affected by every breeze of wind, which necessarily must pen up and elevate some portions of its surface, and cause corresponding depression, at other places, so that an unvarying low-water line will not be found to exist throughout a series of tides on any part even of the ocean itself, however limited the number of low-waters embraced may be. Accordingly, deviations from a truly level line of a few inches occasionally occurred in the observations made at the Dornoch Firth; but these were not of greater extent than could reasonably be traced to the effect of wind, and were found to vary, not only in their amount, but also in their value, being sometimes plus and sometimes minus quantities, causing corresponding variations in the results deduced from the different series of tidal observations that were made. Some of these showed the low-water within a fraction of an inch of being level; while others gave a notable elevation at some of the stations; and others, again, gave a depression below the level line at the very stations where previously there had been a rise.
To illustrate this more fully, we shall give a few examples: Thus, on the 23rd of June (on which day the weather happened to be very calm), the level of low-water at Meikleberry was three-quarters of an inch above that at Portmahomac; and on the next day, the wind blowing fresh from the S.E., the level of low-water at Meikleberry was 3½ inches above that at Portmahomac. Again, a succeeding observation gave the level of Meikleberry three-quarters of an inch below Portmahomac. In the same way, and in similar small degrees, the level between the low-water at Bonar Quarry tide-gauge and at Portmahomac was found to vary. The average of all the observations made, gave the level of low-water at Meikleberry 2·2 inches above that at Portmahomac, and the level of low-water at Bonar Quarry 1·1 inch below the low-water at Portmahomac. Whether these average differences of level be traceable to the effects of prevailing winds, which may be supposed to have exerted a greater influence on the water at the more exposed stations, or to any inaccuracy in the levels, must evidently, from the examples given of the extent and nature of the daily deviations, be a point which we cannot determine; but the result of a lengthened train of observations, notwithstanding the average difference above stated, may fairly be held to be, that the low-water of each tide is practically on the same level at Portmahomac, Meikleberry, and Bonar Quarry; and therefore that the low-water tidal phenomena, throughout the whole extent of the firth, correspond with those of the sea.
But when the results of the observations at Bonar Bridge come to be compared with those made at the seaward station, a very marked difference presents itself; for, while the low-water line is found to be practically level from Portmahomac to Bonar Quarry, a distance of 20 miles, throughout a narrow firth, varying from 1½ mile to 550 feet in breadth at low-water, we find that between the Quarry and Bonar Bridge, a distance of only one mile, there is a rise in the low-water line of spring-tides of no less than 6 feet 6 inches. It was therefore concluded that, in the Dornoch Firth, the point at which the low-water level of spring-tides met the descending current of fresh water, lay somewhere between the Quarry and Bonar Bridge. A different series of observations was made to ascertain the exact point at which this junction takes place, and the result of these observations was, that at low-water of an ordinary spring-tide, rising 14 feet at Meikleberry, the low-water level of the sea meets or intersects the descending fresh-water stream from the Kyle of Sutherland, at a point 1700 yards below Bonar Bridge, or nearly opposite Kincardine Church, and within 60 yards of the Quarry station. Between this point and the bridge, a distance of 1700 yards, there is a rise of 6 feet 6 inches, giving an average slope on the bed of the river of 1 in 784, or 6·7 feet per mile.
In addition to this uniformity in the level of low-water, it was further found that the tidal phenomena of the firth corresponded to that of the adjoining sea, in the outline traced by the passage of the tidal wave, as deduced from observations made at the different stations on the rise and fall of the tide-level between the periods of low and high water. During the period between each low-water or high-water the level of the surface was ever varying, there being no lengthened cessation of ebbing and flowing, the tide-wave being fully developed at the whole of the stations up to Bonar Quarry. The range of tide was indeed increased in the inner part of the firth to the extent of 9 inches at Meikleberry, and 12 inches at Bonar Quarry; that is, when the rise of tide was 12 feet 8 inches at Portmahomac, it was 13 feet 5 inches at Meikleberry, and 13 feet 8 inches at Bonar Quarry—an increase which is due to the momentum of the tidal wave when obstructed by the contracting shores of the firth, and is accounted for by the principle of the conservation of forces.
But if we inquire into the tides at Bonar Bridge, we find that they do not correspond with those of the adjoining sea or of the firth; for taking the tide to which we have already alluded, which rose 13 feet 8 inches at Bonar Quarry, it was found on the same day to rise only one-half of that amount, or 6 feet 10 inches at Bonar Bridge; the difference between the two results being occasioned by the rise on the low-water line of the channel between these two places. The tide on the particular day alluded to rose no less than 6 feet 10 inches at Bonar Quarry before it attained the level of the low-water at Bonar Bridge, when it began to rise at that place also, and afterwards continued to flow nearly uniformly at
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1 Essay towards a First Approximation to a Map of Co-tidal Lines, by the Rev. W. Whewell, Philosophical Transactions, 1833. both places. Fig. 1 is a diagram illustrative of the form of the tide-wave at Meikleberry and Bonar Bridge; the hard line represents the curve formed by the passage of the tidal wave at Meikleberry, and the dotted line shows that at Bonar Bridge. In both cases the vertical space represents the rise of tide, and the horizontal space the elapsed time. From this diagram it will be seen, that while the tide at Meikleberry is symmetrical, and presents a constantly rising or falling outline, the tide at Bonar Bridge represents a long period, extending on some occasions by actual observation to several hours at low-water, nearly unaffected by tidal influence, during which period the water stood almost at the same level. The tidal water admitted into the upper part of the estuary above Bonar Bridge took a considerable time to drain off through the narrow water-way at that place, and hence the water did not attain a permanent low-water level, even long after the tide had ceased to operate in affecting its surface. The observations made to ascertain how far the tidal influence extended up the Kyle of Sutherland were conducted with the same care, and proved that the highest point influenced by the tide was at the junction of the Rivers Oykell and Cassley, 12½ miles above Bonar Bridge.
A further test of the "sea proper" will, it is believed, be found in the existence, at any place of observation within that compartment, of a central point in the vertical range of tide from which the high and low water levels of every tide are very nearly equidistant. The existence of such a point was, it is believed, first determined by Mr James Jardine at the Tay in 1810, and has been observed in the firths of Forth and Dornoch, at the Skerryvore Rocks on the west of Scotland, at the Isle of Man, and in the Mersey. These different series of observations, made at points so far distant from each other, seem to prove the universality of the phenomenon, at least on the shores of this country. But in ascending into the tidal compartment, the rise on the low-water level, which has already been described, destroys at once the symmetry of the tide-wave, as shown in fig. 1, and the existence of any such central point equidistant from the high and low water level of each tide.
The case we have adduced will serve to illustrate the definition we have given of the compartments of rivers. From Portmahomac to Kincardine, near Bonar Quarry, we have all the evidences of what we have termed the "sea proper;" the line traced through the low-water mark at different parts of the firth is practically level; the curve formed by the rise and fall of the tide is symmetrical; there is no lengthened cessation of ebbing and flowing at the period of low water; and the range of tide is unmodified save by the additional rise due to the narrow firth through which the tide-wave passes. From Kincardine to the junction of the Oykell and Cassley, we have proofs no less evident of the modified flow of the tide peculiar to the "tidal compartment." Even at Bonar Bridge, one mile above the quarry, the low-water level is 6 feet 6 inches higher than at the station below. At low-water the tide remains within a few inches of the same level for several hours, and its maximum range is reduced to about one-half of what it is farther seaward, while at the junction of the Oykell and Cassley it disappears altogether. Above this point no tide is known to affect the flow of the stream, which, being free from all tidal influence, may be termed the "river proper."
We must here warn the reader not to suppose that the boundaries we have traced as existing in the Dornoch Firth, and many not equally other places which the writer has investigated, may be determined with the same precision under all circumstances and in every case. The observations to which we have alluded, are supposed to be made at periods when the river is free from floods and the sea unaffected by heavy gales; moreover, the configuration of the bottom and shores of a river and estuary may, in certain cases, render the accurate determination of the boundaries very difficult. All that we assert is, that these compartments do in some measure, more or less defined, exist in all cases; and although not determined with the same careful precision as explained in the case of the Dornoch Firth, we have made observations of a more general character, and with complete success, to define approximately the tidal compartments in many estuaries and rivers in Britain and Ireland.
But there are other data with which the engineer must be furnished before he can advantageously consider the improvement of any part of a river. These data include the determination of its slope, velocity, and discharge, the nature of its bed and banks, and many other particulars. For full details as to the character and extent of such information, and the means of obtaining it, we can only refer the reader to works on the subject of River and Marine Surveying. Neither do we include in the present treatise any sketch or digest, however brief, of the interesting and gradual progress made by philosophers and engineers of the early Italian and French schools, in the theoretical and experimental investigations of the laws which regulate the flow of water in natural and artificial channels, which investigations form the basis of all our practice in hydraulic engineering. These lengthened and laborious experimental researches will be found to be most fully discussed—historically, theoretically, and practically—in the valuable article by Dr Robison on the Theory of Rivers, in this Encyclopedia (see River), and also in the report made by Mr George Rennie to the British Association on the progress and present state of our knowledge of hydraulics as a branch of engineering.
While we do not therefore propose to advert at length either to the theoretical or practical details of the subject, still the whole of river engineering is so connected with, and dependent on, those physical characteristics of rivers which are termed the slope, the hydraulic mean depth, the velocity, and the discharge, that it seems to be indispensable to a proper understanding of the subject that these elements should be defined, and that the relations which subsist between them should be considered. The following definitions will suffice to answer the purpose in view:
1. The slope is the fall on the surface of the river, which is generally expressed in feet or inches per mile. 2. The hydraulic mean depth is the quotient arising from dividing the sectional area of the channel in square feet by the wetted border or perimeter in lineal feet. 3. The mean velocity is that velocity which is common to the whole cross section of a stream, and is represented by the discharge divided by the area of that section.
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1 Report by James Jardine, C.E. 2 Treatise on the Application of Marine Surveying and Hydrometry to the Practice of Civil Engineering, by David Stevenson, C.E., Edinburgh, 1842. 3 Report of the British Association for the Advancement of Science for 1834. The discharge is the quantity of water yielded by the stream in a given time, and is generally stated in cubic feet.
In the practice of engineering it is frequently necessary to consider questions involving the relations which subsist between these different elements, and many formulae have been proposed to facilitate this operation. The Chevalier Dubuat was the first investigator who, by discovering the effects of the friction of fluids on their own particles, and on the bed along which they move, was enabled to apply his theoretical knowledge of hydraulics to practical purposes, and his views and formulae will be found fully discussed in the article River already alluded to. The writer of this article has, however, found that such formulae are not generally applicable, and it seems desirable to lay before the practical engineer the various results given by different formulae when applied under the same circumstances, in order that he may be cautioned as to relying on such a means of computation in cases where great exactness is requisite. In order to ascertain the discharge of a stream or river, the writer has therefore in practice resorted to actual measurement. For this purpose, a situation was selected where the bed was tolerably uniform in its longitudinal and transverse outline. A correct transverse section of the bed or channel was made, and the section was divided into compartments. The surface velocity in the centre of each compartment was then taken by means of floats, or the instrument called the tachometer. These surface velocities were reduced to mean velocities for each compartment by Dubuat's formula:
\[ M = \frac{(\sqrt{V} - 1)^2 + V}{2} \]
or more simply, in cases where great accuracy is not required,
\[ M = 0.8 V \]
where \( V \) = the observed surface velocity in inches per second.
\( M \) = the mean velocity in inches per second.
We have found by means of the tachometer, used at different depths, that this formula expresses accurately the mean velocity of any vertical section of the stream to which the observed surface velocity is applicable. But as the surface velocity on the same cross section is not uniform throughout the width of the stream, it becomes necessary, as already stated, to divide the section into compartments, so as to embrace the maximum and minimum speeds. The areas of the different sections being then multiplied by the corresponding mean velocities obtained by either of the above formulae, the sum of the discharge due to the different compartments is held to give the total discharge of the stream or river.
It is obvious that the accuracy of the result obtained by this process depends on the judgment with which the cross-sectional area is subdivided, and on the care with which the observations are made. The operation is, in many cases, attended with difficulties, and in all with a considerable consumption of time; and many formulae have been proposed to shorten it. The writer has compared the computed discharge given by several of these formulae with the discharge as ascertained by careful observations made in the manner described, and the following result is submitted for the information of engineers.
The formulae subjected to trial were:
I. Formula given by Dr Robison, founded on Dubuat's investigations:
\[ M = \frac{307 (\sqrt{d} - 0.1)}{\sqrt{S} - \text{Hyp. log. of } \sqrt{S + 16}} - 0.3 (\sqrt{d} - 0.1) \]
in which \( M \) = the mean velocity in inches per second,
\( d \) = the hydraulic mean depth in inches,
\( S \) = the reciprocal of the slope of the surface which is the denominator of the fraction expressing the slope, the numerator being always unity (a slope of 1 foot a mile is \( \frac{1}{5280} \), therefore \( \frac{1}{5280} \) = reciprocal for that slope),
Hyp. log. = the common log. of the number to which it is attached, multiplied by \( 2^3026 \).
II. Formula given by Sir John Leslie:
\[ M = \frac{15}{16} \sqrt{af} \]
in which \( M \) = the mean velocity in miles per hour,
\( a \) = the hydraulic mean depth in feet,
\( f \) = the fall on the surface in feet per mile.
III. Formula given by Mr Ellet for calculating discharge of the Mississippi:
\[ V = \frac{8}{10} \sqrt{df} + \frac{df}{20} \]
\[ M = 0.8 V \]
in which \( V \) = the surface velocity in feet per second,
\( d \) = the maximum depth of the river in feet,
\( f \) = the fall on the surface in feet per mile,
\( M \) = the mean velocity in feet per second.
IV. Formula given in Mr Beardmore's tables:
\[ M = \sqrt{a} 2f \times 55 \]
in which \( M \) = mean velocity in feet per mile,
\( a \) = hydraulic mean depth in feet,
\( f \) = fall per mile in feet.
V. In addition to these formulae, the writer also subjected to trial the formula:
\[ M = \frac{(\sqrt{V} - 1)^2 + V}{2} \]
in which \( M \) = the mean velocity in inches per second,
\( V \) = the maximum surface velocity in the axis of the stream in inches per second.
In order to compare these different formulae, a very favourable situation was selected for ascertaining the discharge of a stream by careful measurements of its sectional area and of the velocities at different parts of its surface from the centre to either side, and the result gave a discharge of 1653 cubic feet per minute, which, from various measurements, the writer believes to be a very near approximation to the actual discharge. The slope was also accurately ascertained, and the following are the results:
| Cubic feet | |------------| | Discharge from measurement as above, 1653 per minute. | | 1st. By Robison's formula ................. 2924 do. | | 2nd. By Leslie's do ....................... 2474 do. | | 3rd. By Ellet's do ....................... 2784 do. | | 4th. By Beardmore's do ................... 2335 do. | | 5th. By formula assuming the mean deduced from the centre surface velocity as the mean for the whole section ... 1950 do. |
It will be seen from this statement, that none of the formulae afford a near approximation to the discharge of the small stream to which they were applied.
Again, it was ascertained by the late Dr Anderson, after most carefully dividing the cross section into compartments, that the discharge of the main branch of the Tay at Perth was 147,391 cubic feet per minute. The writer has also ascertained the discharges, as calculated by the different formulae as above, and the following are the results:
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1 See article River; also A System of Mechanical Philosophy, by John Robison, vol. ii., p. 453. 2 Elements of Natural Philosophy, by Professor Leslie, Edinburgh, 1829, vol. i., p. 423. 3 The Mississippi and Ohio Rivers, by Charles Ellet, Philadelphia, 1853. 4 Hydraulic Tables, by Nathaniel Beardmore, C.E., London, 1852. 5 This does not include the Willowgate, nor the Earn. The result of these trials, and others which the writer has had occasion to make, is, that none of the formulas that have been proposed will be found generally applicable. As it is often convenient, however, to be able to approximate to the velocity or discharge due to a given area and fall, the following formula may be applied, and will, in most cases, give a pretty near approximative result, viz.:
\[ x = \frac{y \sqrt{af}}{z} \]
\[ z = \frac{x \times 5280}{60} \]
\[ D = sz \]
in which \( x \) = the mean velocity of the whole section of the stream in miles per hour,
\( y \) = a quotient which is found to vary from 0.65 for small streams under 2000 cubic feet per minute, to 0.9 for large rivers, such as the Clyde or the Tay,
\( a \) = the hydraulic mean depth in feet,
\( f \) = the fall on the surface in feet per mile,
\( z \) = the mean velocity of the whole section of the stream in feet per minute,
\( s \) = the sectional area of the stream in feet; and
\( D \) = the discharge in cubic feet per minute.
It must still be kept in view that the application of any known formula to the determination of the mean velocity and discharge of a river is shown, by experimental inquiry, to afford only a rough approximation; and that if a near approximation is required, it must be obtained by means of observations embracing the velocities at different parts of the cross-sectional area, made in the manner already described.
We must offer the further caution, that those rules whereby the mean velocity is deduced from, or is assumed as bearing any constant ratio to, the surface velocity, do not apply in many situations which are within the influence of the tide. As will be explained more fully hereafter, the fresh water of the river being specifically lighter, is to a certain extent borne up by and floats upon the denser water of the sea. In surveying the Dee at Aberdeen in 1810, Mr Robert Stevenson found that, while there was an outward upper-current of fresh water, there was an inward under-current of salt water; so that, although the upper stratum was constantly running toward the sea, there was a regular rise and fall of the surface produced by the influx of the tidal waters below. Another instance of such an under-current, though not occasioned by the presence of a river, was found to exist in a marked degree at the Cromarty Firth by Mr Alan Stevenson in 1837. The waters of the Cromarty Firth pass to and from the sea through the narrow gorge between the Suters of Cromarty, where the width is about 4500 feet, and the depth about 150 feet. The mean velocity due to the column of water passing this gorge, as deduced from the observed surface velocity, was not sufficient to account for the quantity of water actually passed during each tide, as determined by measuring the cubical capacity of the basin of the firth. This led to the observation of the under-currents through the gorge by means of submerged floats, and it was found that during flood-tides the surface velocity was 1.8 mile per hour; while at the depth of 50 feet the velocity was not less than 4 miles per hour, being an increase of 2.2 miles per hour. During ebb-tide the surface velocity was 2.7 miles per hour, and at 50 feet it was not less than 4.5 miles per hour, being an increase of 1.8 mile per hour.
The existence of these under-currents is due to some obscure causes connected no doubt with the configuration of the bottom, and the circumstances under which the tidal wave approaches and recedes from the shore. The existence of a powerful oceanic under-current during the flood-tide may account for the increased under-velocity of the tide flowing into the Cromarty Firth; and if we suppose a similar rapid under-current to sweep along the coast during the ebb-tide, the tendency would be to draw off the water more quickly from the lower part of the channel between the Suters which forms the mouth of the firth, and thus to increase the velocity at and near the bottom during the ebb-tide, as also indicated by the observations to which we have alluded. It is evident that in all such situations the application of a common or mean velocity, deduced from the observed surface velocity, cannot be relied on as correct.
As the slopes, velocities, and discharges of rivers are so important in all matters connected with the flow of streams, and may be useful for comparison in considering questions of river engineering, we give at the end of this article, in a tabular form, the physical characteristics of different rivers, embracing all the information we have been able to collect, with the sources from whence that information was obtained.
We have considered it necessary to enter thus far into detail, to prepare the way for what is to follow.—First, Because it is quite impossible to consider and design with advantage the improvements of a river without a correct knowledge of its physical characteristics, as developed in the course of such investigations as we have described. Such information cannot in every case be procured with an equal amount of precision, but the more complete and detailed it is, the more confidently and advantageously will the engineer proceed to form his design. Secondly, We have been particular in defining the physical boundaries of rivers, because the remedial means which call for the engineer's consideration in designing improvements on the three compartments which they include, are not less distinct than the different phenomena which have been described as their peculiar characteristics. In proof of this, it may be stated generally, that the works on the "river proper" section consist chiefly in the erection of weirs, by means of which the water is dammed up so as to form stretches of canal in the river's bed, with cuts and locks between the different reaches. The "tidal compartment" embraces a more varied range, including the straightening, widening, or deepening of the courses and beds of rivers, the formation of new cuts, the erection of walls for the guidance of tidal currents, and in some cases the shutting up of subsidiary channels; while the "seaward compartment" embraces all works connected with the improvements or removal of bars and shoals.
On these subjects we shall have to enter at some length; and in treating of them it may be most convenient to consider the question of river navigation under the three following sections, viz.:
1st. The upper compartment, or "river proper."
2nd. The intermediate compartment, or "tidal river;" and
3rd. The lower compartment, or "sea proper."
SECT. III.—THE "RIVER PROPER" DEPARTMENT.
The magnitude of a river is, under certain conditions, proportional to the extent of country which is drained, rivers as will be seen by reference to the table at the end of this portion of the article, and all our ideas regarding rivers, as affording the extent of means of inland navigation, must necessarily be to some extent varied to meet the different physical characteristics of different countries. Thus, in continents we find rivers of great magnitude, fed by the drainage of vast tracts of surrounding land, rolling their contents in a broad, deep current to the ocean, and affording a highway for vessels of the largest class to pursue their course for hundreds of miles into the interior of the country. Of such is the Mississippi, which, according to Mr Ellet, maintains, for a distance of nearly 1200 miles above New Orleans, an average breadth of 3300 feet, and a depth of 115 feet. The Ohio, which joins it at this place, is navigable to Pittsburgh, where the writer of this article has seen from thirty to forty large-sized steamers lying at the quays of that truly inland port, which were all engaged in trading to New Orleans, on the Gulf of Mexico, being a river navigation of upwards of 2000 miles.
In considering the improvement or maintenance of such a navigation as this, the engineer has to deal chiefly with the control of the discharge due to the rains of the district through which the river flows. His difficulty does not so much consist in deficient depth or breadth of navigable channel, as in the magnitude of the floods with which he has to contend, and the provision he has to make for retaining them within such limits as to secure the safety of the surrounding district.
In less extended tracts of country the rivers are proportionally smaller; and when we come to consider our own island, we find that its area and drainage are altogether insufficient to afford depth and breadth of water for extended inland navigation. This will readily be understood when the areas of the basins and the discharges of some of our largest rivers are compared with the Mississippi, to which we have alluded. For example, according to the table to which we have already referred, the Tay drains 2283 square miles, and discharges 274,000 cubic feet per minute; the Clyde drains 946 square miles, and discharges 48,000 cubic feet per minute; the Mississippi drains 1,226,600 square miles, and discharges 76,800,000 cubic feet per minute.
It will not, we believe, be considered inappropriate to the subject we are discussing, to offer a short sketch of what is undoubtedly the most gigantic river navigation in the world, taken from the elaborate work by Mr Charles Ellet, on the Mississippi and the Ohio. It appears, from the information given in that work, that the Mississippi varies from 2200 to 5000 feet in width, the average width being assumed as 3300 feet. It is from 70 to 180 feet in depth, the average being 115 feet. The area of the cross section varies from 105,544 square feet to 268,646 square feet, the average being 200,000 square feet. The length, from its junction with the Ohio to the Gulf of Mexico, is 1178 miles, and its average descent at full water is 3½ inches per mile, and in absence of floods (or during summer and autumn) 2½ inches per mile. The length of the Ohio, from its junction with the Mississippi to Pittsburgh (the head of the navigation for large vessels), is 975 miles, and the average inclination is about 3½ inches per mile. From Pittsburgh to Olean Point the distance is 250 miles, and the inclination 2 feet 10 inches per mile. When the water is high, steamboats have ascended to Olean Point, which is 2400 miles from the Gulf of Mexico; and in doing so, have had to overcome a current which at some places runs with a velocity of 5 miles per hour. This, however, is chiefly in the upper part of the river. Generally speaking, vessels have no difficulty, in the lower or more open part of the stream, in avoiding the strength of the currents by keeping in-shore. But in the Ohio much inconvenience is felt during dry seasons from the currents at certain parts of the river; and the writer has seen a steamer, when deeply loaded, unable to overcome them until assisted by a warp attached to an anchor dropped ahead of the vessel, in the middle of the channel, by which, after considerable detention, she was "warped through the rapid." The discharge of the Mississippi is computed by Mr Ellet, at high water, at 1,280,000 cubic feet per second; and its drainage he estimates at 1,226,600 square miles. When the autumnal rains set in, the river rises above its summer level to the enormous extent of about 40 feet at the mouth of the Ohio, and 20 feet at New Orleans. In investigating the physical characteristics of this mighty stream, Mr Ellet found—
1st. That the average surface velocity in the centre of the river was 5 miles per hour, and occasionally the speed reached 7 miles per hour; 2d. By using under-current floats, he found that the speed of a float, supporting a line of 50 feet long, was always greater than that of the surface float—the average increase of velocity being 2 per cent.; 3d. The results of the experiments made, lead him to conclude that the mean velocity of the Mississippi is about 2 per cent. greater than the mean surface velocity; 4th. In coming to this conclusion, no account is taken of such observations as show remarkable under-currents, the velocity of which were in some places found to be 17 per cent., and 204 per cent. greater than the surface velocities; 5th. While the mass of water which the channel of the Mississippi bears is running downwards with a central velocity, the current next the shore is sometimes found to be running upwards, or in the opposite direction, at the rate of 1 to 2 miles per hour; 6th. While the water is running downwards in the one side of the river, it is often found with an appreciable slope, and visible current running upwards on the other side of the river; 7th. The surface of the river is therefore not a plane, but a peculiarly complicated warped surface, varying from point to point, and inclining alternately from side to side. After considering all the conflicting results derived from his investigations, Mr Ellet, in order to obtain the mean velocity and discharge of the river, employed the formula as already noticed,
\[ V = \frac{8}{10} \sqrt{\frac{df}{20}} \]
\[ M = 0.8 V \]
and \( Ma = D \)
where \( V \) = the velocity of central surface current in feet per second, \( d \) = maximum depth of river in feet at place of observation, \( f \) = slope of surface in feet per mile, \( M \) = the mean velocity in feet per second, \( a \) = area of cross section of river in feet, \( D \) = discharge of river in cubic feet per second.
In discussing the various formulæ for velocities and discharges, we have seen that the formula applied to the Mississippi by Mr Ellet does not apply to such rivers as the Tay, or to smaller water-courses; and until the result which he has given has been compared with the discharge obtained by actual measurement of the velocities at different parts of the cross section, we do not think that the discharge of the Mississippi, which has been calculated by Mr Ellet, can be relied on as accurate.
The chief object of the investigations made by Mr Ellet was the prevention of floods, which have recently increased both in number and extent. This he attributes—
Firstly, To extended cultivation, by which evaporation is supposed to be diminished, the drainage increased, and the floods hurried forward more rapidly into the country below.
Secondly, To the extension of the embankments along the banks of the Mississippi and its tributaries, by which water that was formerly allowed to spread is now confined to the channel of the river.
Thirdly, To what are termed cut-offs, or straight cuts, by which the distance is shortened, and the slope and velocity increased, so that the water is brought down more rapidly from the country above.
1 Sketch of Civil Engineering of North America, by David Stevenson, C.E. Fourthly, To the gradual extension of the delta into the sea, so as to lengthen the lower course of the river, to diminish the slope and velocity, and thus to throw back the water on the land above.
The works suggested for protecting the country against floods are—
First, More sufficient embankments.
Second, The prevention of further cut-offs, or works for straightening the upper parts of the tributaries of the river.
Third, The enlargement of the seaward channels or outlets; and
Fourth, The creation of large artificial reservoirs, by placing dams across the outlets of the lakes or distant tributaries, so as to compensate for the loss of the natural overflow of the water, which is checked by the embankments for protecting the country in the lower part of the river.
An interesting account has been given by Mr Shepherd of certain improvements on the Danube, to which we shall very shortly refer. The navigation of that river was greatly impeded by the constant shifting of its course after every flood. Its channel was divided into numerous branches, and the main object of the improvements was to shut off these lateral branches, and to cause the river to flow in one central channel. This was effected by means of a series of spurs or jetties, made of bundles of brushwood, and thrown out from either side of the river. The brushwood was laid down in its green state, and, taking root, each spur or jetty, after a few years, formed a thick massive hedge, which now prevents the stream from making further ravages on the banks, and confines it to one central channel, scouring out a depth sufficient for navigation. This system of embanking with faggots has been, according to Mr Shepherd, the means of rescuing thousands of acres of land on the Danube, at a cost of not £1 per acre. These improvements have, it appears, been effected in what were the most dangerous parts of the river, and have, it is stated, in connection with an improved organization of pilotage, been of great benefit to the traffic of the Danube, which is now carried on almost uninterruptedly, it being a very rare occurrence to hear of any of the steamers getting aground.
Although it has been considered proper to allude thus briefly to the large continental rivers, yet it will be obvious that the magnitude of such a river as the Mississippi, for example, prevents us from applying the special results and observations of Mr Ellet to rivers in this country. Indeed, the fresh-water or upper compartments of our rivers are so small, and their navigation is so limited, that we have little to say under that head which can be applicable to the British Isles.
Our streams cannot, like the Mississippi or other large rivers, be advantageously navigated in their natural state; and the means employed to render them navigable may be said to consist in throwing dams across their beds, so as to convert them into a succession of narrow lakes or pools, in which the water is dammed up to such a height as to afford sufficient depth for small boats.
In early times this was effected by means of what were called "stanches." Sir William Cubitt states that when he undertook the improvement of the Stour in Essex, there were thirteen stanches along the course of the river. These stanches consisted of two substantial posts, which were fixed in the bed of the river, at a sufficient distance apart to permit a boat to pass easily between them, and connected at the bottom by a cross cill. Upon one of these posts was a beam turning on a hinge or joint, and long enough to span the opening. When the "stanch" was used, the boatmen turned the beam (which was above the level of the water) across the opening, and placed vertically in the stream a number of narrow planks resting against the bottom cill and the swinging beam, thus forming a weir which raised the water in the stream about 5 feet high. The boards were then rapidly withdrawn, the swinging beam was turned back, and all the boats which had been collected above were carried by the flow of water over the shallow below. By repeating this operation at given intervals, the boats were enabled to proceed a distance of about 23 miles in two or three days.
This primitive system, which was at one period very common in England, has been superseded by throwing permanent dams across the river, so as to convert its channel into a series of deep-water reaches, and the boats pass from one reach to the other by means of side-cuts with locks. The same plan has been extensively carried out in many of the smaller rivers in America, and is there called "still-water navigation." It has been executed on a pretty large scale by Sir William Cubitt on the upper part of the Severn, where the river has been divided into four reaches, having a depth of 6 feet, with side-cuts and locks having a lift of 8 feet each. The difficulty attending such an operation is the impediment which the weirs present to the passage of the river during floods; but in the case of the Severn this difficulty seems to have been overcome. Sir William Cubitt says the object of these weirs was to raise the water, and to retain it at a proper height for the navigation of the shallow parts of the river, without opposing such barriers as should prevent the free discharge of flood-water; and that this end has been completely answered, there being a depth of 6 feet of water at all times where there was formerly only a depth of 18 inches, and during floods the backwater does not rise higher than before the establishment of the weirs. A similar result may, he believes, be always attained by making the obliquity of the weirs sufficiently great. The same system has also been adopted by the late Mr Rendel and Mr Beardmore, for the improvement of the navigation of the River Lea, an account of which has been communicated by Mr Beardmore to the Institution of Civil Engineers.
The arguments, however, against canals, in consequence of the greater facilities afforded by railways, seem to apply with equal force to the upper compartments of rivers with their dams and locks; and as it is not likely that such a system of inland navigation will receive much extension, we shall, without further detail, proceed to consider tidal navigations which are more intimately connected with the commercial interests of the British Isles, and consequently occupy a more important position in the hydraulic engineering of this country.
SEC. IV.—TIDAL COMPARTMENTS OF RIVERS.
It is perhaps necessary to preface our observations on this branch of the subject by explaining what is implied by a term such as "tidal navigation," as distinguished from such large fresh-water streams as the Mississippi, or those smaller streams forming the upper compartments of our own rivers, both of which we have been considering. In the former case we saw that the art of navigation is most successfully and extensively practised on the fresh-water streams of the large continental rivers; while in the latter case it was shown that even by the aid of artificial weirs and locks, the largest rivers of this country could only be made navigable for vessels of the smallest class. We learn from this, as has also been stated, that the comparatively limited extent of our isolated country does not afford sufficient area for the collection of so large an amount of rain and spring water as to render our fresh-water streams available for the purposes of navi-
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1 Civil Engineers' and Architects' Journal, vol. xii., p. 321. 2 Transactions of Institution of Civil Engineers, vol. v. 3 Ibid., vol. xiii., p. 241. The amount of fresh water which they discharge varies as the river floods rise and fall; and even at its maximum its effects in the lower portion of our estuaries is but feebly felt, as more fully explained hereafter in section vi., under the head of "Bore." Our rivers, indeed, may be regarded simply as creeks or inlets, formed and kept open, not by the fresh-water stream alone, but mainly by the action of the tide; and may be said to be navigable only when their channels are filled by the influx of water from the ocean. The great agent, therefore, in keeping open and deepening our navigations is to be found in the tidal flow, which not only scours and maintains the sea channels of our rivers, but also increases their depth of water. Nor is this all: another most important advantage derived from the tides is that upward current due to the tidal rise, which, at first checking and ultimately overpowering and reversing the flow of the ebb-stream, carries vessels to their port, far, it may be, into the interior, without the aid of either steam or wind. This is a view of the subject which cannot fail to strike even the most superficial observer, when he sees, on the Thames or Mersey, for example, a vast fleet of vessels of all sizes, and from all countries, hurried on by the silent but powerful energy of the flowing tide. How invaluable is such an agent to the commercial interests of this country! If, indeed, the action of our river-tides were suspended, it might truly be said of the steam power employed on our railways, that its occupation would be gone. Nor need we do more to enforce the wide-spread interests of the subject than remind the reader that the ports of London, Liverpool, and Glasgow, not to name less important places, are entirely dependent on tidal navigation for their existence.
From what has been said as to the physical boundaries of rivers, it will be apparent that the extent to which this tidal influence is felt varies in different situations. Where the slope of the river is gentle, and the channel is comparatively clear and unobstructed, it is felt far up the river, as in the case of the Thames, where it reaches Teddington Weir, 65 miles from the Nore; and in the Tay, where it reaches its junction with the Almond, 35 miles from the bar. In other cases, such as the Lune in Lancashire, or the Dee in Cheshire, the tidal flow is suddenly checked by artificial weirs erected in the bed of the river for the supply of mills. In a third class of rivers the upward flow of the tide is almost neutralized by the existence of natural obstructions, as in the case of the Erne at Ballyshannon, where it flows only about 3, and the Ness, where it flows only about 6 miles up the river.
Now, the great object of the engineer, in dealing with what we have termed the "tidal compartment of a river," is to increase the tidal influence, or, in other words, to facilitate the propagation of the tidal wave through the estuary or river for which he has to design works, and it will be found in the examples we have hereafter to offer that, with proper management, this desirable improvement may be surely accomplished, and its amount accurately determined. But that the subject may be fully understood, it is necessary that we should in the outset explain the nature and laws of "tidal propagation" and "tidal currents"—phenomena attending the tides of our rivers and estuaries which must be duly recognised and estimated in all designs for improvements which are based on sound principles of river engineering.
The tidal wave which enters an estuary is a branch of the great tidal wave of the ocean. Mr Scott Russell was the first experimental inquirer who conducted investigations on the tide wave of estuaries. Mr Russell's observations were made on the Dee in Cheshire, and the Clyde, and the results which he obtained may be briefly stated as follows:
1. The great primary wave of translation differs from every other species of wave in its origin, its phenomena, and its laws. 2. The tide wave is identical with the great primary wave of translation. 3. In a rectangular channel, the velocity with which the tidal wave is propagated is equal to the velocity acquired by a heavy body falling freely by gravity through a height equal to half the depth of the fluid, reckoned from the top of the wave to the bottom of the channel. In a sloping or triangular channel the velocity is that of a gravitating body due to \( \frac{1}{4} \)d of the greatest depth. In a parabolic channel the velocity is that due to \( \frac{3}{8} \)ths or \( \frac{3}{8} \)ths of the greatest depth, according as the channel is convex or concave. And generally the velocity is that due to gravity, acting through a height equal to the depth of the centre of gravity of the transverse section of the channel below the surface of the fluid. 4. The velocity in channels of uniform depth is independent of their breadth. 5. A tidal bore is formed when the water is so shallow that the first waves of flood move with a velocity so much less than that due to the succeeding parts of the tidal wave as to be overtaken by the subsequent parts, or whenever the tide rises so rapidly that the height of the first wave of the tide exceeds the depth of water at that place. 6. A wave of high-water of spring tides travels faster than a wave of high-water of neap tides.
These laws are supposed to apply to the passage of the wave through channels having a pretty uniform depth and form of cross section; but the very irregular outline of the beds of most of our tidal channels renders it almost always difficult, and in many cases impossible, to apply them rigidly to cases which occur in actual practice. The writer may, however, state generally, in corroboration of the correctness of Mr Russell's deductions, that after investigating the tidal phenomena of many estuaries and rivers, he has found that in all cases the quickest propagation of the tidal wave occurs at those places where there is the greatest average depth; but the varying outline of the cross section renders it almost impossible, in most cases, to determine what is the ruling depth for calculating the rates of propagation in any particular section of the river. In the Dornoch Firth, to which we have already alluded, the writer found that the distance of 11 miles between Portmahomac and Meikle-ferry is traversed by the tide-wave in thirty minutes, giving a velocity of 22 miles per hour. The depth of the water of that part of the firth varies from 9 to 50 feet. Between Meikleberry, again, and the Quarry, a distance of 8 miles, where the depth is much less, varying from 6 to 20 feet, the transit of the wave occupies 65 minutes, giving a speed of 6½ miles per hour. Between the Quarry and Bonar Bridge, a distance of 1 mile, the water is comparatively shallow, varying from 1 to 3 feet, and the rise on the bed of the river is very rapid. In consequence of these obstructions, the tide does not appear at Bonar Bridge for an hour and a half after it has appeared at the Quarry, giving a rate of propagation of only two-thirds of a mile per hour. From observations made by the writer at the Dornoch Firth and elsewhere, it appears evident that, in addition to the elements on which the laws of propagation as quoted are based, the slope on the surface of the stream in tidal rivers affects to some extent the rate of propagation, independently either of the depth or cross-sectional form of the channel; but it will be more convenient to notice this at a subsequent part of this treatise.
Now, the obstructions which are most frequently found to operate as retarding influences are, the circuitous routes which of the channels of rivers, inequalities in their beds, the pro- operate in retarding tidal wave. The manner in which such tides flow up an estuary may be explained by a simple illustration. In fig. 3 the letters \(a, b, c, d\) represent a part of the low-water channel of the River Dee, at a place where the estuary is about 8 miles wide, and of tidal consists of extensive sand-banks. In examining minutely bore on the windings of the stream in reference to certain investigations, it was necessary to walk down the right bank of the river at low-water, close to the edge of the channel. While so engaged, the writer crossed at the point \(b\), a hollow in the sand-bank, which, though depressed below the general height of the surrounding surface, was nevertheless quite dry, the lowest part of the track being considerably above the level of the water of the river. Crossing this hollow, the noise of the approaching tide was heard; and expecting to meet the flood forcing its way up the river, he continued to walk on; but seeing no appearance of its approach by the proper channel, and still hearing the noise gradually increasing, and apparently coming from behind, he turned round and perceived a rapid run of water flowing (in the direction shown by the arrow) through the hollow \(d\), which had just been crossed, and emptying itself into the river at \(b\). He immediately hastened back, and after having waded through the newly-formed stream at \(b\), which had attained a depth of 6 or 8 inches, he remained on its upper side to see the result of this unexpected inroad. The water continued to rush through the hollow, rapidly gaining breadth and depth, and at last, after an interval of 2 or 24 minutes from the time at which the noise was first heard, the tide appeared forcing its way up the proper channel of the river with a head or bore of 6 or 8 inches in height.
In this case it is clear, from what has been said as to the slope on the river from Flint to Chester during the early periods of tide, that the level of the water at \(d\) in the diagram would be above that at \(b\). The tide, on arriving at the point \(d\), would be naturally divided into two branches or currents, one proceeding up the natural channel towards \(e\), and the other flowing into the hollow in the sand-bank at \(d\) towards \(e\); and as the level of the water at \(d\) rose, the stream which flowed into the hollow in the sand-bank would gradually rise higher until it surmounted the summit-level at \(e\), after which it would rush from \(e\) to \(b\) without obstruction. The other branch of the tide would in the meantime be forcing its way along the circuitous channel \(d\), which was about a mile in length; and before it reached \(b\), the water at \(d\) had attained a much higher level than at \(b\), and having surmounted the summit-level of the sand-bank at \(e\), continued to flow without obstruction into the channel of the river in the manner represented. Thus in all places where the retarding influences which exist in the regular channel of the river exceed the obstructions in any back lake or swash-way, the tide will flow sooner through the latter than the former, and give rise to an apparent anomaly such as has been described.
The late Admiral Beechey, in his Remarks on the Tidal Bore on Phenomena of the River Severn, published in 1851, gives the Severn.
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**Table:**
| Date | Rise of tide at Flint | Maximum fall from Flint to Chester | |----------|-----------------------|-----------------------------------| | 1839 May 21 | 14 ft. | 3 ft. | | | 15 ft. | 4 ft. | | | 16 ft. | 5 ft. | | | 18 ft. | 6 ft. | | June 10 | 19 ft. | 7 ft. |
| Date | Rise of tide at Glasson | Maximum fall from Glasson to Lancaster | |----------|-------------------------|----------------------------------------| | 1838 Aug 29 | 12 ft. | 1 ft. | | | 31 ft. | 1 ft. | | | Sept 1 | 15 ft. | | | 3 ft. | 10 ft. | | | 5 ft. | 23 ft. | | | | 23 ft. |
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1. The writer has found, that in all cases the heaping up of the water increases with the rise of tide, being greatest in spring and least in neap tides; as will be seen from the following tabular views of the maximum difference of level between the surface of the water at Flint and Chester on the Dee, and (to offer another example) at Glasson and Lancaster on the Lune, during the flow of tides of various amounts of vertical range. The following interesting account of the bore on that river:
"The bore," he says, "is not dangerous to boats if afloat in the middle of the river; and it is the common practice up the Severn to row the boats out to the centre of the stream on the approach of the bore, and put their head to the wave; but if this precaution be not taken, and the boats are allowed to remain at the edge of the shore, they are liable to be swamped or stove, as the waves break with great violence along the banks as it proceeds; but towards the centre of the river, if the water be not very shallow, the wave is smooth and unbroken. Before the arrival of the bore, the stream runs down the river, and the altitude of the water at a distance from the sea is quite stationary; but on the arrival of the bore, the water instantly rises according to the height of the breast of the wave, and the stream turns and follows the wave up the river, although it had but a few minutes before been running down at a rapid rate; and this change of stream is effected without any breaking wave. When there is a heavy fresh down the river, and the stream is running at the rate of four or more miles an hour, the upward stream hangs for several minutes after the bore has passed, not being able to overcome at the moment the impetus of the ebbing water; but when it has once turned upwards, it attains its maximum speed in the first half hour of the tide. When the reaches of the river are straight, the bore travels evenly up the river, but at the turnings it is thrown off towards the further side, where it rises higher than in the straight reaches; thence it recoils and impinges upon the opposite shore, and so, like a disturbed pendulum, it oscillates from side to side, and only regains its steady course when the reaches lengthen. The highest tide of the year rolled up the Severn on the 1st of December. There was about 2 feet of water above the ordinary summer-level in the river, and the morning was calm and favourable to the phenomenon. The stream at low-water ran down at the rate of 2½ miles (geographical) per hour, until the time when the bore came rolling up the river with a breast from 5 to 6 feet high at the sides, and 3 feet 6 inches in the centre. The wave was glassy smooth; and as it advanced towards a spectator stationed at Stonebench, a singular effect was produced by the distorted surface of the wave reflecting the rising sun, and brilliantly illuminating the stems and branches of the wood skirting the river as the bore passed along—an effect which greatly enhanced the interest of the phenomenon, which is at all times an object of curiosity. The stream turned up the instant after the bore passed, and ran at the rate of 3½ miles per hour, which was about half the average rate of the bore, the speed of which varied from 12 to 7 miles per hour, averaging 8 between Stonebench and Gloucester." Admiral Beechey further says, "that the effect of a fresh, or a certain depth of water in the river, upon the advance of the bore is remarkable. At dry periods the great obstruction to the progress of the bore lies between Sharpness and Bollowpool, and at such times the many dry sand-banks prevent the bore attaining a rate greater than about 4 miles an hour; but when the river is under the influence of freshes, and the water raised and covering some of the banks, it appears to roll on at a rate of 10 miles an hour in opposition to the stream, which runs down at the rate of upwards of 4 miles an hour."
But the passage of the tidal-wave through an estuary or river, must not be mistaken for what is called the "tide current," which is a totally different phenomenon. The tidal-wave which we have been describing as passing through the lower part of the Dornoch Firth, for example, at the rate of 22 miles per hour, is not the current due to the flowing tide by which vessels are carried across the bar, and borne onward to their destination. That current flows with a velocity which at the Dornoch Firth does not exceed 4 or 5 miles per hour; a velocity which, indeed, is not often exceeded, excepting in such rapid tideways as the Severn, at the New Passage, where the velocity is said to reach 9 miles per hour; and in the Pentland Firth, where Captain Otter measured a velocity during ebb-tide of no less than 10½ nautical miles per hour, being, so far as we know, the greatest tide velocity on record. The laws of the propagation of the tidal wave, to which we first alluded, depend, as explained, on circumstances somewhat obscure; but the velocity of the tide current, or that current which flows into our rivers, and affects the transit of shipping, is due entirely to the slope or fall on the surface of the water. The amount of this slope has been shown to be dependent on the rapidity with which the tide rises, and the amount of obstruction presented to its propagation up the river. The more rapid the rise of tide, and the greater the obstruction to its flow, the higher will the tide-wave at certain parts of a river or estuary be heaped up. A head of water is thus formed whose height is due to the rapidity of the rise of the tide and the obstruction to its progress; and a flow of water having a velocity due to that head is generated up the river or estuary, and this flow of water is what we term the tide current.
This is probably the most convenient place to notice some facts of great importance in river engineering, which, by increasing the rate of propagation, we deduce from these considerations, as to the nature of tidal propagation and tide currents. The obstructions which we have alluded retard the rate of propagation, whereas the but by raising the head, they increase the velocity of the velocity of tide currents. Now, as the aim, and, if successful, the tidal effect of all engineering works, is to increase the rate of currents, tidal propagation, no less certainly will they tend to lessen the heaping up of water in the lower reaches, and at the same time to decrease the velocity of the tide currents. In cases where these currents are found to act prejudicially by producing a bore, or by bringing up sand from the lower parts of the estuary, or where they are inconveniently rapid for navigation, we are thus, while increasing the propagation of the tidal wave, enabled to check their energy, and thus to effect an important improvement.
Another important circumstance is worthy of notice at this place. It is well known that the momentum of the high-column of water, flowing up the gradually contracting and water not rising channel of a river, causes the level of high-water to stand higher than in the open ocean or in the lower reaches. This is accounted for, as already stated, on the principle of the conservation of forces. The height to which the water is thus raised depends on the quantity of water thrown in by the tide during a given time, the elevation being greatest at spring, and smallest at neap tides. At the Dee, for example, the writer found that the high-water of spring tides at Chester was 14 inches higher than that at Connal's Quay; while at neap tides the difference of level was only 4 inches. Now, the effect of engineering works, as will be more fully detailed hereafter, is not only to produce a free propagation of the tide, but to admit a larger body of tidal water; and it has been contended that such operations must necessarily cause the tide to rise higher, and it has been attempted to be shown that they might in some situations occasion inconvenience, and even injury to property, in consequence of the overflowing of the river's banks. After the most careful observation, however, the writer has not been able to detect that such operations have in any case had the effect of appreciably raising the level of the high-water line. Although the tide in improved rivers begins to flow earlier, and a much larger body of
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1 Report to the Admiralty on the Severn Improvement Bill, 1849, by Captain Vetch. 2 Admiralty Survey of Pentland Firth, by Captain Otter, Admiralty coasting pilot. water is thrown up the river, still, in conformity with the views already stated, the velocity with which the water flows is decreased, so that the momentum of the column of water remains nearly the same, or at least is not so notably altered as sensibly to increase the height to which the high-water rises; and by this fortunate compensative action our rivers, though their beds are opened up and improved, do not inundate our towns or even overflow our quays, but quietly keep within their original limits.
The removal of all obstacles to the flow of the tide is the object, as already stated, to which the engineer has chiefly to direct his attention in designing improvements in the department of navigation now under consideration; and it may be stated, that in order to form a satisfactory opinion on this matter, it is essential to have an accurate survey, showing the depths of water and the breadths of channel throughout the whole extent of the river, and also to ascertain the amount of tidal range, the velocity of the currents, the rise on the bed, and the nature of the materials of which the bottom and banks are composed. Possessed of this information, he is in a position to consider to what extent the bed of the river may with advantage be deepened and widened, and the currents directed by means of walls; also if subsidiary channels may with safety be shut up, or new cuts be made for the passage of the river, or whether or not irregularities in the width which injuriously affect the currents may be corrected. In all these matters the engineer must, in each particular case, be guided by experience. While it is therefore impossible, in such circumstances to specify works which shall be of universal application, it is nevertheless quite within the range of sound engineering advice to point out generally the works which are most likely to effect improvements, and to direct the reader to cases in which such works have proved successful; and this is all that we propose to do in the remarks we have to offer on this part of our subject.
With reference to these operations, then, it may be stated, that all obstructions which prevent the extension of the tidal influence up the river may safely be taken away, and their removal may confidently be expected to be followed by highly beneficial effects. It is necessary to remark, however, that the removal of artificial weirs erected for the purposes of manufacture is, in many cases, attended with difficulty, arising from the value of the interests involved, which are sometimes so great that the abolition of such erections cannot be effected without large compensation. The weirs on the Dee in Cheshire, and the Lune in Lancashire, are instances of this, being productive of much injury; while in both cases the interests affected are so important, and the consequences so serious, as hitherto to have operated as an effectual barrier to their removal. The removal of existing quays and other works of long standing, as in the case of the Thames, the Tyne, and the Wear, is also for the same reason difficult, and works must therefore be designed for such localities which shall not injuriously affect existing interests. But all natural weirs or shoals, consisting of fixed rock or hard gravel, which cannot be disturbed by the action of the current, as well as all projections into the stream, where unattended by the difficulties alluded to, should at once be removed. Whenever it is possible, divided currents should be united into one stream. The channel, where it is necessary, should be guided by longitudinal walls, and the river's bed should be deepened to the full extent compatible with a due amount of slope being left on the surface.
These may be said to be the safest and most beneficial works which can be adopted in designing river improvements, their effect being to cause the currents of flood and ebb tide to flow always in one channel, and thus to exert their full and combined power in keeping open one navigable track. The manner in which they are executed demands a few remarks; and we shall treat the different works under the heads of:
1. Removal of lateral obstructions. 2. Closing subsidiary channels. 3. Dredging. 4. Excavation. 5. River walls; and 6. Scouring.
1. Removal of Lateral Obstructions.
Under the "Removal of Lateral Obstructions" may be classed all those works which have for their object the formation of proper outlines for the banks or sides of the river. In the early history of river engineering it was not uncommon to construct jetties or groins projecting from the banks on either side, with the view of narrowing the stream and producing a greater scouring power to operate on the bottom. It is no doubt true, that such projections have the effect of producing a local acceleration of the currents, and in soft bottoms a corresponding increase of depth in their immediate vicinity. But this increase of velocity and depth being due entirely to the obstruction and consequent raising of the level of the water caused by the jetty, is strictly local. Whenever the water passes the head of the jetty, it expands into the greater width of bed, the head is reduced, a stagnation or eddy takes places, and a bank or shoal is formed,—a result which invariably follows the projection of any obstruction or foreign body into a stream having a soft bottom. As an aggravated instance of the effect of such obstructions, we may refer to the case of a vessel of about 170 tons, which, in consequence of the breaking of a towline, grounded at the side of the River Tay when there was some flood in the river. The effect is shown in fig. 4, where the vessel is represented at \(a\) as lying in a pool which was scoured to the depth of about 10 feet in the course of a few tides; and the gravel thus excavated by the current, acting on the grounded vessel, and amounting to upwards of 1000 tons, was deposited in the form of a bank, 5 feet above low-water, immediately below the pool, as shown in hatched lines. A similar effect, though varying in degree, occurs in all rivers confined by jetties. The beds of rivers so treated consist of an alternation of shoals nearly dry at low-water, and pools of a depth far greater than is actually requisite, instead of presenting, as they ought to do, a regular bottom and a uniform depth of water available for the purposes of navigation. Examples of the prejudicial effect of jetties are to be met with in the history of the Clyde, the Ribble, the Dee in Cheshire, the Tay, and, the writer believes, with little or no exception, in every situation where the system of contracting, or even directing the currents by means of such works, has been generally adopted. From the Clyde, the Ribble, and the Tay they have been entirely removed. The writer has invariably found, that whenever jetties existed, their entire or partial removal formed one of the first steps towards an improvement of the navigation, and this course has, in all cases which have come within his experience, been followed by good results. In some instances, where the river is contracted by the projection of quays or by the natural formation of the banks, it is desirable, where it can be done consistently with existing interests, to enlarge the cross-sectional area, in order to reduce the velocity of the currents and prevent disturbance of the tidal flow. 2. Closing Subsidiary Channels.
The next work to be noticed is the closing of what we term subsidiary channels. These are channels, or, as they are sometimes called, back lakes, caused by islands which divide the stream and reduce its scouring power. The consequence is, that instead of flowing in one broad, deep, navigable bed, kept open by the whole available scouring power, the river is divided into two shallow channels, neither of them affording a good navigation, while frequently a ford or shallow is occasioned both above and below the island by the disturbance which occurs at the junction of the divided currents. On the Tay and the Lune several such secondary channels were, with much advantage to the navigation, closed up by means of embankments formed of gravel dredged from the river, while the other or principal channel was enlarged and deepened, so as fully to compensate for the closing of the smaller channel, and assimilate its cross-sectional area to the rest of the navigable track.
3. Dredging.
The introduction of mechanical appliances for the purpose of excavating materials under water, raising them to the surface, and depositing them in barges, was an important era in canal and river engineering. The first employment of machinery to effect this important object is, like the discovery of the canal lock, claimed alike for Holland and Italy, in both of which countries dredging is believed to have been practised before it was introduced into Britain. The moving power at first employed in conducting the process was manual labour, but in all large works dredging is now performed by steam, and is probably the most effective and generally applicable means of improvement at the command of the engineer. The Dutch, at a very early period, employed what is termed the "bag and spoon" dredge for cleaning their canals. It consisted of a ring of iron about 2 feet in diameter, flattened and steeled for about one-third of its circumference; to this ring a bag of strong leather was attached by means of thongs, and the whole apparatus was fixed to a long pole, which, on being used, was lowered to the bottom from the end of a barge moored in the canal or river. A rope made fast to the iron ring was then wound up by a windlass placed at the other end of the barge, and the spoon was thus dragged along the bottom, and was guided in its progress by a man who held the pole. When the spoon reached the end of the barge where the windlass was placed, the winding was still continued, and it was raised to the surface, bringing with it the stuff excavated, and deposited in the bag during its progress along the bottom. The windlass being still wrought, the whole was raised to the gunwale of the barge, and the bag being emptied, was again lowered and hauled back to the opposite end of the barge for another supply. This system is slow, and only adapted to a limited depth of water and a soft bottom. It has, however, been generally employed in canals, and was much used in the Thames; and the writer, in one situation where, from want of space and other peculiarities, more perfect mechanical means could not be employed, used it to a pretty large extent, the quantity raised being about 135,000 tons. The process, although tedious, was very convenient, and the cost of raising the materials did not exceed £2½d. per ton. Another plan practised at an early period was to moor two large barges, one on either side of the river; between them was slung an iron bucket or box, attached to both barges by chains wound round the barrels of a powerful crab-winch in one barge, and round a capstan in the other. The bucket was lowered at the side of the barge in which was the capstan, and being drawn across the bottom by the crab in the opposite barge, was raised and emptied; after which it was again lowered, and hauled across by the capstan for a repetition of the process. But in all large operations these and other primitive appliances have, as already stated, been superseded by the steam-dredge, which was first employed, it is believed, in deepening the Wear at Sunderland, about the year 1796. This machine was made for Mr Grinshaw by Bolton and Watt. Receiving improvements from Mr Hughes, Mr Rennie, Mr Jessop, and others, the steam-dredge, as now generally constructed, is a most efficient machine, excavating and raising materials from the depths of 15 to 20 feet of water, at a cost not very different from that at which the same work could be performed on dry land.
For details as to the construction of steam-dredges, we have to refer to the articles in Weale's Quarterly Papers, already quoted. As to the nature and extent of work performed by them, we may state generally, that almost all materials, excepting rock or very large boulders, may be dredged with ease. Loose gravel is probably the most favourable material to work in; but a powerful dredge will readily break up and raise indurated beds of gravel, clay, and boulders. In such cases it is usual to alternate on the bucket-frame, a bucket of sheet-iron for raising the stuff, with a rake or pronged instrument for disturbing the bottom. Hand-dredges have been used by Messrs Stevenson at several harbours, by means of which, even disintegrated or rotten rock has been easily raised; and the writer believes, that in very many cases the surfaces of submerged rocks near the mouths of harbours may, by means of such machines, be broken up and removed, so as to obtain in certain situations a considerable increase of depth, without recourse to coffer-dams, which, on exposed coasts, involve great expense and sea risk, as well as interruption to the trade. These small dredges are worked by eight or ten men, and cost about £350.
A well-constructed steam-dredge of 16 horse-power will, under favourable circumstances, raise about 140 tons of stuff per hour. The excavated materials are first discharged into lighters or barges, and then deposited in any convenient position, where they are sufficiently removed from the risk of being carried off by floods, and again thrown into the bed of the river.
In some cases the discharge is made into hopper punts or barges, which are floated out to sea, and the stuff is dropped in deep water. The cost of steam-dredging varies according to the nature of the materials and the circumstances of working, as regulated by the tides and the distance of deposit. It has, in the writer's own experience, varied from 4½d. to 6½d. per ton, or from 5½d. to 8½d. per cubic yard, including all expenses. We believe that in no place has steam-dredging been more extensively used than in the Clyde, where the navigable depth has been increased and is maintained mainly by that process. The following details as to the dredging on that river are given in a communication made to the Institute of France by the late Mr William Bald, who acted as resident engineer on the Clyde. Mr Bald says, that annual dredging to the amount of from 160,000 to 180,000 tons was necessary at the time he wrote, in order to maintain the navigable depth of water in the Clyde in the 18 miles from Glasgow seawards. In execut-
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1 Encyclopaedia of Civil Engineering, by Edward Cressy, London, 1847; "The Dredging Machine," Weale's Quarterly Papers, part I., London, 1843; The Improvement of the Port of London, by R. Dodd, Engineer, 1798. 2 It was found on the Tay that 18 cubic feet of gravel weighed 1 ton. 3 Civil Engineers' and Architects' Journal for August 1845. All these dredges had governors, which regulated the speed to about 28 strokes per minute in ordinary working stuff. The average pressure in boilers was about 3½ lb. per square inch. In general, 14 buckets were discharged per minute. The speed of the buckets on the frames of dredges Nos. 1, 2, 3, and 4, was 48 feet 5 inches per minute, and that on No. 5, 49 feet 8 inches per minute. They consumed from 15½ to 18 lb. of coal per horsepower per hour. The following is a statement of the amount of work which was performed by these dredges, and the expense of the process:
| Year ending | Amount expended | Work executed | Rate per Cubic Yard | |-------------|----------------|---------------|--------------------| | December 25, 1841 | £11,841 18s 2d | £218,110 | £1 1 | | 24, 1842 | £13,612 11s 3d | £313,810 | £0 10½ | | 23, 1843 | £9,742 7s 6d | £294,440 | £0 8 | | 21, 1844 | £10,659 3s 8d | £317,660 | £0 8 |
Tabular View of the Dredging of the Wear at Sunderland in 1842-46.
| Date | Total Quantity Raised per Annum | Expenditure in Labour for Raising and Depositing per Annum | Expenditure in Fuel per Annum | Expenditure in Labour for Repairs per Annum | Expenditure in Materials for Repairs per Annum | Total Expenditure per Annum | Average cost per Ton on the Year's Expenditure | |------|--------------------------------|----------------------------------------------------------|-------------------------------|--------------------------------------------|-----------------------------------------------|---------------------------------|-----------------------------------------------| | 1842 | 128,245 tons | £922 1s 2d | £111 0s 0d | £754 16s 0d | £704 17s 11d | £2492 15s 1d | £4·665 | | 1843 | 141,325 tons | £879 16s 0d | £70 0s 0d | £603 13s 4d | £786 13s 11d | £2240 3s 3d | £3·804 | | 1844 | 90,980 tons | £567 13s 4d | £65 5s 9d | £259 2s 1d | £663 9s 10d | £1456 11s 0d | £3·842 | | 1845 | 101,075 tons | £721 9s 0d | £65 7s 6d | £336 8s 0d | £527 7s 10d | £1651 12s 4d | £3·921 | | 1846 | 140,350 tons | £724 5s 4d | £58 2s 5d | £500 17s 2d | £520 3s 2d | £1803 8s 1d | £3·083 |
Hence the average cost per ton on five years' work: For raising and depositing at sea = £1·628 For fuel = £0·149 For labour in repairs = £0·943 For materials to ditto = £1·243
Average total Expenditure = £3·869
Mr Murray gives the above tabular view of the dredging of the Wear at Sunderland, which is also an interesting record of the quantity and cost of material raised by a dredging-machine; but this view is not given by way of comparison with the preceding, as there is little analogy between the cases. The contracted state of the Clyde, the frequent interruptions to which the work was subject by the constant passage of vessels, and the expense of removing and depositing the stuff, necessarily increased the cost of executing the work in that situation.
In river-dredging two systems are pursued; one plan consists in excavating a series of longitudinal furrows parallel to the axis of the stream, the other in dredging cross furrows from side to side of the river. It is found that inequalities are left between the longitudinal furrows, when that system is practised, which do not occur to the same extent in side or cross-dredging; and the writer has invariably found cross-dredging to leave the most uniform bottom. To explain the difference between the two systems of dredging, it may be stated, that in either case the dredge is moored from the head and stern by chains about 250 fathoms in length. These chains in improved dredges are wound round windlasses worked by the engine, so that the vessel can be moved ahead or astern, by simply throwing them into or out of gear. In longitudinal dredging, the vessel is worked forward by the head chain, while the buckets are at the same time performing the excavation; so that a longitudinal trench is made in the bottom of the river. When the dredge has proceeded a certain length, it is stopped and permitted to drop down and commence a new longitudinal furrow parallel to the former one. In cross-dredging, on the other hand, the vessel is supplied with two additional moorings, one at either side, and there
chains are, like the head and stern chains, wound round barrels wrought by the engine. In commencing to work by cross-dredging, we may suppose the vessel to be at one side of the channel to be excavated. The bucket-frame is set in motion, but instead of the dredge being drawn forward by the head chain, she is drawn to the opposite side of the river by the side chain, and having reached the extent of her work in that direction, she is then drawn a few feet forward by the head chain; and the bucket-frame being yet in motion, the vessel is hauled back again by the opposite side chains to the side from whence she started. By means of this transverse motion of the dredge, a series of cross furrows is made; she takes out the whole excavation from side to side as she goes on, and leaves no protuberances such as are found to exist between the furrows of longitudinal dredging, even where it is executed with great care. The two systems will be best explained by reference to the annexed cut (fig. 5), where AB represents the head and stern moorings, and DC the side moorings; the arc of represents the course of the vessel in cross-dredging; while in longitudinal dredging, as already ex- plained, she is drawn forward towards A, and again dropped down to commence a new longitudinal furrow.
In some cases, however, the bottom is found to be too hard to be dredged until it has been to some extent loosened and broken up. Thus at Newry, Mr Rennie, after blasting the bottom in a depth of from 6 to 8 feet at low-water, then removed the material by dredging, at an expense of from 4s. to 5s. per cubic yard. The same process was adopted by Messrs Stevenson at the bar of the Erne at Ballyshannon, where, in a situation exposed to a heavy sea, large quantities of boulder stones were blasted, and afterwards raised by a dredger worked by hand, at a cost of about 10s. 6d. per cubic yard. But the most extensive application of blasting, preparatory to dredging, of which the writer is aware, was that on the works for improving the Severn, by Sir William Cubitt, of which an interesting and instructive account is given by Mr George Edwards, in a paper addressed to the Institution of Civil Engineers, from which the following particulars are taken:
"It appears that a succession of marl beds, varying from 100 yards to half a mile in length, were found in the channel of the Severn, which proved too hard for being dredged, the whole quantity that could be raised being only 50 or 60 tons per day; while the machinery of the dredges employed was constantly giving way. Attempts were first made to drive iron rods into the marl bed, and to break it up; a second attempt was made to loosen it by dragging across its surface an instrument like a strong plough. But these plans proving unsuccessful, it was determined to blast the whole surface to be operated on. The marl was very dense, its weight being 146 lb. per cubic foot; and it was determined to drill perpendicular bores, 6 feet apart, to the depth of 2 feet below the level of the bottom to be dredged out. The bores were made in the following manner, from floating rafts moored in the river:—Pipes of \( \frac{3}{8} \)-inch wrought-iron, \( \frac{3}{8} \) inches diameter, were driven a few inches into the marl. Through these pipes holes were bored, first with a \( \frac{1}{4} \)-inch jumper, and then with an anger. The holes were bored 2 feet below the proposed bottom of the dredging, as it was expected that each shot would dislocate or break in pieces a mass of marl of a conical form, of which the bore-hole would be the centre and its bottom the apex; so that the adjoining shots would leave between them a pyramidal piece of marl, where the powder would have produced little or no effect. By carrying the shot-holes lower than the intended dredging, the apex only of this pyramid was left to be removed; and in practice this was found to form but a small impediment. Fig. 6 is a section, and fig. 7 a plan of the bore-holes; the inner dotted circles represent the diameters of the broken spaces at the level of the bottom of dredging. The cartridges were formed in the ordinary way, with canvas, and fired with Pickford's fuse. The weight of powder used for bore-holes of 4 feet, 4 feet 6 inches, and 5 feet, were respectively 2 lb., 3 lb., and 4 lb. The effect of the shot was generally to lift the pipes a few inches, which were secured by ropes to the rafts. Mr Edwards says that not one in a hundred shots missed fire, and these shots were generally saved by the following singular expedient:—The pointed end of an iron bar, \( \frac{3}{8} \)-inch diameter, was made red-hot, and being put quickly through the water, and driven through the tamponing as rapidly as possible, was in nine cases out of ten sufficiently hot to ignite the gunpowder and fire the shot.
"The cost of each shot is calculated as follows:
| Use of material | £0 1 0 | |----------------|-------| | Labour | 0 3 3 | | Pitched bag for charge | 0 0 0 | | 3 lb. of powder at 4d. | 0 1 4 | | 15 feet of patent fuse at \( \frac{1}{4} \)ths of a penny | 0 0 9 | | Pitch, tallow, twine, coals, &c. | 0 0 4 |
Cost per shot | £0 7 0 |
Each shot loosened and prepared for dredging about 4 cubic yards; so that the cost for blasting was 1s. 9d. per yard. The cost of dredging the material, after it had been thus prepared, was 2s. 3d.; making the whole charge for removing the marl 4s. per cubic yard."
4. Excavation.
But there are cases where the bottom cannot be advantageously operated on by any of the means we have mentioned, and where it is necessary to have recourse to other appliances for its removal, such as the diving-bell or diving-helmet, and coffer-dams. The diving-bell has, in conjunction with dredging, been much used on the Clyde, and Mr Bald gives the following account of the operation as conducted on that river:
"Between Erskine Ferry and the New Shot Isle the bed of the Clyde, for a distance of 2000 yards, was greatly encumbered with stones and stone boulders, which were highly injurious to vessels if they grounded there; and frequently large ships, in being tugged through this part of the river-channel, had their copper bottoms injured when they touched the rocky channel-bed. In deepening and clearing this part of the river, two diving-bells were employed, and one, and sometimes two, steam-dredgers. The clearing and deepening of this channel was exceedingly severe on the machinery and working-gear of the steam-dredgers; the speed of the engines was therefore governed by the nature of the material in the bottom; and although the iron-work frequently gave way, yet spare links and buckets being always ready to replace those
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1 "Account of Blasting on the Severn," by George Edwards, C.E. (Trans. of Institution of Civil Engineers, vol. iv., p. 361). 2 Clay weighs about 100 lb., and sandstone about 155 lb. per cubic foot. which broke, there was little interruption to the continuous working of the dredgers. When the dredgers had cleared away the material which covered the boulders in the bottom of the channel, the diving-bell boats were worked over the ground so cleared, removing all the larger boulders; and when that part of the channel had been cleared of them, the dredgers went again over the same bottom, removing all the lighter material from the heads of the lower boulders, preparatory to the bells commencing again; and these operations were continued until the necessary depth was attained.
"The buckets of the steam-dredgers, in working along the bottom, always slipped over the head of the large boulders, which the diving-bells alone could lift and remove. Some of these masses of trap or whinstone were 4 and 5 tons in weight, and from their rounded forms and smooth surfaces, it was evident that they had been brought from some distance. Some of them were of sandstone, but they were more angular than the trap boulders. Quantities of these boulders, lifted from the bed of the channel, might be seen lying along the sides of the river; and many of them had since been split and broken up by gunpowder for repairing the river dykes. The tops of some of the large stone boulders lifted from the bed of the channel were found grooved to a depth of about an inch or more, by the ship's keels having been rubbing over them; and metallic particles were distinctly to be seen upon their surface. In removing these stone boulders from the bed of the channel, the diving-bell men found numerous fragments of copper and iron which had been torn off the ship's bottoms and keels by the large stones; but latterly this had not been the case, as great progress had been made in the removal of the boulders, and the deepening of the channel."
Large isolated masses of stone have also been removed from many rivers by fixing lousies in them, and raising them by floatation. On the Tay this was done to some extent, and one boulder of 50 tons was raised from the river by that means. Where a large area and considerable depth of solid rock has to be removed, coffer-dams are doubtless the best means of executing the work; but the chief difficulty in employing dams in the narrow channels of rivers is the obstruction which they necessarily present to the passage of floods and also to shipping. It is therefore a matter of high importance to reduce their bulk to the smallest possible limits. With this view the writer designed a coffer-dam for the works of the River Ribble, dams which consisted of two rows of iron rods, 3 feet apart, jumped into the rocky bottom, and supporting two linings of planking, the intermediate space being filled with clay, and the whole structure being stayed from the inside, so as to present no obstruction beyond the outer line of the dam. Three dams of this construction were formed in the Ribble; and by means of them, a bed of rock, 300 yards in length, and of a maximum depth of 13 feet 6 inches, was successfully excavated. The maximum depth of water at high-water against the dam was 16 feet, but in very high floods of the river the whole dam was sometimes completely submerged; but on the water subsiding, it was found that the iron rods, on which alone its stability depended, although only jumped 15 inches into the rock, were not drawn from their fixtures. As this construction of dam completely overcomes the difficulty of fixtures in a hard bottom, where piles cannot be driven, and offers very little obstruction to the navigation; and moreover, as it has been successfully used on a large scale, and seems to fulfil all the conditions demanded in such a situation, it may be perhaps considered generally applicable to situations where there is a hard bottom and limited space. The sketch (fig. 8) shows, by an elevation, section, and plan, the manner in which it was constructed.
5. River-Walls.
In open estuaries filled with sand-banks, the courses of rivers are liable to constant alteration, due to every change in the tides or winds. The accompanying woodcut (fig. 9) of the River Lune illustrates this remark; the several dotted lines represent the variation of the channel during the period of a few years. This tendency to wander is common to all rivers when left undirected to work their way through a
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1 "Description of a Coffer-Dam adapted to a Hard Bottom," by David Stevenson, C.E. (Trans. of Inst. of Civil Engineers, vol. iii., p. 277). tract of sand; and the evils attending such a state of matters are generally of a serious nature, proceeding mainly from a constant abrasion or wasting of the sand-banks. This abrading action, operating during every flood and ebb tide, sets loose a large amount of floating sand, which is drifted to and fro, and deposited in some new situation. A channel
which is constantly shifting its course never remains sufficiently long in one position to form for itself a properly defined bed, but is in fact always in a transition state; the sand which is worn from the concave being thrown to the convex side of the stream, while some portion of the floating materials, carried to and fro during this process of perpetual change, is often deposited, and forms shoals in the middle of the fairway. A river left in this state of nature cannot possibly attain the maximum depth due to the natural scour of the tidal currents, as their power is expended in abrading and removing the sand-banks through which the stream flows, and not, as it ought to be, in deepening and scouring its bed. In such cases what is wanted is to secure a permanent channel, by guiding the first of the flood and the last of the ebb tide by means of walls, so that the strength of the currents may constantly operate on the same line of channel. In this way, it is obvious that not only will the advantage of a permanent navigable track be obtained, but the constant action of the currents of flood and ebb tide flowing in the same channel, will secure a much greater permanent depth than they could possibly do if permitted to wander at random through the estuary, sometimes operating in the same channel, and at other times directly opposed to each other.
Questions have been raised as to the comparative advantages of straight and curved walls for directing a channel. It is believed that, in most cases, the direction of such walls is necessarily determined, not by any abstract consideration as to the superiority of straight or curved walls, but chiefly by the relative positions of the points between which the stream is to be conducted, and the outline and geological formation of the shores and banks of the estuary that intervene between those points. The consideration of such matters may render it expedient, according to the special circumstances of the locality, to adopt walls having concave, straight, or convex outlines, as shown in figs. 10, 11, and 12.
Viewed as a purely abstract question, it may, we think, be safely affirmed, that a stream is most likely to follow a permanent course when directed by a concave wall, as shown in fig. 10, in which the axis of the stream is represented by the dotted line. Dr Young observes that the centrifugal force in curved channels has a tendency to draw the greater portion of the water to the concave side, and thus the greatest scouring power, and consequently the greatest depth of the stream, will be found upon that side. In a channel directed by straight walls (fig. 11), the current has no such decided bias for either wall, and is consequently easily thrown across from side to side. A wall, on the other hand, having a convex outline, as shown in fig. 12, is (especially if the radius of curvature be small) still less suitable as a guide, as the line of wall diverges from the direction of the axis of the current. These remarks are not hypothetical, as the writer has found that their correctness has been verified by cases in actual practice. There is doubtless some disadvantage in the deep water being on one side of the channel, as more particularly shown in the cross section, fig. 13. It would be more convenient for navigation were the deep water in the centre; but it is found that the current invariably adheres to one or other of the walls, and it is better that the channel should keep constantly to one wall, than that it should alternate from side to side, as is more apt to be the case in absolutely straight channels.
The direction and extent of river-walls must, however, be carefully considered by the engineer with reference to existing circumstances, and every case must be judged per se. But we think it will be found safe, in executing such works, to adhere as closely as possible to the following general rules:
First, The channel through open estuaries should, in all cases where funds will admit of it, be guided by double walls. In cases, however, where the estuary is bounded by a hard beach, presenting a favourable line of direction, a single wall may occasionally be found sufficient. All curves which it may be necessary to introduce should be of as large a radius as possible, and should, if practicable, be tangential to each other, or to the straight parts of the line with which they are connected.
Second, The walls should not be raised to a higher level above the low-water line than is absolutely necessary for the purpose of conducting the early and late currents of the not prevent the tide at high-water from flowing on either side of them and filling the estuary.
Third, River-walls should, during their erection, be pushed forward with vigour, and not in a desultory, timid manner; the effect of such a course being to increase the depth of water in which the wall has to be made, and the amount of stone required for its construction.
Fourth, It will be found that such walls as we have been describing will be most advantageously formed of rough rubble stones, backed with clay and gravel, in the manner shown in fig. 15.
It was found by Mr Park, under whose immediate directions, as local engineer, the walls on the River Ribble, which are about 12 miles in length, were constructed, that their foundations, with few exceptions, did not sink more than a few feet below the sand. He found that it was advantageous to mix clay in the internal core of the wall; and after the materials were deposited, it was necessary from time to time, in certain places, to add additional stones to make up slips, before attempting to pitch the top or the face of the slope. Walls somewhat similar have also been largely introduced on the Clyde by Mr Walker.
6. Scouring.
The removal of hard portions of the bed of a river by dredging or coffer-dams, and the direction of the channel by low walls, are operations which are in themselves improvements; but they further operate beneficially in causing the currents to scour the softer parts of the river's bed, so that it sometimes happens that by dredging a few hundred yards of hard material from a river's bed, or erecting a short wall, thousands of tons of soft materials are scoured away by the action of the current. In all river improvements this is an effect which should be fully taken into consideration by the engineer, especially in forming estimates; and its importance will be apparent on inspecting the section of the River Lune (Plate II.). By dredging the upper shoals of that river, which are marked in hatched lines in the section, the whole lower part of the river was deepened by the natural scour, without entailing any expense in its removal. To facilitate this scour, a species of harrow has sometimes been applied, which is drawn to and fro by a tug-steamer across the bank to be removed. This system was extensively employed by Captain Denham in opening the Victoria Channel at the Mersey; it was also employed by Messrs Stevenson at the Tay; but it is obvious that it can only be advantageously used where there is deep water in the immediate neighbourhood of the bank to be removed, in which the sand and mud disturbed by the harrow, and carried off by the current, may be deposited. The process of scouring has, in some situations, to the knowledge of the writer, continued in operation for many years after the completion of the original work, the low-water level of the river continuing gradually to sink; and as this process goes on, it sometimes happens that hard portions of the bottom originally covered become gradually exposed. Such obstructions are, in fact, hard portions of the bed brought to light, in consequence of the improvement of the river, and must not be mistaken for accumulations due to ill-regulated currents. It is necessary, however, that such hard portions should be removed as soon as they appear, otherwise they disturb the currents and occasion shoals. Whenever the depth due to the currents acting in their improved direction has been reached, such obstructions will cease to present themselves.
The effect of works executed according to the principles indicated is,—First, to fix the navigable track in a defined course; second, to deepen the bed of the river; third, to reduce the slope, and lower the low-water level; and fourth, to increase the duration of tidal influence and the quantity of tidal water in the river. The benefits to navigation are threefold—First, greater depth of water; second, a properly defined channel; and, third, a greater length of time during which, in consequence of the presence of the tide, the river is navigable.
The writer has, before leaving this part of the subject, to state that the works specified are believed to be those most generally applicable. All of them may not be applicable in every case; and there may be special cases which render it expedient to adopt works of a somewhat different, and, in some respects, apparently antagonistic character,—such, for example, as the contraction of channels by means of quay-walls.
SECT. V.—APPLICATION OF THESE WORKS IN PRACTICE.
We come now to give a sketch of navigation improvements, executed in accordance with these general views, and to show their application in practice, and the effect produced by them. The first example to which we shall allude is the River Tay, improved under the advice and direction of Messrs Stevenson of Edinburgh; as we know of no instance in which the improvements effected by particular works are more fully and satisfactorily demonstrated by a comparison
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1 Mr Rendel, in his address as president of the Institution of Civil Engineers in 1852, says,—"At the present moment changes are taking place in the Thames and most of the principal rivers, which afford invaluable opportunity for observations on the effects rivers can produce by their own action, and also on what is done by the passage of steam-vessels in keeping the lighter silt constantly in motion."
2 Admiral Beechey, in his Observations on the Tides of the River Severn, mentions a fact which it is proper to record. He says, "While upon the subject of the low-water line, it may here be remarked, that the inverse of the ordinary effect of the spring-tide occurs in the river above Lidney. From Lidney downwards to the sea, the low-water at springs follows the general rule of being lower at such times than at the neaps; but above Lidney the reverse takes place, the low-water at the springs being higher than at the neaps. This, no doubt, is occasioned by the tide at springs throwing more water into the river than can escape before the return of the following tide." of observations made previously and subsequently to their execution, than in the case of that navigation; where the changes were brought about in so short a time, and were so marked as to leave no doubt, even to superficial observers, of their attainment, and no difficulty, by the use of proper means, in ascertaining their amount.
The River Tay, with its numerous tributaries, as stated in the table at the end of this treatise, receives the drainage-water of a district of Scotland amounting to 2283 square miles, as measured on Arrowsmith's Map. Its mean discharge has been ascertained to be 274,000 cubic feet, or 7645 tons of water per minute. It is navigable as far as Perth, which is 22 miles from Dundee, and 32 from the German Ocean. The different points on the river, hereafter to be referred to, will be seen in the small chart given in Plate II.; and we propose, in this particular case, to enter somewhat into detail as to the nature of the obstructions to the navigation, the means employed for their removal, and the effects produced by the works on the tidal currents; as the remarks made will, it is believed, serve to illustrate the subject of river improvement generally.
Before the commencement of the works, certain ridges, called "fords," stretched across the bed of the river at different points between Perth and Newburgh, and obstructed the passage to such a degree, that vessels drawing from 10 to 11 feet could not, during the highest tides, make their way up to Perth without great difficulty. The depth of water on these fords, the most objectionable of which were six in number, varied from 1 foot 9 inches to 2 feet 6 inches at low, and 11 feet 9 inches to 14 feet at high water of spring tides; so that the regulating navigable depth, under the most favourable circumstances, could not be reckoned at more than 11 feet. In addition to the shallowness of the water, many detached boulder stones lay scattered over the bottom. Numerous "fishing cairns," or collections of stones and gravel, had also been laid down, without regard to any object but the special one in which the salmon-fishers were interested, and in many cases they formed very prominent and dangerous obstructions to vessels. The chief disadvantages experienced by vessels in the unimproved state of the river was the risk of their being detained by grounding, or being otherwise obstructed at these defective places, so as to lose the tide at Perth—a misfortune which, at times when the tides were falling from springs to neaps, often led to the necessity either of lightening the vessel, or of detaining her till the succeeding springs afforded sufficient depth for passing the fords. The great object aimed at, therefore, was to remove every cause of detention, and facilitate the propagation of the tidal wave in the upper part of the river, so that inward-bound vessels might take the first of the flood to enable them to reach Perth in one tide. Nor was it, indeed, less important to remove every obstacle that might prevent outward-bound vessels from reaching Newburgh, and the more open and deep parts of the navigation, before low-water of the tide with which they left Perth.
The works undertaken by the harbour commissioners of Perth for the purpose of remedying the evils alluded to, and which extended over six working seasons, may be briefly described as follows:
1st. The fords, and many intermediate shallows, were deepened by steam-dredging; and the system of harrowing was employed in some of the softer banks in the lower part of the river. The large detached boulders and "fishing cairns," which obstructed the passage of vessels, were also removed.
2d. Three subsidiary channels, or offshoots from the main stream, at Sleepless, Darry, and Balhepurn islands, the positions of which will be seen on the plan, were shut up by embankments formed of the produce of the dredging, so as to confine the whole of the water to the navigable channel.
3d. In some places the banks on either side of the river beyond low-water mark, where much contracted, were excavated, in order to equalize the currents, by allowing sufficient space for the free passage of the water; and this was more especially done on the shores opposite Sleepless and Darry islands, where the shutting up of the secondary channels rendered it more necessary.
The benefit to the navigation in consequence of the completion of these works has been of a twofold kind; for not only has the depth of water been materially increased by actual deepening of the water-way, and the removal of numerous obstructions from the bed of the river, but a clearer and freer passage has been made for the flow of the tide, which now begins to rise at Perth much sooner than before; and as the time of high-water is unaltered, the advantages of increased depth due to the presence of the tide is proportionally increased throughout the whole range of the navigation; or, in other words, the duration of tidal influence has been prolonged.
The depths at the shallowest places are now pretty nearly equalized, being 5 feet at low and 15 feet at high water, of ordinary spring tides, instead, as formerly, of 1 foot 9 inches at low and 11 feet at high water. Steamers of small draught of water can now therefore ply at low-water, and vessels drawing 14 feet can now come up to Perth in one tide with ease and safety.
In obtaining the requisite data, both as to the design and execution of these works, minute tidal observations were made at various times during a period of ten years, from 1833 to 1844 inclusive, throughout the River and Firth of Tay, at the following stations,—viz., Dundee, which is marked No. 1 on the plan, Plate II.; Balmerino, No. 2; Flisk Point, No. 3; Balmreich Castle, No. 4; Newburgh, No. 5; Carpow, No. 6; Kinsauns, No. 7; and Perth tide harbour, No. 8. The general results deduced from these observations are given in the following tables, and show, by the favourable change which has been effected in the tidal phenomena of the estuary, that the works executed fully answered the intended end:
1. Propagation of Tidal Wave.
The following table of elapsed times, between arrival of the tide-wave, or commencement of the tidal flow, at the following stations, during spring tides in 1833 and 1844, shows the rate of its propagation:
| Station | Time (h.m.) | Distance in Miles | Rate of Tide-Wave in Miles per Hour | |------------------|-------------|-------------------|------------------------------------| | Dundee to Balmerino | 0 15 | 5 00 | 18 75 | | Balmerino to Flisk Point | 0 29 | 2 93 | 6 06 | | Flisk Point to Balmreich | 0 25 | 2 04 | 4 69 | | Balmreich to Newburgh | 0 53 | 3 42 | 3 86 | | Newburgh to Perth (tide harbour) | 2 30 | 8 56 | 3 42 |
The result of observations made in 1842, 1843, and 1844, on spring tides, give the same velocity, as above stated, between Dundee and Newburgh, and the following rates between Newburgh and Perth:
| Station | Time (h.m.) | Distance in Miles | Rate of Tide-Wave in Miles per Hour | |------------------|-------------|-------------------|------------------------------------| | Newburgh to Carpow | 0 25 | 1 33 | 3 17 | | Carpow to Kinsauns | 0 53 | 4 92 | 5 35 | | Kinsauns to Perth (tide harbour) | 0 20 | 2 32 | 6 93 |
Giving, as a mean for the whole distance from Newburgh to Perth in 1844... | 1 40 | 8 56 | 5 13 | Time from Newburgh to Perth in 1833... | 2 30 | 8 56 | 3 42 |
Thus showing an increase in the velocity of the tide-wave in the upper part of the river, which was improved, of more than 1 mile per hour as the result of the improvements.
The difference of the time in neap tides between Newburgh and Perth in 1844, was 1 h. 53 m. 2. High-Water Level.
The levels of the surface of high-water at different stations throughout the river have been found to be unchanged, and the following results refer to the years 1833 and 1844:
From Fisk Point to Balmreich there is a fall of 5 in. Balmreich to Newburgh there is a rise of 7½ in. Newburgh to Perth (tide harbour) there is a rise of 18 in.
From Fisk to Balmreich there is a fall of 2½ in. Balmreich to Newburgh there is a rise of 6 in. Newburgh to Perth (tide harbour) there is a rise of 12 in.
3. Low-Water Level.
Rise on the Surface of Low-water (Spring Tides) in 1833.
| Ft. in. | Distance in Miles | Rate of Spring Tide in Inches per Mile | |--------|------------------|--------------------------------------| | Fisk to Balmreich | 0 4 | 2 04 | 1 95 | 4 69 | | Balmreich to Newburgh | 2 8 | 3 42 | 9 35 | 3 85 | | Newburgh to Perth (tide harbour) | 4 0 | 8 56 | 5 06 | 3 42 |
Rise on the Low-water of Spring Tides in 1844.
| Ft. in. | Distance in Miles | Rate of Spring Tide in Inches per Mile | |--------|------------------|--------------------------------------| | Newburgh to Carpow | 0 5 | 1 33 | 3 75 | 3 17 | | Carpow to Perth | 1 7 | 7 23 | 2 63 | ... | | Hence from Newburgh to Perth, 1844, the rise is | 2 0 | 8 56 | 2 80 | 5 13 |
The result of the observations of 1844 thus gives a depression on the level of the low-water mark on the gauge of two feet at Perth tide harbour.
4. Duration of Flood and Ebb.
The results of observations in 1833 and 1844 at Newburgh show that the duration of flood and ebb tides at that place are unchanged. The times are as follows:
| Spring Tides flowed | 4 20 | |---------------------|-----| | ... ebbed | 7 20 | | Neap Tides flowed | 4 30 | | ... ebbed | 6 45 |
At Perth in 1833: - Spring Tides flowed 2 20 - ... ebbed 7 0 - Neap Tides flowed 3 15 - ... ebbed 7 0
At Perth in 1844: - Spring Tides flowed 3 10 - ... ebbed 7 0 - Neap Tides flowed 3 10 - ... ebbed 7 0
Increase of duration of Flood in springs at Perth 0 60
It will be observed from these tables that important changes have taken place:
First, The fall on the surface of the river from the tide basin at Perth to Newburgh in the year 1833 was 4 feet, but after the works were executed it was only 2 feet.
Second, In 1833 the passage of the tidal wave from Newburgh to Perth (8 56 miles) occupied 2 hours 30 minutes, being at the rate of 3 42 miles per hour; but it is now propagated between the same places in 1 hour 40 minutes, being at the rate of 5 13 miles per hour,—giving a decrease in the time of 50 minutes, and an increase in the speed of the first wave of flood of more than 1 ½ mile per hour, since the commencement of the works.
Third, The spring tides in 1833 at Perth flowed 2 hours 20 minutes, and ebbed 7 hours; but now the tide flows 3 hours 10 minutes, and ebbs 7 hours,—being an increase in the duration of flood of 50 minutes.
The works on the Forth, also executed under the direction of Messrs Stevenson, produced changes on the tidal phenomena, which, in connection with those described on the Tay, are interesting and instructive as regards the propagation of the tide, and therefore we shall briefly allude to them. The river between Stirling and Alloa is very circuitous, the distance by the navigation being 10½ miles, while in the direct line it measures only 5 miles. The navigation was found to be impeded by seven fords or shallows which occur between Alloa and Stirling, and are composed of boulder stones, varying from a few pounds to several tons in weight, embedded in clay.
It was determined, in the first instance, to remove two of these obstructions, viz., the "Town" and the "Abbey" fords, which lie nearest to Stirling, and having the smallest depth of water, form the greatest obstruction to the free passage of vessels. The works were commenced at the lower end of the Abbey ford, and were carried regularly upwards. The new channel excavated through this ford was about 500 yards in length and 75 feet in breadth, and was deepened in some places about 3 feet 6 inches.
Previous to the commencement of the work, tide-gauges were erected in the positions marked 1, 2, 3, and 4, in fig. 16, on which a series of observations was made for the purpose of establishing the original tidal phenomena of the river. After the Abbey ford was cut through, farther observations were made on the same gauges; and it is to a comparison of these two sets of observations that we desire specially to refer. It is necessary to explain that gauge No. 1 is at Stirling quay, No. 2 about 500 yards farther down, No. 3 at the top of the Abbey ford, and No. 4 immediately below it. It will therefore be understood that the Abbey ford, through which a canal was cut, lies between gauges Nos. 3 and 4. The whole of the gauges were placed on the same level, so that their readings might be more easily compared; and the following are the results obtained with reference to the level of the low-water line:
| Levels of Low-water Line | Gauge No. 1 | Gauge No. 2 | Gauge No. 3 | Gauge No. 4 | |--------------------------|-------------|-------------|-------------|-------------| | In 1847 the low-water line was found to stand at the following levels | 2 0 | 5 0 | 3 5 | 5 6 | | In 1840 | 2 0 | 3 6 | 4 6 | 5 0 | | Depression | 0 0 | 1 6 | 0 9 | 0 6 |
From this tabular statement, we find that the low-water level at No. 4, which is below the site of the works, remains unaltered, but that it has fallen 1 foot 6 in. at the top of the Abbey ford (through which the cut has been made). It further appears that the formation of this cut has drained off the water, and lowered the surface 9 inches at gauge No. 2, and 6 inches at gauge No. 1, which is at Stirling. The former and present low-water lines and bed of the river are represented in fig. 16, in which is also shown the amount of excavation on the Abbey Ford by hatched lines. This general depression of the level has of course altered the slopes or inclinations formed by the surface of low-water; the slope between 4 and 3 being decreased, while the inclinations between 3 and 2, and between 2 and 1, have been increased in the following ratios:
| Inclination | Dist. | 1847 | 1848 | Difference | |------------|------|------|------|-----------| | | Feet | Inches per Mile | Inches per Mile | | | Inclination between 4 and 3 | 1550 | 122-5 | 61-3 | -61-2 | | Do. do. 3 and 2 | 3050 | 5-19 | 29-77 | +15-58 | | Do. do. 2 and 1 | 1490 | 11-31 | 22-62 | +11-31 |
Again, these changes on the low-water line have produced corresponding alterations on the velocities of the first wave of flood, which are found to be as follows:
| Velocities | 1847 | 1848 | Difference | |------------|------|------|-----------| | | Minutes | Minutes | Minutes | | Time occupied by first wave of tide in passing between gauges Nos. 4 and 3 | 24 | 8 | -16 | | Do. do. Nos. 3 and 2 | 6 | 11 | +5 | | Do. do. Nos. 2 and 1 | 6 | 8 | +2 | | Do. do. Nos. 4 and 1 | 36 | 28 | -8 |
From this it appears that between Nos. 4 and 3 there is an acceleration of 16 minutes, while between 3 and 1 there is a retardation of 8 minutes, leaving the difference, or 8 minutes, as the actual amount of acceleration at Stirling, due to the removal of the ford and the lowering of the low-water level 6 inches at that place. The rates of propagation in miles per hour are as follows:
| Rates of Propagation | Miles per hour | Miles per hour | Difference | |---------------------|---------------|---------------|-----------| | Rates of propagation between Nos. 4 and 3 | 6-5 | 2-2 | +4-35 | | Do. do. Nos. 3 and 2 | 5-77 | 3-0 | -2-77 | | Do. do. Nos. 2 and 1 | 2-65 | 1-87 | -0-78 |
These observations and results seem to throw some additional light on the circumstances which modify the propagation of the tidal wave. The table of the results obtained at the Tay shows that the decreased inclination of the low-water lines of that river was attended by an acceleration of the velocity of the tidal wave; and the above observations further show that a retardation has attended an increased inclination of the low-water line of the upper part of the Forth. From the foregoing tabular statements, it will be seen that between gauges 4 and 3, where the slope has been decreased, the propagation has been accelerated; while between 3 and 2, where, from the state of the works when the observations were made, it is found to have been increased, the rate of propagation had been sensibly retarded.
It is worthy of remark, however, that the rates of propagation do not, either at the Tay or Forth, bear any constant relation to the slopes, but are modified by other circumstances; in proof of which, it will be found that the rate of propagation at the Forth between gauges 4 and 3, where the slope is 61-3 inches per mile, is actually greater than between gauges 2 and 1, where it is only 22-62 inches per mile. The circumstances of the Forth at this particular place are somewhat peculiar. Before the Abbey ford was cut through, it acted as a dam extending across the river, and had the effect of increasing the depth at low-water all the way up to Stirling. By cutting the channel through the ford, however, not only has the water been drained off and rendered shallow, but its surface has been broken by the projection of boulders from the bottom, which formerly were entirely covered; and while this effect has taken place in the upper part of the river, a comparatively smooth cut, with regular sides and bottom, has been formed in the Abbey ford, through which the river flows at low-water in a body of considerable depth. The writer therefore attributes the slow propagation of the tide between 2 and 1 to the shallowness of the water and the very rugged state of the bottom, which is in many places completely studded with boulders, rising some above the surface at low-water, and others to within a few inches of it; while the high velocity up the steep slope of the ford is to be attributed—1st, To the depth of water caused by the whole river being made to pass through a comparatively narrow channel; 2d, To the rectangular cross section of the cut; and 3d, To the smoothness of the sides and bottom. At the Firth of Dornoch, again, as already noticed, between the Quarry and Bonar Bridge, a distance of 1 mile, although the water is shallow and the bottom rough, it is not, on the whole, more so than between gauges 1 and 2 on the Forth; but at the Dornoch the slope on that mile is no less than 6 feet 6 inches, and the rate of propagation is only two-thirds of a mile per hour. Moreover, it was found that the tide did not begin to show at Bonar until it had risen 6 feet 6 inches on the gauge at the Quarry, being the exact difference of level between the two points of observation.
These various results as to slopes and rates of propagation, as well as others which have come under the writer's notice, seem to justify the following deductions as to the propagation of the tide-wave in rivers with sloping surfaces and irregular bottoms:—1st, That a decrease of slope is followed by an acceleration of the rate of propagation of the tidal wave. 2d, That an increase of slope is followed by a retardation of the rate of propagation. 3d, That the rate of propagation does not bear any constant relation to the amount of slope, although it is to some extent modified by it. 4th, That while the rate of propagation in rivers is in some measure due to the depth of water, it is nevertheless influenced by the slope of the surface, the form of the channel, and the obstructions protruding from the sides or bottom. 5th, That if not in all cases, at least when there are steep slopes and shallow water, as at the Dornoch Firth, the level of the crest of the wave must rise to the level of the surface of the water (or perhaps the bed of the river) above it, before a progressive motion takes place; and, 6th, That, from the difficulty of dealing with so many variable elements, it is impossible in most rivers to determine the ruling circumstances which can be held as regulating the rate of tidal propagation.
The Clyde affords a striking proof of the extent to which Clyde river improvements may be carried. So insignificant was the stream in its natural state, that Smeaton, in 1775, proposed to erect a dam with locks in the lower part of the river, and to convert it into a tidal canal. In 1775, however, Golburne surveyed the river, and although he found that as far down as Kilpatrick the depth of water was only 2 feet, he nevertheless recommended the construction of a series of jetties from either side, for the purpose of narrowing and deepening the stream; and this may be held as the commencement of the improvement of the River Clyde, which originally barely afforded depth of water for larger craft than flat boats, but which now, as our readers know, admits vessels of large draught up to Glasgow Bridge. The reader must be cautioned from supposing, however, that this result has been attained by means of the jetties which were erected under the advice of Golburne. It was soon discovered that the object could not be gained by such It was not until the ends of the jetties were connected by longitudinal walls, and until dredging-machines were extensively employed, that the Clyde improvements began to assume an importance commensurate with the vast commercial interests of the city of Glasgow and surrounding districts. The works on the Clyde have latterly been under the direction of Mr Walker, and have mainly consisted in forming longitudinal walls, dredging, and increasing the width of certain parts of the channel, which had in the early stage of the improvements been contracted to an injurious extent.
The Ribble in Lancashire, the improvements of which were designed by Messrs Stevenson, presents an example of a great amount of additional depth having been obtained in a comparatively short space of time. That river, according to Mr Park, who conducted, as resident engineer, the greater part of the works, has a course of 82 miles, and drains 900 square miles of the counties of York and Lancaster. The formation of the bed in which it flows rendered the state of the tidal compartment previous to the improvements very defective. The bottom in the lower part of the river consists of loose sand; while that of the upper reach is alternately compact gravel and sandstone rock. About half a mile below Preston, in particular, it was found that a solid ridge of sandstone, extending to 300 yards in length, stretched quite across the channel. Its surface was from 3 to 5 feet higher than the general bed of the river both above and below it, and so prominent an obstruction did it form, that the higher parts of the rock were occasionally left dry during the long droughts of summer. The propagation of the tidal wave, and free flow of the currents, were checked on approaching it; while the power of the tidal and fresh-water scours was in a great measure neutralized and rendered almost unavailable in keeping open the upper and lower stretches of the navigation; so that its influence in obstructing the river resembled that of a great artificial weir stretching across the stream. In proof of this, it may be stated that the ordinary rise of spring tides at Lytham, which is 12 miles seaward of Preston, is about 19 feet, and that of neap tides is 14 feet, while at Preston, prior to the operations, the rise of spring-tides did not exceed 6 feet, and neap tides of 13 or 14 feet rise at Lytham did not reach Preston at all. The removal of the rock which encumbered the bed, was naturally viewed as the most urgent and important work for effecting an improvement in the tidal phenomena and general depth of water. To this, therefore, the Navigation Company first directed its attention, and in the course of eighteen months, succeeded in excavating a channel through the solid rock 300 yards in length, and in some places 13 feet 6 inches in depth. This operation was successfully accomplished, at an expense not exceeding £10,000, by means of a coffer-dam of the construction already shown in fig. 8. In addition to this work, about 480,000 tons of gravel and sand have been removed from the upper part of the river by dredging; and 9 miles of low rubble walling (formed, in so far as it was available for the purpose, of the rock excavated from the bed of the river) have been constructed in accordance with the sketch shown in fig. 15, for guiding the current in the lower channel.
The effect of the different works that have been executed has been to increase the tidal range at Preston about 5 feet, and to accelerate the propagation of the tidal wave nearly an hour; and vessels, to which the navigation may be said to have been previously closed, now come up to the quays of Preston with comparative ease and safety.
The works on the Lune in Lancashire were executed by Messrs Stevenson, under the direction of the Admiralty. They extended over a period of four years, and consisted in removing foreshore by dredging, shutting up subsidiary channels, and erecting river walls; the whole operation costing under £10,000. The sketch of the Lune in fig. 9 shows the varying state of the channel in its original condition, which was regulated by means of a single rubble wall, as the funds did not admit of a double wall being erected. It was necessary also, from the configuration of the shores, that the channel should follow the convex side of the wall which should, if possible, be avoided, as some difficulty occurs in maintaining the channel always close to the wall,—a difficulty which can only be removed by the formation of a second wall; and we mention this as an example of the desirableness, as already stated, of forming double walls in all cases when the funds at disposal will admit of it. Fig. 17 represents the gradual depression of the tidal lines since the works commenced: the upper line shows the surface of the river in 1838, the intermediate line in 1848, and the lower line in 1851. The effect of the works has been to increase the depth of water up to the quays at Lancaster about 4 feet, and to prolong the duration of the tidal influence at that place thirty minutes in neap, and one hour and a half in spring tides; so that vessels can approach and leave Lancaster much earlier than formerly, while the improved channel is navigated with much greater ease.
It is unnecessary to give farther examples of navigations which have been benefited by means of works constructed on the principles which we have indicated. We doubt not that such cases could be quoted, although we do not possess sufficient details to enable us to do so; and we close this part of our subject by stating that the further extension of these works in such rivers as the Tay, the Ribble, or the Lune, and their application in many other cases, would be followed by a greatly increased improvement of their navigation.
Instances might be referred to where a course of treatment opposed to that which we have recommended has not been followed by similar favourable results; but we deem it sufficient to confine this treatise to an exposition of the correct principles of river improvement, without discussing erroneous practice or its baneful results; the more so as these have been most fully and ably treated by Mr E. K. Calver, R.N., whose investigations into the former and present state of some of our tidal rivers are of great value to the hydraulic engineer.
**SECT. VI.—SITUATIONS WHERE THE PRINCIPLES OF IMPROVEMENT RECOMMENDED ARE NOT APPLICABLE.**
We have further to state, that in some situations the principles of improvement which we have advanced will be found to be of very limited application. Such cases indeed are rarely to be met with, but still it is necessary to notice them. We allude to rivers the tidal or intermediate compartments of which are, from natural causes, of very small extent. In illustration of what we mean, we may refer to
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1 Captain Sir Edward Belcher, while engaged in making the Admiralty survey of the Ribble, found that on one occasion the tide at Lytham rose 25 feet 7½ inches.
2 *The Conservation and Improvement of Tidal Rivers*, by E. K. Calver, R.N., London, Weale, 1833. the Erne in Donegal, which has a tidal capacity of only 2½ miles, extending from the bar up to the town of Ballyshannon, where the tidal flow is terminated by the "Salmon Leap," a perpendicular rise in the bed of the river of about 15 feet in height. This waterfall forms the limit of the tidal flow, beyond which it could not, without works of a gigantic character, be extended.
Another case is the Ness, which has a short course of about 2 miles, from a little above the town of Inverness to the Beauly Firth, at Kessock Roads. The difficulties attending the navigation of this river are mainly the prevailing outward currents due to the physical conformation of the bed of the Ness, which may be shortly described, as it illustrates generally a class of rivers which are very difficult to improve:—1st, The rise of ordinary spring tides at the mouth of the river is 14 feet. 2d, The distance to which the influence of such tides extends is only about 2 miles, which includes the whole tidal compartment of the river. 3d, The slope or inclination of the low-water line of this tidal compartment is no less than 7 feet per mile, and the tide takes from two to three hours to make its way up the first mile. 4th, The natural result of such a state of matters is, that no tidal current is generated at the mouth and propagated up the stream, and consequently the phenomenon of a current due to flood-tide may be said to be almost unknown.
Under these circumstances, the barrier to the free navigation of the River Ness is the absence of a tidal current or in-draught, to aid the entrance of vessels from Kessock Roads, and assist their progress up to the quays. This is at present effected by help of men and horses against the nearly constant downward current, which varies in strength with the amount of water discharged by the River Ness, during its frequent heavy floods.
This absence of sufficient internal capacity and gentleness of inclination to admit of the generation of tidal currents is strikingly exemplified in the two rivers to which we have alluded, and naturally leads us to offer some general remarks in passing on the subject of the "backwater" and the "slopes" of rivers. In most, if not in all cases, it will be found (as more particularly noticed hereafter in section 8, in treating of bars) that it is of the highest importance to maintain unimpaired the full tidal capacity, and to be careful to make no reduction of its amount without obtaining an equivalent in the low-water section, to compensate for any reduction which it may be found advisable to make at or near the high-water line.
The subject of the reduction of backwater has given rise to various questions, which have occupied the attention of the engineer; but as every case must be judged on its own merits, and no two situations are exactly alike, it would be unprofitable to enter upon the discussion of the various arguments that have been adduced with reference to particular localities. All we can do is to lay down the general principle, that the more the tidal influence can be extended, and the larger the amount of backwater that can be obtained, the greater will be the benefit conferred on the navigation from the bar upwards; provided always that such increased scouring power is, by judicious works, placed under proper regulation. The question as to the possibility of excluding the tide from any part of an estuary, without injury to the outer channels, is a wide subject, as will be seen from our merely stating some of the considerations which may be held to determine the peculiar circumstances in which the exclusion of water may be compensated. These are, the configuration of the banks and bed of the estuary, the simultaneous levels of the surface of the water at different periods of the tide throughout the estuary, the velocities of the surface and under-currents at different periods of tide, and the times of ebbing and flowing, together with many other more minute data peculiar to each case, which it is not possible to specify in a general summary.
The existence of a moderate amount of fall or slope on slopes of the low-water line of a river is a hopeful feature in its capacities for improvement; while on the other hand, such a slope as that on the Ness proves a great barrier to its extended improvement as a tidal river; for it is obvious, that to obtain on that river a slope sufficiently gentle for easy navigation, it would be necessary to lower its bed so great an extent, and to execute works of such magnitude, as to render it inexpedient to entertain such a project.
The consideration of the proper slope is important in river engineering. Dubuat considers 1 in 500,000 to be the smallest possible inclination that can be given to a canal to produce sensible motion. It will be found, on inspecting the table at the end of this treatise, that the slopes of tidal rivers vary from a few inches to several feet per mile. As a general rule, we should say that the engineer may calculate on reducing the slopes of tidal navigations to 4 inches per mile \(= \frac{1}{25}\); and that they should not, if possible, exceed 10 inches per mile \(= \frac{1}{25}\).
Directly connected with the slope is the velocity of streams, Velocities—an important matter as affecting navigation, for it cannot be conducted with advantage in situations where the velocity of the currents is very great. The velocities of tidal currents in some places are very great; as, for example, in the Pentland Firth, where Captain Otter measured a velocity of 10½ miles per hour, and in the Severn, where it was found to be 9 miles per hour. From 2 to 3 miles per hour is, however, a very common velocity on many of the rivers in this country, and it is found to present no inconvenience to the navigation of vessels. The following are the velocities of the currents in different rivers, with their authorities. The whole of them are surface velocities:
| Name | Per Hour | Authority | |-----------------------|----------|-----------------| | Mississippi | | | | Clyde, between Glasgow and junction of Cart, during ebb | 0 1576 | W. Bald. | | Do., flood | 0 771 | Do. | | Do., at narrow places during floods | 3 1148 | Do. | | Do., during high floods below Glasgow harbour, ebb | 2 1613 | Do. | | Do., near Stonebench, flood, spring tide | 4 950 | Admiral Beechy. |
| Name | Per Hour | Authority | |-----------------------|----------|-----------------| | Severn, near Stonebench, flood, spring tide | 3 12 | Admiral Beechy. | | Wear, spring tide, ebb | 1 ½ to 2 | J. Murray, C.E. | | Do., neap tides, ebb | 1 to 1 | Do. | | Do., good tides, ebb | 1 to 2 | Do. | | Tay at Buddonness, sp. tides | 2 to 2½ | North Sea Pilot. | | Do., at Perth | 3 09 | Messrs Stevenson. | | Willowgate at Perth | 1 35 | Do. | | Dornoch Firth, Melkie ferry, flood | 2 63 | Do. | | Do., ebb | 2 55 | Do. | | Tayat Mugdrum, flood and ebb | 2 to 2½ | Do. | | Thames | 2 to 2 | G. Rennie. |
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1 See Report of Tidal Harbour Commission, by D. Stevenson, C.E., and Joseph Maynard, R.N., in Admiralty Reports for 9th March, 1847. 2 The slope of the river Niagara at the rapids, immediately above the far-famed "Falls," is said to be 50 feet in half a mile, or 1 in 52-8. SECT. VII.—WORKS FOR ACCOMMODATION OF VESSELS.
The works we have described are for facilitating the ingress and egress of vessels. In addition to this, it is necessary to provide for their accommodation. For this purpose it is desirable, where local circumstances admit of it, that it should be possible to withdraw them from the action of the river currents which, during heavy floods accompanied by ice, are often very destructive to shipping.
This is accomplished in a simple manner by forming what are termed tide-basins, which are artificial cuts retiring from the stream having their sides bounded by quays or wharves, into which vessels may be withdrawn, but where they are still liable to take the ground at low-water. The object is accomplished more effectually by means of wet docks, for details of which the reader is referred to the article on that subject. In many situations, however, especially where the river is wide, and affords ample room, as in the case of the Foyle at Londonderry, for example, the berthing for vessels is afforded by means of lines of quays formed along the shore.
Such quays constitute an important part of all harbours which are formed in tidal rivers; and in illustration of some of the various methods of construction adopted in such cases we submit the following cross sections. Fig. 18 shows the timber wharfage constructed by Mr Smith at Belfast, which is composed of a facing of timber-work secured by iron ties fixed to piles, the space behind the face-work being filled up, and the roadway formed at the top. Fig. 19 is a plan showing the positions of the piles and ties. Sometimes a similar face-work is employed, backed by a wall of concrete; and iron plates have also been used for the facing, instead of planking. Figs. 20 and 21 are a section and elevation of the quays of Londonderry, designed and executed by Messrs Stevenson. At this place the ground is very soft, and in order as much as possible to reduce the weight, the front compartment of the wharf next the river is left open. Figs. 22 and 23, again, are sections of the stone wharves now being constructed from a design by Mr Walker, at Glasgow, under the superintendence of Mr Ure. Fig. 22 is the section adapted to a clay bottom; and fig. 23 is that which is adopted when the bottom consists of sand. In both cases the depth of water in front of the quays is 20 feet at low-water, and is intended to accommodate merchant vessels of the largest class. These examples furnish an illustration of the means employed for providing wharfage on tidal rivers; the details of their construction must be studied in treatises on such branches of engineering construction as Carpentry, Masonry, Piling, Foundations, Mortar, and Quay Walls.
The engineer is often called on to construct swing-bridges in connection with navigations, but for particulars as to such works, reference is made to the article IRON BRIDGE.
SECT. VIII.—"SEA PROPER" DEPARTMENT OF RIVERS.
Having considered the treatment of rivers from their source to the ocean embracing the upper or "river proper," and the intermediate or "tidal compartment," we have now to direct attention to what we have termed the "sea proper" compartment, which, in the sense we have attached to it, may be said to embrace the phenomena connected with the flow of rivers or bodies of tidal water into the sea.
In some instances, such, for example, as the Forth, the junction of the river with the sea occurs without giving rise to any very perceptible or marked phenomena; the one seems to glide naturally into and be mingled with the other, without producing any apparent disturbance of the currents or change on the bed of the channel. But such cases are exceptions; and, generally speaking, we may safely affirm that the junction of a river with the sea gives rise to what is termed a "bar,"—the most difficult subject with which the hydraulic engineer has to grapple, and the nature and cause of which we have now to discuss.
A bar, then, is the name applied to that shallow part of a channel which occurs at the junction of a river or estuary with the sea. On either side of it—that is, both seaward and landward of it—there may be ample depth of water for all purposes of navigation, but the bar forms the regulating navigable depth, and no passage over it can be obtained until the tide has risen sufficiently high to enable vessels to cross it. The depth at low-water on the bars of some of our rivers is as follows:
- The Mersey has a depth of from 9 to 10 feet at low-water. - Tyne: 6 to 7 feet. - Wear: 3 to 4 feet. - Ribble: 7 to 8 feet. - Tay: 16 to 18 feet.
And while these limited depths exist on the bar, there is in all of these cases ample depth within, or landward, for vessels of the largest class to lie afloat at all times of tide.
Many theories have been propounded to account for the phenomenon of the bar. Some have advocated the idea that bars are composed of materials held in suspension by the river, and deposited so soon as its current is checked by meeting the still water of the ocean. But this theory, at all events as regards sea bars, of which we are now treating, is disproved by the facts of the Dornoch Firth, to which we have already alluded. The bar at that place occurs at a point 14 miles seaward of the point at which the river enters the sea. The idea that a bar of such magnitude as that at the Dornoch Firth, could be formed by the detritus brought down by the small rivers Oykell and Cassily, is wholly untenable, and is indeed contradicted by the fact that the bar and adjoining banks are composed of pure sand; and hence the writer attributed its formation, when he examined the firth in 1842, entirely to the action of the sea. We find that Mr Ellet, though founding his opinion on totally different premises, also comes to the conclusion that the bars of the Mississippi were not due to materials deposited by the outgoing stream. In explaining his views, he writes as follows:—"The velocity of the river is not destroyed, nor very sensibly diminished, at the bars. When the river was rising, but still far from being at full height, I measured the velocity of the current on the bar of the Passala Loutre, and found it to vary, at different times and places, from 3 feet to 3½ feet per second, or from 2 miles to 2½ miles per hour. I measured it also repeatedly on the south-west bar, and found it there 3 feet per second, or about 2 miles per hour. But there are many parts of the river where the speed of the current does not exceed 2½ miles, or even 2 miles per hour, in times of flood, and where it is, notwithstanding, more than 100 feet deep. In fact, on testing the velocity of the south-west pass, 4 miles above the bar, and in 5 fathoms water, I found the current to be but 2 miles per hour,—precisely the same as it was under like circumstances of wind and tide on the bar. The current of the Mississippi sweeps over the bars at the mouths of the passes, and at periods of flood many miles out into the gulf, with a velocity almost undiminished by its contact with the waters of the gulf." He therefore concludes that there is in the Mississippi no retardation of the river's velocity on the bar to account for any deposit due to such a cause. Another theory attributes bars to the want of sufficient scouring power; but when we find bars existing at the mouths of such rivers as the Mississippi, we cannot attach much importance to such a suggestion. Another theory attributes the absence of a bar to "the presence of a nearly equal duration of the period of the ebb and flow in the lower reach of the river accompanied by an extremely gentle inclination of its surface at low water." To refer again to the Dornoch Firth, we have an equal duration of the ebb and flow throughout the firth, and a surface practically level, and yet we have as perfect a specimen of a bar at the Gizzen Briggs, at the mouth of the firth, as can possibly be imagined. We cannot, therefore, in endeavouring to account for the existence of bars, or the exemption from them, accept the explanations to which we have alluded.
The bars with which we have to do in this country may be said to be of two kinds; one class of bars is due to the hard formation of the bottom, which occurs in some situations; the other class is due to the action of certain elements, on the soft matters of which the bottom in other places is composed. Of the first class are such bars as that at Ballyshannon in Ireland, or at the entrance of Loch Fleet in Sutherlandshire, both of which the writer has had occasion professionally to examine. The bar at Loch Fleet, for example, is composed of boulder stones firmly imbedded in a mass of indurated gravel, and is obviously a continuation of a bed of similar formation which seems to traverse the coast at that place. The consequence is, that no scouring power can prove available in deepening the channel. Such bars being entirely due to the hardness of the bottom, are generally comparatively easily treated by the engineer, and an encouraging prospect is held out that their removal will be attended with permanent benefit, since, by excavating a channel through them, the engineer at the same time removes the evil and its cause.
In the other class are comprehended those sand-bars which occur at the mouths of the firths of Dornoch and Tay, and of the Tyne, Wear, Mersey, Ribble, and other tidal rivers and estuaries; and it is to the formation of these capricious and troublesome accumulations that the theories to which we have alluded apply. The true source of all such bars is to be found, as already stated, in the action of the sea. The natural effect of the sea is to throw up sand, and form a continuous line of beach across the mouths of all our tidal rivers and inlets; while, again, the flow of the tidal and fresh-water currents tends to maintain an open channel through the beach. In this way the antagonistic action of the waves of the sea on the one hand, and the currents of the estuary or river on the other, produce the well-known feature of a submerged beach or sand-bank, extending from shore to shore across our inlets, having a deeper channel through them, which channel is termed the "bar." This explanation is due to the Abbot Castelli, who, in his work on the Mensuration of Running Waters, written in the beginning of the seventeenth century, gives the following clear announcement of his views:—"As to the other point of the great stoppage of ports, I hold that all proceedeth from the violence of the sea, which being sometimes disturbed by winds, especially at the time of the waters flowing, doth continually raise from its bottom immense heaps of sand, carrying them by the tide and force of the waves into the lake; it not having on its part any strength of current that may raise and carry them away, they sink to the bottom, and so choke up the ports. And that this effect happeneth in this manner, we have most frequent experience thereof along the sea-coasts; and I have observed in Tuscany, on the Roman shores, and in the kingdom of Naples, that when a river falleth into the sea, there is always seen in the sea itself, at the place of the river's outlet, the resemblance, as
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1 *Treatise on the Improvement of the Navigation of Rivers*, by W. A. Brooks. 2 *The Mensuration of Running Waters*, by Don Benedetto Castelli, Abbot of St Benedetto Aloysio, and professor of the mathematics to Pope Urban VIII. in Rome; translated by Thomas Salisbury, Esq., London, 1661. it were, of a half-moon, or a great shelf of settled sand under water, much higher than the rest of the shore, and it is called in Tuscany il cavallino, and here, in Venice, lo scanto; the which cometh to be cut by the current of the river, one while on the right side, another while on the left, and sometimes in the midst, according as the wind fits. And a like effect I have observed in certain little rilles of water along the Lake of Bolsena, with no other difference save that of small and great.
"Now whoso well considereth this effect, plainly seeth that it proceeds from no other than from the contrariety of the stream of the river to the impetus of the sea-waves; seeing that great abundance of sand, which the sea continually throws upon the shore, cometh to be driven into the sea by the stream of the river, and in that place where these two impediments meet with equal force, the sand setteth under water, and thereupon is made that same shelf or cavallino; the which, if the river carry water, and that any considerable store of it shall be thereby cut and broken, one while in one place, and the other while in another, as hath been said, according as the wind blows; and through that channel it is that vessels fall down into the sea, and again make to the river, as into a port. But if the water of the river shall not be continual, or shall be weak, in that case the force of the sea wind shall drive such a quantity of sand into the mouth of the port and of the river as shall wholly choke it up. And hereupon there are seen along the sea-side very many lakes and meers which at certain times of the year abound with waters, and the lakes bear down that inclosure, and run into the sea.
"Now it is necessary to make the like reflections on our ports of Venise, Malamocco, Bandolo, and Chioggia, which in a certain sense are no other than creeks, mouths, and openings of the shore that parts the lake from the main sea; and therefore I hold that if the waters in the lake were plentiful, they would have strength to scour the mouths of the ports thoroughly and with great force; but the water in the lake failing, the sea will, without any opposition, bring such a drift of sand into the ports, that if it doth not wholly choke them up, it shall render them at least unprofitable and impassible for barks and great vessels."
The conditions under which such accumulations are formed the writer holds to be,—1st, The presence of sand or shingle, or other easily moved material; 2d, Water of a depth so limited as to admit of the waves during storms acting on the bottom; and 3d, Such an exposure as shall allow of waves being generated of sufficient size to operate on the submerged materials.
In confirmation of this opinion, we may once more refer to the Dornoch Firth. The Oykell joins it at a point about a mile below Bonar Bridge, but we find no indication of what may be termed a bar throughout the whole of the sheltered part of the firth, which extends for 12 miles seaward of that point, until we reach the outer portion which is exposed to the unbroken sea of the Moray Firth, and there we find an extensive sand-bank, forming as it were, a continuation of the shore on either side, and stretching quite across the mouth of the firth, with the bar in the centre of it. But the fact, that in all such bar-rivers and estuaries the depth is often found to be seriously diminished after heavy seas, is beyond doubt, and serves as a further confirmation of the correctness of the theory for which we are contending.
The same reasoning may explain why, in such a case as the Firth of Forth, for example, no bar exists. The Firth of Forth is an inlet or arm of the sea, of great width and depth, the seas entering it do not act on the bottom, so as to cause a heaping up of the material of which it is composed, in the same manner as in a shallow sea. This great natural depth continues as the Forth gradually contracts; and before the necessary conditions for the formation of a bar occur—namely, sufficiently shallow water and presence of sand—the sea is so land-locked that waves of sufficient size to produce the necessary effect cannot be generated. There is, in fact, in the Forth that gradual diminution of depth, and increase of shelter, which combine to produce the phenomenon of a river without a bar.
We must also notice a cause for the formation of bars advanced by Mr Ellet. Although it is not applicable to the rivers in this country, still, from the observations he has made, we think it likely that his theory may be held to account for the bar of the Mississippi. It is founded on Relative the fact, that at the junction of a river with the sea the specific fresh water flows in a stratum above, and distinct from, the gravities of salt water, for some distance after entering the ocean. This fresh weight of fresh water being 1000, while that of salt is 1026.
Before noticing Mr Ellet's theory, however, we may state, Experiments that so far as we are aware, the first observations made on this subject were those instituted by the late Mr Robert Stevenson, of Edinburgh, on the River Dee in Aberdeenshire, in the summer of the year 1812, while engaged in surveying that river with reference to a disputed right of salmon-fishing. Mr Stevenson, in his report on that subject, states that, by means of an instrument devised for that purpose, he ascertained that the salt or tidal water of the ocean flowed up the channel of the River Dee, and also up Footdee and Torryburn, in a distinct stratum, next to the bottom and under the fresh water of the river, which, owing to the specific gravity being less, floated upon it, continuing perfectly fresh, and flowing in its usual course towards the sea, the only change discoverable being in its level, which was raised by the salt water forcing its way under it. The tidal water so forced up contained salt; and when the specific gravities of specimens from the bottom were tried, they were found to possess the greater degree of specific gravity due to salt water, while the surface specimens were found to be specifically unaltered.
Similar observations have been made by the writer of this article in several places, with the same results. The appearance of fresh water floating on the surface of the sea is no doubt familiar to most persons. It occurs at the mouths of many of our rivers, and is most apparent when they are in flood, from the brown tinge given to the water, which is easily discoverable for many miles at sea. It is well known on our coasts to the crews of the welled smacks employed in cod-fishing, who invariably lose a great portion of their live stock if they happen to encounter what they term "a fresh," which is believed by them to be a brackish portion of the sea, caused by the imperfect admixture of fresh water discharged from rivers in flood. On this subject the following passage from the work of Father Manuel Rodriguez, a Spanish Jesuit, is interesting, and its correctness, as regards the extent to which the influence of large rivers is felt, has since been corroborated by the investigations of Colonel Sabine. "This river," says Rodriguez, speaking of the Amazon, "is like a tree; its roots enter as far into the sea as into the land. It communicates to it a flavour, so that at 80 leagues within the sea its waters are seen, and taste sweet, and in a semicircle of 100 leagues in circumference they form a gulf not the least degree brackish, so that sailors call it the fresh sea."
But to return to the Mississippi: Mr Ellet, in the following extract, says:—"The river water does not mix suddenly..." with the sea, but rises upon it, floats over it, and rushes far out into the gulf on the top of the dense sea water, by which it is buoyed up. I tested this repeatedly, and found uniformly a column of fresh water, nearly 7 feet deep, in the gulf, entirely outside of the land, and salt water at a depth of 8 feet from the surface, and extending thence to the bottom. The river does not come down with a certain normal depth and speed, and encounter the gulf at the bar. No such process takes place. There is no sudden destruction of velocity, or consequent deposit of suspended silt. But the water of the Mississippi does not move over the surface of the gulf at a speed of 3 feet per second without imparting a portion of its motion to the sea. The fresh water and the salt water take the same direction towards the sea, and with nearly the same velocity, but yet keep separate. This state of things clearly cannot exist at the bottom; for as the river water is for ever coming forward, if the salt water all flowed towards the gulf, it would all be carried out, and river water would take its place. Salt water must come in from some quarter, to supply the current of sea water that is for ever setting towards the gulf, beneath the water discharged by the river. This salt water can only come from the sea, and can only come in along the bottom. It is, in fact, an eddy that is here at work, the movements being in a vertical instead of a horizontal plane.
Now, the question is, How does this account for the existence of the bar? The fresh water running out cannot produce deposit, for it has velocity enough to sweep away a foundation of coarse gravel. The outpouring salt water, immediately beneath the fresh, cannot produce deposit, because it also has a velocity seaward strong enough to remove anything that is brought down the Mississippi. The salt water that is coming in might produce, and I doubt not does produce, a deposit, for it passes over the soft muddy bottom of the gulf, and moves into the river, and along the bar, at a very slow rate. According to these facts, and this reasoning, there must be usually on the bar three distinct strata: 1st, Fresh water, running out at top, found by experiment on the S.W. bar to have a velocity of 3 feet per second. 2d, Salt water below the fresh, also running out with nearly the same velocity as at top; and 3d, Salt water coming in slowly along the bottom, and apparently a sheet of salt water between that running out and that coming in, which will be without motion.
"But as already said, and as is obvious, all the sea water that comes in must go out again. It comes in along the bottom, and it must go out between the column of salt water coming in and that of the fresh water going out. Each particle of salt water, therefore, must change its direction and position in elevation. It must pass from an inward-bound lower stratum to an outward-bound upper stratum. But in passing through this change of motion, its velocity up stream must be neutralized. It passes, to use a technical term, the dead point. At this point it may cease to bear its whole burden of mud, which it has brought from the gulf further forward. It leaves it, or a portion of it, at the turning-point. This turning-point is the place where the bar for the time being is in process of formation. But as the upper and lower strata are moving in opposite directions, the intermediate column must of necessity have a rotatory motion; that motion must be shared by the lower column of salt water, and this turning-point must therefore be found at the same time at different places along the bar."
Mr Ellet gives an interesting detail of his experiments on the saltness and freshness of the water, as taken from different depths, and also of the means he took to ascertain the strength and directions of the two under-currents referred to in his ingenious theory of the formation of the bar; but the details are too long to give in this sketch of his researches.
We have seen that bars, in some situations, are formed by the hard strata of which the bottom is composed; that in other places they are due to the waves of the sea; and that in the case of the Mississippi Mr Ellet attributes the phenomenon entirely to the eddy caused by an under-current inwards.
The removal of hard bars is, as already noticed, likely in most cases to result in a successful issue; but the treatment of those bars which are due to the waves of the sea, with which class of phenomena we have chiefly to do in this country, is an operation not only more difficult to deal with, but far more uncertain in its results.
From what has been said, the reader will see that we believe the depth of water, on such bars as are caused by the waves of the sea, to be in some degree proportional to the amount of scour produced by the tidal currents, which cross them four times in every twenty-four hours. If this assumption be correct, it is obvious that the principle which should guide us in all our considerations as to increasing, or even maintaining, the depth upon sea-bars, is the preservation of a sufficient amount of tidal water to counteract the tendency of the sea to heap up detritus at the mouths of our harbours. These two agents, the waves and the tidal scour, are constantly opposed the one to the other: a storm from the sea, or a heavy flood from the land, occasionally causes the one or the other to have the ascendancy; but this is only temporary. A variation in the depth of water on the bars of our harbours, caused by such temporary disturbances, may occasionally occur; but nevertheless, unless some work of magnitude is formed, so as to alter permanently the natural disposition of matters, no pilot has any difficulty in fairly estimating what is the general navigable depth over the bar of any of our seaports.
That the beds of the upper parts of rivers are scoured, compared with their depth maintained by the flow of the fresh-water stream, is not to be questioned; and it is also beyond doubt that in many situations the upper portions of the tidal compartments of rivers are kept open in a great measure by the fresh-water stream; but it is no less certain that the estuaries which would assign the depth of water in the lower parts of tidal rivers, and also through estuaries and across bars, to any other cause than the action of tidal water as the chief agent, are erroneous. We think this will be apparent by a reference to some of the investigations which have from time to time been made to ascertain the amount of the river or fresh water, as compared to the volume of the tidal water of some of our firths and estuaries.
By means of a series of careful observations and measurements made at the Cromarty Firth in 1837, to which reference has already been made, Mr Alan Stevenson found that the River Conon, when highly flooded (a state of matters which of course occurs only occasionally), discharges during twelve hours a quantity which is only equal to \( \frac{1}{4} \)th part of the water which passes out of the firth at every ordinary spring tide, and \( \frac{1}{4} \)th of that which passes out at neap tides. While in its summer-water state, the produce of the river is reduced to \( \frac{1}{4} \)th of the discharge of the firth in spring, and \( \frac{1}{4} \)th of the discharge in neap tides; a quantity too small to affect appreciably either the velocity of the currents of the firth or their scouring power. It has often been argued, that in situations where the velocity of the ebb exceeds that of the flood tide, the excess is due to the increased quantity of water passing out with the ebb, the volume of the ebbing waters being assumed to be augmented by the amount discharged by the river. But this
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1 This is in harmony with Venturi's well-known experiments, from which he found, that a body of water in motion leads or drags with it the particles of water at rest with which it may be in contact. is wholly disproved in the case of the Cromarty Firth; for while the increased quantity due to the river is seen to be only from \( \frac{1}{4} \) to \( \frac{1}{8} \), the average velocity at the flood-tide at that place was found to be 2-9 miles per hour, while that of the ebb was 3-6; an increase which is in all probability due to the tide beyond the Suters falling more rapidly than it rises, and thus producing a greater head and more rapid current on the ebb, but is assuredly not due to any augmentation of water from the discharge of the Conon.
The Tay presents another example of the disproportion between the tidal and river waters. That river, as gauged by Mr Leslie when in flood, was found, including the Earn, to discharge 969,340 cubic feet per minute. Mr Walker, in his report to the trustees of Dundee harbour, assumes the discharge in round numbers at one million cubic feet per minute, or 240,000,000 during four hours, and arrives at the following conclusion:—“To compare the above with the effect of the tidal water at Dundee, I assume 15,000 acres as the average area (above Dundee) of the reservoir or estuary during the first four hours of the ebbing tide, and the vertical fall of tide during these four hours to be 11 feet. This will give 7,187,400,000 cubic feet, or thirty times the 240 millions of river water. To compare the effect upon the bar, the area of the river between Dundee and the bar must be added; and the tidal water upon the bar will then be upwards of forty times the river water.”
It will be apparent that an important question is suggested as to the manner in which such works as we have recommended for improving the upper parts of rivers may operate in assisting or retarding the scour of the bar. We have no hesitation in replying, that if executed in the manner we have indicated, they will improve the higher part of the river without prejudicially affecting the bar, and in certain cases they will operate beneficially on the bar also.
The construction of piers for improving the entrances of rivers, as in the case, for example, of the Wear at Sunderland, is more properly included in the subject of harbours. Such works seem to us to act beneficially, not so much by increasing the depth on the bar, as by limiting the extent of shoal water at the entrance to the river. In its natural state such a river as the Wear flows across the beach from high to low water in a broad and shallow channel, the direction of which is ever changing. It thus forms a long bar or shoal, with broken water throughout its whole extent. But the projection of piers across the beach affords shelter from the waves, and admits of a navigable channel being excavated and maintained; and after a vessel crosses the short bar, which occurs at or near the pierheads, she not only gets into deeper water, but has the additional advantage arising from the shelter afforded by the piers. To this extent piers in such situations are highly advantageous. They further act beneficially in directing the flow of the tidal currents in a fixed channel across the beach and bar, and, in connection with an increase of tidal capacity in the interior, such as we have mentioned as the result of the works on some rivers, they cannot fail, if judiciously designed, to operate beneficially by maintaining an increased depth of water on the bar.
In certain situations where the coasts are faced with gravel or shingle beaches, accumulations or bars may be lessened by means of groynes, so formed as to intercept the gravel, and either retain it, or lead it past the harbour’s mouth into an adjoining bay. The writer has in several situations recommended the adoption of such works; and Mr Walker has applied them with success at the harbour of Newhaven in Kent, where, in conjunction with increased backwater, due to deepening and removal of obstructions, the depth on the bar has been materially increased.
Such twofold schemes as have for their ostensible object schemes the improvement of rivers and the formation of land, have for gaining generally been unsuccessful in benefiting navigation. We land and do not affirm that river-works, constructed on the principle improving which we have advocated, have not the effect of making not general land, in the particular sense in which we shall afterwards rally complain it; but we do state that land-making is no part of possible sound river engineering. Judiciously designed works may, as we shall presently explain, have the effect of reclaiming and protecting land, while at the same time they, as their primary object, benefit navigation; but we know of no case where the interests of navigation have been promoted by any measure which has for its object the conversion of large tracts of tide-covered sands into cultivated fields.
We refer to the Dee in Cheshire, as an aggravated instance of the incompatibility of the two interests. The outline of this river is shown in Plate L, from a survey by Messrs Stevenson, made in 1838. The River Dee Company, incorporated by act of Parliament in 1732, have from time to time reclaimed from the upper part of the estuary a large tract of land, extending to about 4000 acres, which is now in full cultivation; and alongside of this gradually gained territory the river has been conducted from Chester to near Flint, in a narrow canal of about 8 miles in length, and 400 feet in width. A considerable portion of land has also been reclaimed on the Flintshire side of the estuary, though not by the proprietors of the Dee Company; and it is believed that the aggregate amount which has from first to last been gained from the sea is about 7000 acres. Now it is well authenticated, that previous to the commencement of the land-making operations on that river, there was a depth of not less than a fathom at low water of spring tides up as far as Burtonhead, and that there was an anchorage for vessels of the largest size opposite to Parkgate, the positions of which places are marked on the plan. But when the writer surveyed the Dee in 1838, the depth of 6 feet was not found for more than 6 miles below Burtonhead, the low-water features of the estuary having been forced to that extent further seawards by the extensive reclamation of land in the upper part of the estuary, and the consequent diminution of the tidal scour. It cannot, we think, be disputed, that the effect of the works executed on the River Dee, whatever may have been the anticipations of their projectors, has been to shut out the sea, and form land at the expense of the navigation. They are designed, in fact, in direct opposition to the general principle laid down in the present treatise, which provides for the admission of the greatest possible quantity of tidal water.
It seems also equally clear that an increase of tidal water must inevitably attend the execution of such works as we tidal water have proposed; but in proof of this assumption, and in contrast produced to the case of the Dee, it may be well for the reader’s information to cite examples which have occurred in practice. Proceeding on actual calculations of comparative sections of the River Tay before and after the operations, the writer found that by the lowering of the low-water line, consequent on the improved state of the river, an additional quantity of sea water, amounting on an average to not less than 1,000,000 cubic yards, or 760,560 tons, is during every tide propelled into and again withdrawn from that part of the river which lies above Newburgh. This quantity is equal to two hours’ discharge of the Tay in its ordinary state; and it therefore follows that the additional tidal discharge for one year is equal to two months’ constant ordinary flow of the River Tay. In the same way it was found by
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1 Great Britain Coasting Pilot, by Captain Greenville Collins, hydrographer in ordinary to the King’s most excellent Majesty, London, 1767; Reports to the Admiralty, by Captain Washington; Report of Tidal Harbour Commissioners; Report by Messrs Stevenson, 1839. calculation that the additional amount of tidal water admitted every tide into the Lune above Heaton, in consequence of the operations, was 736,278 cubic yards. To estimate truly the beneficial effects of these important changes on the scouring power of the Tay and the Lune, it must be kept in view, that in both cases the increased volume of tidal water, being obtained by the enlargement of the low-water channel, operates during every tide, and thus may be held to produce the maximum amount of benefit; for it is obvious that a cubic yard of low-water area, gained in the low-water channel, and filled by every tide, is very much more valuable than a similar amount of space gained at or near high-water of spring tides, which is filled only at remote intervals.
The tendency, then, of the whole system of works recommended in this treatise is to lower the low-water line, and to admit an increased amount of tidal water to act on the low-water channel. But the depression of the low-water line, particularly when the river is confined by walls in the lower part of an estuary, conveys the impression that a great rise has taken place in the level of the adjoining sand-banks, and it has consequently been thought that the erection of river walls is inconsistent with the principles of non-exclusion of tide-water which we have recommended; but we are enabled to show that this is not the case. In its natural state, the channel of such an estuary as the Lune or the Ribble, as already explained in section iv., is subject to constant change of position. The writer has seen many acres of marsh or grass land in such estuaries carried off by the waves, and the solid matter of which they were composed scattered over the shores and sand-banks. Now, the effect of fixing the channel by means of walls, in the manner which has been recommended, is to form one permanent navigable track; and the banks on either side, being no longer subject to the periodical inroads of the river or tides, gradually rise in elevation until they are capable of producing vegetation, and ultimately become what are termed marsh lands. When a river channel has been thus fixed and confined by walls, it has been ascertained by repeated observation that the tidal water comes up the channel in a comparatively pure state, instead of being loaded with particles abraded from the sand-banks and marshes. It has also been found that the process of deposit at the sides of an estuary so improved goes on very slowly after it has reached a certain stage; for the materials deposited on the upper parts of the banks are, as afterwards more particularly described, exceedingly fine, and are carried only by the highest tides, which seldom reach those elevated portions of the shores. From all these considerations we infer that the effect of river-walls upon an estuary is to prevent the constant disturbance of the materials of which the banks are composed, but not to occasion additional accumulations.
The writer had an opportunity, in the case of the Lune, of testing by actual measurement in how far the raising of the banks, caused by the erection of walls, was due merely to a new disposition of the materials which originally filled the bed of the estuary, or to additional foreign matters deposited in consequence of the operations; and the following results are instructive, being, it is believed, the only observations that have been made to determine the state of the sand-banks of an estuary after the river has been improved, as compared with their former condition.
The rubble walls and other works constructed on the Lune caused, as might have been expected, a very considerable alteration in the position and form of the sand-banks in the estuary; and this alteration, in connection with the depression of from 2 to 3 feet in the low-water level of the river, was apt to lead a casual observer to suppose that a great accumulation of sand had taken place, and consequently that a corresponding amount of backwater had been excluded. The writer was authorized by the Admiralty to make such observations as were necessary to determine the true state of the case. Figure 24 represents the changes that were produced by the works. Over the whole area which is represented as covered by sand a deposit had taken place, the banks being higher than formerly; whereas the whole area included in hatched lines had been scoured, the banks having been lowered. A careful calculation was made, founded on numerous sections taken in 1838, before the works commenced, and in 1851, after their completion. The result of this investigation was, that after the completion of the works, the amount of deposit on the space shown as sand in the cut was 3,070,146 cubic yards; while the amount of scour on the
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1 The mere cubic contents dredged from a ford or shoal often form no measure of the gain of tidal water due to the operations, because the removal of such an obstruction has the effect of lowering the low-water line for a considerable distance up the river, the extent to which the influence of the works extends depending on the amount of fall; and the whole of the wedge-shaped space included between the old and new low-water lines is a clear gain of tide-water, and the cubic contents of this space generally greatly exceed the cubic quantity of materials removed from the ford by dredging. space shown by hatched lines was 2,810,449 cubic yards; giving an excess of deposit of 259,697 cubic yards. But the amount stated as having been scoured does not include what has been taken away below Glasson and Basil points; and which has doubtless been deposited in the bank above. The survey of 1838 did not afford data for ascertaining the amount of what had been scoured from below Glasson with sufficient accuracy to admit of its being included in the foregoing calculations. But an amount of scouring was ascertained to have actually occurred at that place, which was amply sufficient to counterbalance the surplus of 259,697 cubic yards of deposit, as given in the above statement.
Such a result, we think, may indeed be expected; for it is difficult to conceive in what way parallel walls formed in an estuary can operate either in bringing down additional alluvial matters from the river above, or in bringing up additional detritus from without the bar.
Holding these views, and supported by the actual observations made in the case of the Lune, we therefore conclude—1st, That works executed in accordance with the principles laid down do not necessarily produce additional accumulation of matter, but simply alter the disposition of the existing materials of which the bed of the estuary was originally composed. 2nd, By deepening the navigable track they admit of a large accession of water to act upon the low-water channel during all tides, and at the most favourable period of the tide. 3rd, That the depth of water on such bars as are produced by the action of the waves may be maintained, and even increased, by means of the works which have been described as applicable to the intermediate or tidal compartments of rivers.
While treating of deposits, this is probably the proper place to observe, that the size of detrital particles which can be carried by a current depends on the velocity of the stream, the nature of the bottom along which the detritus is moved, as well as the shape of the particles of which the detritus itself is composed, and is altogether a subject so dependent on special circumstances, that there is great difficulty in laying down rules which can be generally applicable. The following are the results of experiments made by Bossut, Dubuat, and others, on the size of detrital particles which streams flowing with different velocities are said to be capable of carrying:
- 3 ins. per sec. = 0·170 mile per hour will just begin to work on fine clay. - 6 " = 0·340 do., will lift fine sand. - 8 " = 0·545 do., will lift sand as coarse as linseed. - 12 " = 0·819 do., will sweep along fine gravel. - 24 " = 1·638 do., will roll along rounded pebbles 1 inch diameter.
The only recent experiments made on this subject are those of Mr T. Logan, C.E., given in the Proceedings of the Royal Society of Edinburgh, vol. iii., p. 475, which were made with a stream seldom exceeding half an inch in depth; and are as follows:
| Nature of Materials | Rate of sinking in water | Current required to move | |---------------------|-------------------------|--------------------------| | Brick-clay when mixed with water, and allowed to settle for half an hour | 565 feet per minute | 15 feet per minute, 1·70 mile per hour | | Fresh-water sand | 10 feet per minute | 40 feet per minute, 4·54 mile per hour | | Sea sand | 11·707 feet per minute | 60 feet per minute, 7·52 mile per hour | | Rounded pebbles about the size of peas | 60 feet per minute | 120 feet per minute, 1·37 mile per hour | | Vegetable soil | 60 feet per minute | 60 feet per minute, 0·55 mile per hour |
Brick-clay in its natural state was not moved by a current of 123 feet per minute, or 1·45 mile per hour.
We give these results as they have been stated by their authors; at the same time it is necessary to say that, for the reason above mentioned, we consider their application in practice to be very uncertain. Regarding the subject in a general point of view, however, certain laws as to the transmission and deposition of detritus will be found applicable to certain situations. On this subject Sir H. De la Beche says:—"Where the velocity of a river is sufficient to produce attrition of the substances which it has either torn up, collected by undermining its banks, or which have fallen into it, they gradually become more easy of transport, and found next would, if the force of the current continued always the same, be forced forward until the river delivered itself into the sea; but as the velocity of a current greatly depends on the fall of the river, the transport is regulated by the inclination of the river's bed. Now it is well known that this inclination varies materially even in the same river; so that it may be able to carry detritus to one situation, but may be unable to transport it further under ordinary circumstances, in consequence of diminished velocity. As a general fact, it may be fairly stated that rivers, where their courses are short and rapid, bear down pebbles to the seas near them, as in the case of the Maritime Alps, &c.; but that where their courses are long, and change from rapid to slow, they deposit the pebbles where the force of the stream diminishes, and finally transport mere sand or mud to their mouths, as is the case with the Rhone, Po, Danube, Ganges, &c."
This holds true in the case of such rivers as those to which Sir H. De la Beche refers; but it will be found that the case is exactly reversed in tidal estuaries. There the heavier sands and deposits are found at the mouth of the estuary, and the particles are lighter as we recede inwards.
The writer has tested this on several occasions, more particularly in the Dee, the Ribble, the Lune, the Wear, the Forth, and the Tay, by agitating equal quantities of sand and deposit (taken from different parts of the tidal estuary) in equal quantities of water, and observing the time which elapsed in each case before the materials were deposited and the water assumed a state of purity. The result of these observations proved that the sand of outer or seaward banks was composed of large particles, which were held in suspension only a few seconds, and that in the inner parts of the estuary the deposit decreased in weight, and that generally it decreased from low to high water, where the silt was exceedingly fine, and remained in suspension in some cases even for hours after the agitation of the water. The following statement by Mr William Bald of experiments made on materials taken from different parts of the bed of the Clyde, shows the variety of materials found in the same stream, and is a valuable record of the weight of the deposits which form the beds of our tidal rivers:
| Deposits | Lbs. to cubic feet | No. of cubic ft. to the Ton | |----------|-------------------|---------------------------| | Fine sand and a few pebbles laid in the box, loose, not pressed, nearly dry | 87 | 26 | | Do, do, pressed | 92 | 24 | | Mud at White Inch, dry, and firmly packed; also very fine sand and much | 97 | 23 | | Wet mud, rather compact and firm, well pressed into the box | 115 | 19 | | Wet, fine shingle gravel, well pressed | 134 | 18 | | Wet running mud | 142 | 18·1 | | Sharp dry sand deposit in harbour | 92 | 24·3 | | Port-Glasgow Bank (sand) wet, pressed into a box | 1204 | 18·6 | | Sand opposite Erskine House, wet, pressed | 116 | 19·3 | | Alluvial earth, pressed | 93 | 24 | | Loose | 67 | 32 |
The writer of this article found the gravel of the Tay to be 18 feet to the ton.
1 De la Beche's Geological Manual. 2 Trans. of Institution of Civil Engineers, vol. v., p. 330. The quantity of solid matter carried or held in suspension by rivers has also been made the subject of observation; but the different observers whose remarks have come under our notice have stated their results in different ways, some giving the weight and others the bulk of detritus. But assuming 18 cubic feet of solid matter to weigh a ton, we think the following table presents a fair view of the cubic measure of solid matter, and the ratios of volume and weight in each case. In submitting this table, we must observe that the discrepancies in the statements are so great, that further observations are necessary before any satisfactory conclusion can be arrived at; but we give the results as they have been stated by their respective authorities:
| Name of River | Cubic inches of solid matter per every cubic yard of water | Ratios of volume of solid matter to volume of water | Ratios of weight of solid matter to weight of water | |---------------|-------------------------------------------------------------|---------------------------------------------------|---------------------------------------------------| | Mississippi, mean... | 15-5 | 3777 | 3777 | | Irrawaddy, in flood... | 11-71 | 3777 | 3777 | | Do., ordinary state | 4-3 | 3777 | 3777 | | Rhine, in flood... | 1-67 | 3777 | 3777 | | Do., ordinary state | 1-13 | 3777 | 3777 | | Do., mean... | 1-5 | 3777 | 3777 | | Mersey, flood-tide... | 29 | 3777 | 3777 | | Do., ebb-tide... | 33 | 3777 | 3777 |
From this table it will be seen that the Rhine, as compared to the others, is exceedingly pure; while the waters of the Mersey, on the other hand, hold in suspension a very large amount. It must be kept in view, however, that the source from whence the sedimentary matter in the Mersey is derived is very different from any of the other cases mentioned in the table. The main part of the solid matter in suspension in the Mersey, and indeed in all our tidal rivers, is sand, stirred up by the flowing tide, which is deposited again during the ebb-tide. The sedimentary matters in such rivers as the Mississippi or the Irrawaddy, on the other hand, are borne down from the low tracts of alluvial country through which it flows, and form a constant and consequently increasing deposit at the mouth of the river.
In all cases where the tidal currents across the mouths of such rivers are languid or altogether absent, as in the Mississippi, the Nile, the Danube, and other continental rivers, the deposits brought down are not carried away, but form deltas, which collect with greater or less rapidity in proportion to the quantity of material brought down and the depth of water in which it is deposited. Mr Ellet computes the delta of the Mississippi at 40,000 square miles in extent, its average length from north to south being 500 miles. Assuming the sedimentary matter brought down at $\frac{1}{3}$th of the volume of water, and the discharge of the river at 21,000,000,000,000 cubic feet per annum, he estimates that this vast accretion of deposited stuff must have formed at an average rate of 1 mile in 99 years, giving a period for its entire formation of something like 45,000 years! Sir H. De la Beche has, however, with reason, suggested that deltas would increase most rapidly at the first period of their formation, on account of the greater declivity of the river, and the supposition that the detritus from the interior would become gradually less, from the equalization of levels and the fewer asperities that agents have to act on; and thus it seems impossible to calculate from the present rate of accretion the time which the whole mass has taken to accumulate.
In concluding this treatise, we have to point out in what way, and to what extent, river improvements conducted on property benefited the principles advocated benefit adjoining property; for it is obviously highly important if the two objects of river and improvement can be carried on simultaneously, and we think that to a certain extent this is perfectly practicable. The attempts of proprietors to protect the foreshores of their lands from the encroachments of rivers in tidal estuaries are often attended with great expense; and if those efforts prove for some time effectual in warding off the approach of the channel, the land speedily takes on vegetation, and is fit for pasturage. But the tenure by which such property is held is very slight; and the spot which today affords grazing for cattle may in a few tides become the navigable channel of the river. Now, it is obvious that the perfect protection from such encroachments afforded by the training and guiding of the low-water channel by longitudinal walls, adds materially to the value of the adjoining property; for not only is the land beyond high-water mark completely protected from encroachment, but the marsh lands bordering the estuary become in fact permanent property, and not an ever-changing benefit held for one year and probably lost the next. Marsh lands so protected from waste are still, it is true, liable to be flooded by high tides, a circumstance, however, which is considered by some persons not injurious, but rather beneficial, for marsh pasture lands.
The process of reclamation in all such cases goes on very slowly after it has reached a certain stage, because, as the banks rise, they are more seldom covered by the tide, and the materials deposited on the inner and higher parts of the banks are, as already stated, exceedingly fine, and are carried only by the highest tides, which seldom reach them. Mr Park has found on the Ribble that the first indications of vegetation appear when the banks are elevated 12 feet above the ordnance datum-line, which is the mean level of the sea. This height corresponds at the Ribble to about the level of high-water of neap tides. Mr Gordon also found, that in the Norfolk estuary "the samphire began to settle on the sands, which the neap tides just cover," and that "grass began to grow about one foot above the samphire level." Such marsh lands, if left unprotected, must remain for ever liable to be covered during high floods or tides, and therefore cannot be said to be available as arable lands, without the erection of considerable works for the purpose of protecting them from floods, and providing for their effectual drainage. As the erection of such works, however, forms no part of river improvement, we allude to them in this place only for the purpose of remarking, that in all cases they should be erected with caution. There are situations in which the erection of embankments for protecting land may be injurious to the interests of navigation; there are others in which such works, if judiciously laid out, may be harmless; but their
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1 Mr Ellet says that the sedimentary matter transported by the Mississippi forms $\frac{1}{3}$th part of the volume discharged by the river. (Ellet, On the Ohio and Mississippi.)—Mr T. Logan, C.E., in Pegu, states in a paper on the Delta of the Irrawaddy, read before the Royal Society of Edinburgh, session 1857, that the waters of the Irrawaddy contained $\frac{1}{3}$th part of their weight of sediment during floods, and $\frac{1}{3}$th part of their weight when the river was in a low state, and gives the mean deposit as 8 inches per cubic yard.—Mr Leonard Horner found that the water of the Rhine at Bonn contained from $\frac{1}{3}$th part of its weight during floods to $\frac{1}{3}$th part of its weight in a low state. (Transactions of Science and Art, 1855.) Captain Denham found that the tidal water of the Mersey contained 29 cubic inches of solid matter in every cubic yard during flood-tides, and 33 cubic inches in every cubic yard during ebb-tide. (Observations on the Mersey, by Captain H. M. Denham, R.N.; Liverpool, 1840.)—Mr Lyell says, Mr Hartsecker computed the Rhine to contain most flooded, 1 inch in 100 of mud in suspension, whereas observations by Sir Charles Lyell show that the turbid waters of the Yellow River in China contained earthy matter in the proportion of 1 to 200. Manfredi, the celebrated Italian hydrographer, conceived the average proportion of sediment in all running water to be $\frac{1}{3}$th. Some writers, on the contrary, as De Maillet, have declared the most turbid waters to contain far less sediment than any of the above estimates would import; and there is so much contradiction and inconsistency in the facts and speculations hitherto promulgated on the subject, that we must wait for additional experiments before we can form any opinion on the subject. (Principles of Geology, by Charles Lyell, F.R.S., London, 1830, vol. i., p. 247.)
2 Report on Norfolk Estuary, by L. D. B. Gordon, C.E., Glasgow, 1856. Navigation, Inland.
Rivers, effect in any case can only be determined by a careful consideration of the special circumstances of the locality in which they are erected. We know many cases where the interests of navigation have been sacrificed by unwarrantable encroachment; and, on the other hand, instances are not wanting where even important works have been embarrassed and crippled by an over-cautious regard to the principle of non-encroachment on the high-water line.
With reference more particularly to the operations of landowners, it is notorious that in many cases attempts to reclaim or protect property have led to serious and costly legal proceedings between landowners and the local conservators of navigations; and this we are sensible has in some instances arisen from a feeling, on the part of the landowners, that their operations could not be regarded as prejudicial. The local conservators, on the other hand, have generally no means of knowing what the ultimate intentions of the landowners are until their operations have proceeded so far as to render it impossible, if the interest of navigation require it, to stop or to remove the works without considerable loss. A difference of opinion has thus been raised, which has too often ended in an expensive lawsuit. We have long held the opinion that it would in many, if not in all, of our estuaries, be most desirable to have a line of conservation marked out by the Admiralty (without whose authority no encroachment can be made within high-water mark) for the regulation of all works for the protection of land. Were such a line defined, the landowners could then with confidence, and without risk of challenge, enter on such works within the line of conservation as they considered necessary for the protection of their property, and a source of much difference of opinion and expensive litigation would be at once removed. We had hoped that the Tidal Harbour Commission, who have been enabled, through the exertions of Captain Washington, the hydrographer of the Admiralty, who was one of the commission, to give in their printed reports so valuable a fund of information on our tidal harbours, would have terminated their labours by pointing out and recommending some such system as we have suggested of defining lines of conservation for all the important rivers and estuaries of the country. It is obvious, however, that were such a duty to be performed, it must be committed to a duly qualified commission, acting most naturally under the Admiralty, and so composed that the protection of navigation, and the interests of landowners or trustees for public works, should be fully represented, the whole of its members being actuated by one common desire to do what is best for the community at large.
The following is a statement of ratios between the discharges of certain rivers during low-water and when in flood; but it must be kept in view, as stated in treating of the formula for calculating the discharge, that its determination is a difficult problem; so that the results stated with reference to the discharge of different rivers must be received with this caution as to their accuracy.
| Name of River | Length in miles | Area of drainage in square miles | Ordinary discharge per minute cubic feet | Mean Discharge Cubic ft. per min. | Flood | Part of River where slope occurs | Length of River affected by tide in miles | Depth on bar at lowwater in feet | Authority | |---------------|----------------|---------------------------------|----------------------------------------|-------------------------------|------|---------------------------------|-------------------------------------|----------------|------------| | Clyde | 48,000 | 194,000 | 1 to 40 | | | | | | N. Beardmore's Hyd. Tables. | | Conon | 7,959 | 216,589 | 1, 27:2 | | | | | | Messrs Stevenson, C.E., Edin. | | Earn | 54,000 | 215,600 | 1, 39 | | | | | | A. Nimmo, C.E. | | Ganges | 12,420,000 | 29,652,480 | 1, 24 | | | | | | J. Ure, C.E., Glasgow. | | Irrawaddy | 4,500,000 | 45,000,000 | 1, 10:0 | | | | | | Messrs Stevenson, Edin. | | Mississippi | 39,554,000 | 76,800,000 | 1, 1:0 | | | | | | E. K. Calver, R.N. | | Nile | 1,388,000 | 13,200,000 | 1, 9:5 | | | | | | J. Gibb. | | Tay | 218,000 | 753,740 | 1, 3:4 | | | | | | Messrs Stevenson, Edin. | | Thames | 80,220 | 475,000 | 1, 5:9 | | | | | | Messrs Stevenson, Edin. |
The high ratio on the Conon may be due to the steepness of its bed, and the absence of any natural lake or reservoir on its course to act as a regulator.
The quantities in the following table represent the discharges of the rivers in their ordinary state.
Physical Characteristics of Rivers.
| Name of River | Length in miles | Area of drainage in square miles | Ordinary discharge per minute cubic feet | Mean Discharge Cubic ft. per min. | Flood | Part of River where slope occurs | Length of River affected by tide in miles | Depth on bar at lowwater in feet | Authority | |---------------|----------------|---------------------------------|----------------------------------------|-------------------------------|------|---------------------------------|-------------------------------------|----------------|------------| | Amazon | 4000 | 400 | | | | | | | N. Beardmore's Hyd. Tables. | | Annan | 35 | 65 | | | | | | | Messrs Stevenson, C.E., Edin. | | Boyne | 60 | 180,000 | 257 | | | | | | A. Nimmo, C.E. | | Clyde | 98 | 945 | 48,000 | | | | | | J. Ure, C.E., Glasgow. | | Conon | 35 | 700 | 10,675 | | | | | | Messrs Stevenson, Edin. | | Coquet | 41 | 10,675 | | | | | | | E. K. Calver, R.N. | | Dee, Aberdeen | 87 | 755 | 10,675 | | | | | | J. Gibb. | | Dee, Chester | 85 | 620 | | | | | | | Messrs Stevenson, Edin. | | Forth | 63 | 452 | 29,285 | | | | | | Messrs Stevenson, Edin. | | Foyle | 55 | 1100 | 31,500 | | | | | | Captain Washington. | | Ganges | 1680 | 432,480 | 12,420,000 | | | | | | Messrs Stevenson. | | Irrawaddy | 4,500,000 | 105 | | | | | | | T. Logan, C.E. | | Lune | 50 | 2070 | | | | | | | Messrs Stevenson. | | Mersey | 70 | 1,748 | | | | | | | Captain Denham, R.N. | | Mississippi | 4000 | 39,554,000 | 32:5 | | | | | | C. Ellet. | | Nile | 2240 | 1,286,000 | 26 | | | | | | Messrs Stevenson. | | Nith | 45 | 500 | | | | | | | Messrs Stevenson. | ### NAVIGATION LAWS
The, of which some notice has been given under the articles COMMERCE and ENGLAND, and which were considered obnoxious to the interests of commercial enterprise, were in effect repealed in 1854. An act to admit foreign ships to the coasting trade of this country (17th Vict., cap. 5) was passed March 23, 1854; and an act to amend and consolidate the acts relating to merchant shipping (17th and 18th Vict., cap. 104) on the 10th August 1854. The latter act received some amendment, particularly with regard to the erection and maintenance of colonial lighthouses (18th and 19th Vict., cap. 91) in 1855.
### NAVIGATOR'S ISLANDS, or SAMOAN ISLANDS, a group in the Pacific, lying to the N.E. of the Friendly Islands, between S. Lat. 13° 30' and 14° 30', and W. Long. 168° and 173°. They are eight in number, and three of them are of considerable size. The largest of the group is Savaii, which is upwards of 200 miles in circuit, and the principal others are Maona, Pala, and Oyalava. They are for the most part mountainous and of volcanic formation. The soil is very rich and fertile, and the mountains are thickly covered with wood to their very summits. The trees are chiefly evergreens, and are remarkable for the beauty and variety of their appearance. Palms, cocoa-nut trees, breadfruit trees, banyans, sugar-canes, pine-apples, potatoes, coffee, yams, and tobacco are among the productions of these islands. The climate is variable, and heavy rains fall during the winter. No indigenous quadrupeds are found; but horses, cattle, and swine, which have been introduced from other places, thrive well and increase largely. Fowls are numerous, and the surrounding parts of the ocean abound in fish. The inhabitants are superior in appearance to most of the tribes of the South Sea Islands. They are stout, well-proportioned, and of a dark-brown complexion, and the men are in general better looking than the women. In character, they are generally good-natured, hospitable, and affectionate, and they show great respect for the aged. They are intelligent, and display considerable ingenuity in making their canoes and houses; but they are indolent, fond of pleasure, covetous, and deceitful. Their language is smooth and liquid, and in it alone of Polynesian tongues the sibilant sound occurs. The Navigator's Islands were first visited by missionaries in 1830 from Otaheite; and since 1836 missionaries have been sent out directly from Europe, by whose means a great number of the inhabitants have been converted to Christianity. They are also beginning to pursue the employments of trade, and to learn the use of money. Cocoa-nut oil is the principal article of export, and the imports are cotton, calicoes, fire-arms, ammunition, &c., supplied chiefly by American whalers. The whole area of the islands is 2650 square miles; and the estimates of the population vary from 50,000 to 160,000.
| Name of River | Length in miles | Area of drainage in square miles | Ordinary discharge per minute in cubic feet | Slope of river where slope occurs | Part of River where slope occurs | Length of River affected by tide, in miles | Depth on bar at low-water, in feet | Authority | |---------------|----------------|----------------------------------|------------------------------------------|---------------------------------|----------------------------------------|-------------------------------------|----------------|-------------| | Rhine | 700 | 88,553 | 3,260,000 | 445 | Source to Roichemau | Johnston's Physical Atlas. | | Rhone | 550 | 38,329 | | | Reichenau to Constance | Johnston's Physical Atlas. | | Ribble, before works | 80 | 850 | 139,935 | 159 | Basle to Basle | P. Park, C.E. Preston. | | Severn | 180 | 8,560 | | | Lower 221 miles | Remarks on the Tidal Phenomena of the River Severn, by Capt. F. W. Beechey, R.N. | | Tay, before works | 160 | 2,283 | 274,000 | 120 | Diglis Weir to Upton Bridge | Messrs Stevenson. | | Tweed | 100 | 680 | | | Upton Bridge to Mythe Br. | John Murray, C.E. | | Thames | 2041 | 5,000 | 102,000 | 204 | Lockaloe to Toddlington | Messrs Rennie, C.E. | | Tweed | 100 | 1,870 | | | Leader to Kelso | A. Peterman, F.G.S. | | Tyne | 80 | 1,100 | | | Kelso to sea | E. K. Calver, R.N. | | Wear | 58 | 437 | | | New Bridge to sea | Thos. Meik, C.E., Sunderland. |
(d. s.—N.)