NAVIGATION.

PROB. V. Given the latitudes of two places, and their distance; to find the course and difference of longitude.

Example. A ship from St Mary's, in latitude 36° 57' N., longitude 25° 9' W., sailed on a direct course between the north and east 1162 miles, and was then by observation in latitude 49° 57' N. Required the course steered, and longitude come to.

By Construction.

Fig. 30.

Make AB equal to 780, and AD equal to 1081; draw BC, DE perpendicular to AD; make AC equal to 1162, and through AC 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.

To find the difference of longitude.

PROB. VI. Given the latitudes of two places, and the departure; to find the course, distance, and difference of longitude.

Example. From Aberdeen, in latitude 57° 9' N., longitude 2° 8' W., a ship sailed between the south and east till her departure was 146 miles, and latitude come to 53° 32' N. Required the course and distance run, and longitude come to.

Fig. 31.

By Construction.

With the difference of latitude 217 m. and departure 146 m. construct the triangle ABC; make AD equal to 382, draw DE parallel to BC, and produce AC to E; then the course BAC will measure 33° 56', the distance AC 261, and the difference of longitude DE 257.

By Calculation.

To find the course.

To find the distance.

To find the difference of longitude.

PROB. VII. Given one latitude, distance, and departure; to find the other latitude, course, and difference of longitude.

Example. A ship from Naples, in latitude 40° 51' N., longitude 14° 14' E., sailed 252 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 come to.

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 cords, 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 come to 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° 30' E.

Fig. 32.

By Calculation.

To find the course.

To find the difference of latitude.

PROB. VIII. Given one latitude, course, and difference of longitude; to find the other latitude and distance.

Example. A ship from Tercera, in latitude 38° 45' N., longitude 27° 6' W., sailed on a direct course, which, when corrected, was N. 32° E., and is found by observation to be in longitude 18° 24' W. Required the latitude come to, and distance sailed.

By Construction.

Fig. 33.

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° 46'; 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 the meridional difference of latitude.

To find the distance.

PROB. IX. To find the difference of longitude made good upon compound courses.

Rule I. With the several courses and distances complete the traverse table, and find the difference of latitude, departure, and course made good, and the latitude come to, as in Traverse Sailing. Find also the meridional difference of latitude.

Now 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 left, will give the ship's present longitude.

Example. A ship from Port St Julian, in latitude 49° 10' S. longitude 68° 44' W. sailed as follows: E. S. E. 53 miles, S. E. by S. 74 miles, E. by N. 68 miles, S. E. by E.

1/2 E. 47 miles, and E. 84 miles. Required the ship's pre-Mercator's sent place. Sailing.

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 II. Complete the traverse table as before, to which annex five columns. Now, with the latitude left, 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 is 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 Prob. IV. which place in the fourth or fifth columns, according as the coast 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. 1. Four days ago we took our departure from Faro Head, in latitude 58° 40' N. and longitude 4° 50' W. and since have sailed as follows: N. W. 32 miles, W. 69 miles, W. N. W. 93 miles, W. by S. 77 miles, S. W. 58 miles, and W. 1/2 S. 49 miles. Required our present latitude and longitude.

Ex. 2. A ship from latitude 78° 15' N. longitude 28° 14' E. sailed the following courses and distances. The latitude come to is required, and the longitude, by both methods; the bearing and distance of Hacluit's headland, in latitude 79° 55' N. longitude 11° 55' E. is also required.

CHAP. VII.—CONTAINING THE METHOD OF RESOLVING THE SEVERAL PROBLEMS OF MERCATOR'S SAILING, BY THE ASSISTANCE OF A TABLE OF LOGARITHMIC TANGENTS.

PROB. I. Given one latitude, distance, and difference of longitude; to find the course and other latitude.

RULE. To the arithmetical complement of the logarithm of the distance add the logarithm of the difference of lon-

gitude in minutes, and the log. cosine of the given latitude; the sum, rejecting radius, will be the log. sine of the approximate course.

To the given latitude taken as a course in the traverse table, and half the difference of longitude in a distance column, the corresponding departure will be the first correction of the course, which is subtractive if the given latitude is the least of the two; otherwise additive.

In Table A, under the complement of the course, and

Method of opposite to the first correction in the side column, is the resolving second correction. In the same table find the number answering to the course at the top, and difference of longitude in the side column; and such part of this number being taken as is found in Table B opposite to the given

latitude, will be the third correction. Now these two corrections subtracted from the course corrected by the first correction will give the true course. Oblique Sailing.

Now, the course and distance being known, the difference of latitude is found as formerly.

Example. From latitude 50° N. a ship sailed 290 miles between the south and west, and differed her longitude 5°. Required the course, and latitude come to.

This problem was proposed and resolved by Mr Robert Hues, in his Treatise on the Globes, printed at London in the year 1639, p. 181.

It was afterwards proposed by Dr Halley, in the second volume of the Miscellanea Curiosa, p. 35, in the following words:

A ship sails from a given latitude, and, having run a certain number of leagues, has altered her longitude by a given angle; it is required to find the course steered. And he then adds: The solution hereof would be very acceptable, if not to the public, at least to the author of this tract, being likely to open some further light into the mysteries of geometry.

Since that time, this problem has been solved in an indirect manner, by several writers on navigation, and others; as Monsieur Bouguer, in his Nouveau Traité de Navigation; Mr Robertson, in the second volume of his Elements of Navigation; Mr Emerson, in his Theory of Navigation, which accompanies his Mathematical Principles of Geography; Mr Israel Lyons, in the Nautical Almanac for 1772; and Monsieur Bezout, in his Traité de Navigation; and lately, Baron Maseres, with the assistance of Mr Attwood, has given the first direct solution of this problem. For a comparison of the various solutions which have hitherto been made of this problem, the reader is referred

to that by Dr Mackay, in the fourth and sixth volumes of Baron Maseres' Scriptores Logarithmici.

CHAP. VIII.—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, &c.

Ex. 1. At 11^th A. M. the Girdle Ness bore W. N. W., and at 2^nd 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 N, E, S, W, 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 Girdle Ness. Now the distances CF, HF will measure 19.1 and 26.5 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 CFH is 3 points; to find the distances CF, HF.

To find the distance CF.

To find the distance FH.

Ex. 2. Running up Channel E. by S. per compass at the rate of 5 knots an hour. At 11^th A. M. the Eddystone

Current Sailing. lighthouse bore N. by E. $\backslash frac\{1\}\{2\}$ E., and the Start Point N. E. by E. $\backslash frac\{1\}\{2\}$ E.; and at 4 P. M. the Eddystone bore N. W. by N., and the Start N. $\backslash frac\{3\}\{4\}$ E. Required the distance and bearing of the Start from the Eddystone, the variation being $2\backslash frac\{1\}\{2\}$ points W.

By Construction.

Let the point C represent the first station, from which draw the N. by E. $\backslash frac\{1\}\{2\}$ E. line CA, the N. E. by E. $\backslash frac\{1\}\{2\}$ E. 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. E. by N. line DA intersecting CA in A, which represents the Eddystone; and from the same point draw the N. $\backslash frac\{3\}\{4\}$ E. line DB cutting CB in B, which therefore represents the Start. Now the distance AB applied to the scale will measure 22.9, and the bearing per compass BAF will measure $73\backslash frac\{1\}\{2\}^\backslash circ$.

Fig. 35.

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. IX.—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 on each tack.

By Construction.

Draw the N. W. line CA (fig. 36) equal to 64 miles; from C draw CB W. by S., and from A draw AD parallel thereto, and in an opposite direction; bisect 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 $36\backslash frac\{1\}\{2\}$ miles, and BD = 2 CB = $72\backslash frac\{1\}\{2\}$ miles.

Fig. 36.

CHAP. X.—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 ebbings and flowings 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 di-

rectly 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 a-head 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.

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. $2\backslash frac\{1\}\{2\}$ miles an hour. Required the course and distance made good.

By Construction.

Draw the N. N. E. line CA (fig. 37) equal to $18\; \backslash times\; 8\; =\; 144$ miles; and from A draw AB parallel to the N. W. by W. rhumb, and equal to $18\; \backslash times\; 2\backslash frac\{1\}\{2\}\; =\; 45$ miles; now BC being joined will be the distance, and NCB the course. The first of these will measure 159 miles and the second $6^\backslash circ\; 23\text{'}$.

Fig. 37.

Ex. 2. A ship from latitude $38^\backslash circ\; 20\text{'}\; N.$ sailed 24 hours in a current setting N. W. by N., and by account is in latitude $38^\backslash circ\; 42\text{'}\; N.$, having made 44 miles of casting; but the latitude by observation is $38^\backslash circ\; 58\text{'}\; N.$ Required the course and distance made good, and the drift of the current.

By Construction.

Make CE (fig. 38) 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 $41\backslash frac\{1\}\{2\}^\backslash circ$.

Fig. 38.

CHAP. XI.—INSTRUMENTS PROPOSED TO SOLVE THE VARIOUS PROBLEMS IN SAILING, INDEPENDENT OF CALCULATION.

Various methods besides those already given have been proposed, to save the trouble of calculation. One of these methods is by means of an instrument composed of rulers, so disposed as to form a right-angled triangle, having numbers in a regular progression marked on their sides. These instruments are made of different materials, such as paper, wood, brass, &c. and are differently constructed, according to the fancy of the inventor. A number of other instruments, very differently constructed, have been proposed for the same purpose; of these, the rectangular instrument by the late A. Mackay, LL. D. F. R. S. E. &c. is one of the best. It is seldom, however, that any of them are used.

The charts usually employed in the practice of navigation are of two kinds, namely, Plane and Mercator's Charts. The first of these is adapted to represent a portion of the earth's surface near the equator, and the last for all portions of the earth's surface. For a particular description of these, see the article CHART; and as these are particularly described under the above article, it is therefore sufficient in this place to describe their use.

Use of the Plane Chart.

PROB. I. To find the latitude of a place on the chart.
RULE. Take the least distance between the given place and the nearest parallel of latitude; now this distance applied the same way on the graduated meridian, from the extremity of the parallel, will give the latitude of the proposed place.

Thus the distance between Bonavista and the parallel of 15°, being laid from that parallel upon 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, and take the nearest distance between the centre of any of the compasses on the chart and the edge of the ruler; move this extent along so as one point of the compass may touch the edge of the ruler, and the straight line joining their points may be perpendicular thereto; then will the other point show the course. The interval between the places, being applied to the scale, will give the required distance.

Thus the course from Palma to St Vincent will be found to be about S. S. W. $\backslash frac\{3\}\{4\}$ W. and the distance 13$\backslash frac\{1\}\{2\}$°, 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 place sailed from parallel to the rhumb, expressing the given course; take the distance from the scale, and lay it off from the given place by the edge of the ruler; and it will give the point representing the ship's present place.

Thus, suppose a ship had sailed S. W. by W. 160 miles from Cape Palmas; then, by proceeding as above, it will be found that she is in latitude 2° 57' N.

The various other problems that may be resolved by means of this chart require no further explanation, being only the construction of the remaining problems in Plane Sailing on the 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 by this chart, is performed exactly in the same manner as in the plane chart, which see.

PROB. I. To find the distance between two given places on the chart.

CASE I. 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 II. When the given places are under the same parallel.

RULE. If that parallel be the equator, the difference or sum of their longitudes is the distance; otherwise, take half the interval between the places, lay it off upwards and

downwards on the meridian from the given parallel, and Of finding the intercepted degrees will be the distance between the the Latitude and Longitude at Sea.

Or, take an equal extent of a few degrees from the meridian on each side of the parallel, and the number of extents, and parts of an extent, contained between the places, being multiplied by the length of an extent, will give the required distance.

CASE III. 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 two places, and move one point of the compass along the edge of the ruler until the other point just touches a parallel; then the distance between the place where the point of the compass rested by the edge of the ruler, and the point of intersection of the ruler and parallel, being applied to the equator, will give the distance between the places in degrees and parts of a degree, which multiplied by 60 will reduce it to miles.

PROB. II. Given the latitude and longitude in, to find the ship's place on 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 a known 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, and its intersection with the parallel of latitude arrived at will be the ship's present place.

PROB. IV. Given the latitude of the place left, and the course and distance sailed, to find the ship's present place on the chart.

RULE. The ruler being laid over the place sailed from, and in the direction of the given course, take the distance sailed from the equator, put one point of the compass at the intersection of any parallel with the ruler, and the other point of the compass will reach to a certain place by the edge of the ruler. Now this point remaining in the same position, 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 will be known, and the intersection of the corresponding parallel with the edge of the ruler will be the ship's present place.

The other problems of Mercator's Sailing may be very easily resolved by this chart; but as they are of less use than those given, they are therefore omitted, and may serve as an exercise to the student.

BOOK II.

CONTAINING THE METHOD OF FINDING THE LATITUDE AND LONGITUDE OF A SHIP AT SEA, AND THE VARIATION OF THE COMPASS.

CHAP. I.—METHOD OF FINDING THE LATITUDE AT SEA.

SECT. I.—Of Hadley's Quadrant.

Hadley's quadrant is the chief instrument in use at present for observing altitudes at sea. The form of this instrument, according to the present mode of construction, is an octagonal sector of a circle, and therefore contains 45 degrees; but because of the double reflection, the limb

finding is divided into 90 degrees. SEE ASTRONOMY and QUADRANT. Fig. 39 represents a quadrant of the common construction, of which the following are the principal parts.

1. 1. ABC, the frame of the quadrant.
2. 2. BC, the arch or limb.
3. 3. D, the index; $a,\; b$, the subdividing scale.
4. 4. E, the index-glass.
5. 5. F, the fore horizon-glass.
6. 6. G, the back horizon-glass.
7. 7. K, the coloured or dark glasses.
8. 8. HI, the vanes or sights.

Of the Frame of the Quadrant.

The frame of the quadrant consists of an arch BC, firmly attached to the two radii AB, AC, which are bound together by the braces LM, in order to strengthen it, and prevent it from warping.

Of the Index D.

The index is a flat bar of brass, and turns on the centre of the octant: at the lower end of the index there is an oblong opening; to one side of this opening the vernier scale is fixed, to subdivide the divisions of the arch; at the end of the index there is a piece of brass, which bends under the arch, carrying a spring to make the subdividing scale lie close to the divisions. It is also furnished with a screw to fix the index in any desired position. The best instruments have an adjusting screw fitted to the index, that it may be moved more slowly, and with greater regularity and accuracy, than by the hand. It is proper, however, to observe, that the index must be previously fixed near its right position by the above-mentioned screw.

Of the Index-Glass E.

Upon the index, and near its axis of motion, is fixed a plane speculum, or mirror of glass quicksilvered. It is set in a brass frame, and is placed so that its face is perpen-

icular 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; and the intention of this glass is to receive the image of the sun, or any other object, and reflect it upon either of the two horizon-glasses, according to the nature of the observation.

The brass frame with the glass is fixed to the index by the screw $c$; the other screw serves to replace it in a perpendicular position, if by any accident it has been deranged.

Of the Horizon-Glasses F, G.

On the radius AB of the octant are two small speculums: the surface of the upper one is parallel to the index-glass, and that of the lower one perpendicular thereto, when 0 on the index coincides with 0 on the limb. These mirrors receive the reflected rays, and transmit them to the observer.

The horizon-glasses are not entirely quicksilvered; the upper one F is only silvered on its lower half, or that next the plane of the quadrant, the other 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. The other horizon-glass is silvered at both ends. In the middle there is a transparent slit, through which the horizon may be seen.

Each of these 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 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 one 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.

Of the Coloured Glasses K.

There are usually three coloured classes, two of which are tinged red and the other green. They are used to prevent the solar rays from hurting the eye at the time of observation. These glasses are set in a frame, which turns on a centre, so that they may be used separately or together as the brightness of the sun may require. The green glass is particularly useful in observations of the moon; it may be also used in observations of the sun, if that object be very faint. In the fore observation, these glasses are fixed as in fig. 39; but when the back observation is used, they are removed to N.

Of the two Sight Vanes, H, I.

Each of these vanes is a perforated piece of brass, designed to direct the sight parallel to the plane of the quadrant. That which is fixed at I is used for the fore, and the other for the back observation. 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.

Of the Divisions on the Limb of the Quadrant.

The limb of the quadrant is divided from right to left into 90 primary divisions, which are to be considered as degrees, and each degree is subdivided into three equal parts, which are therefore of 20 minutes each: the intermediate minutes are obtained by means of the scale of divisions at the end of the index.

Of the Vernier, or Subdividing Scale.

The dividing scale contains a space equal to 21 divisions of the limb, and is divided into 20 equal parts. Hence the difference between a division on the dividing scale and a division on the limb is one twentieth of a division on the limb, or one minute. The degree and minute pointed out by the dividing scale may be easily found thus.

Observe what minute on the dividing scale coincides with a division on the limb; this division being added to the degree and part of a degree on the limb, immediately preceding the first division on the dividing scale, will be the degree and minute required.

Thus, suppose the fourteenth minute on the dividing scale coincided with a division on the limb, and that the preceding division on the limb to 0 on the vernier was $56^\{\backslash circ\}\; 40\text{'}$; hence the division shown by the vernier is $56^\{\backslash circ\}\; 54\text{'}$. A magnifying glass will assist the observer to read off the coinciding divisions with more accuracy.

Adjustments of Hadley's Quadrant.

The adjustments of the quadrant consist in placing the mirrors perpendicular to the plane of the instrument. The fore horizon-glass must be set parallel to the speculum, and the planes of the speculum and back horizon-glass produced must be perpendicular to each other when the index is at 0.

ADJUSTMENT I. To set the index-glass perpendicular to the plane of the quadrant.

Set the index towards the middle of the limb, and hold the quadrant so that its plane may be 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 that seen directly, the speculum is perpendicular to the plane of the instrument. If they do not appear in the same plane, the error is to be rectified by altering the position of the screws behind the frame of the glass.

ADJUSTMENT II. To set the fore horizon-glass perpendicular to the plane of the instrument.

Set the index to 0; hold the plane of the quadrant parallel to the horizon; direct the sight to the horizon, and if the horizons seen directly and by reflection are apparently in the same straight line, the fore horizon-glass is perpendicular to the plane of the instrument; if not, one of the horizons will appear higher than the other. Now if the horizon seen by reflection is higher than that seen directly, release the nearest screw in the pedestal of the glass, and screw up that on the farther side, till the direct and reflected horizons appear to make one continued straight line. But if the reflected horizon is lower than that seen directly, unscrew the farthest, and screw up the nearest screw till the coincidence of the horizons is perfect, observing to leave both screws equally tight, and the fore horizon-glass will be perpendicular to the plane of the quadrant.

ADJUSTMENT III. To set the fore horizon-glass parallel to the index-glass, the index being at 0.

Set 0 on the index exactly to 0 on the limb, and fix it in that position by the screw at the under side; hold the plane of the quadrant in a vertical position, and direct the sight to a well-defined part of the horizon; then if the horizon seen in the silvered part coincides with that seen through the transparent part, the horizon-glass is adjusted; but if the horizons do not coincide, unscrew the milled screw in the middle of the lever on the other side of the quadrant, and turn the nut at the end of the lever until both horizons coincide, and fix the lever in this position by tightening the milled screw.

As the position of the glass is liable to be altered by fixing the lever, it will therefore be necessary to re-examine it; and if the horizons do not coincide, it will be necessary

either to repeat the adjustment, or rather to find the error of adjustment, or, as it is usually called, the index error; which may be done thus:

Direct the sight to the horizon, and move the index until the reflected horizon coincides with that seen directly; then the difference between 0 on the limb and 0 on the vernier is the index error; which is additive when the beginning of the vernier is to the right of 0 on the limb, otherwise subtractive.

ADJUSTMENT IV. To set the back horizon-glass perpendicular to the plane of the instrument.

Put the index to 0; hold the plane of the quadrant parallel to the horizon, and direct the sight to the horizon through the back-sight vane. Now if the reflected horizon is in the same straight line with that seen through the transparent part, the glass is perpendicular to the plane of the instrument. If the horizons do not unite, turn the sunk screws in the pedestal of the glass until they are apparently in the same straight line.

ADJUSTMENT V. To set the back horizon-glass perpendicular to the plane of the index-glass produced, the index being at 0.

Let the index be put as much to the right of 0 as twice the dip of the horizon amounts to; hold the quadrant in a vertical position, and apply the eye to the back vane; then if the reflected horizon coincides with that seen directly, the glass is adjusted; if they do not coincide, the screw in the middle of the lever on the other side of the quadrant must be released, and the nut at its extremity turned till both horizons coincide. It may be observed, that the reflected horizon will be inverted; that is, the sea will be apparently uppermost and the sky lowermost.

This method of adjustment is esteemed troublesome, and is often found to be very difficult to perform at sea, on which account the method of observation by the back horizon-glass is seldom or never used.

Use of Hadley's Quadrant.

The altitude of any object is determined by the position of the index on the limb, when by reflection that object appears to be in contact with the horizon.

If the object whose altitude is to be observed be the sun, and if 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 may 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 in the silvered part of the glass adjacent to the line of separation of the two parts.

There are two different methods of taking observations with the quadrant. In the first of these the face of the observer is directed towards that part of the horizon immediately under the sun, and is therefore called the fore observation. In the other method, the observer's back is to the sun, and it is hence called the back observation. This last method of observation is to be used only when the horizon under the sun is obscured, or rendered indistinct by fog or any other impediment.

In taking the sun's altitude, whether by the fore or back observation, the observer must turn the quadrant about upon the axis of vision, and at the same time turn himself about upon his heel, so as to keep the sun always in that part of the horizon-glass which is at the same distance as the eye from the plane of the quadrant. In this way the reflected sun will describe an arch of a parallel circle round the true sun, the convex side of which will be downwards in the fore observation and upwards in the back; and consequently, when, by moving the index, the lowest point of

Of finding the arch in the fore observation, or highest in the back, is the Latitude and Longitude at Sea.

made to touch the horizon, the quadrant will stand in a vertical plane, and the altitude above the visible horizon will be properly observed. The reason of these operations may be thus explained: The image of the sun being always kept in the axis of vision, the index will always show on the quadrant the distance between the sun and any object seen directly which its image appears to touch; therefore, as long as the index remains unmoved, the image of the sun will describe an arch everywhere equidistant from the sun in the heavens, and consequently a parallel circle about the sun as a pole. Such a translation of the sun's image can only be produced by the quadrant's being turned about upon a line drawn from the eye to the sun as an axis. A motion of rotation upon this line may be resolved into two, one upon the axis of vision, and the other upon a line on the quadrant perpendicular to the axis of vision; and consequently a proper combination of these two motions will keep the image of the sun constantly in the axis of vision, and cause both jointly to run over a parallel circle about the sun in the heavens; but when the quadrant is vertical, a line thereon perpendicular to the axis of vision becomes a vertical axis; and as a small motion of the quadrant is all that is wanted, it will never differ much in practice from a vertical axis. The observer is directed to perform two motions rather than the single one equivalent to them on a line drawn from the eye to the sun; because we are not capable, while looking towards the horizon, of judging how to turn the quadrant about upon the elevated line going to the sun as an axis, by any other means than by combining the two motions above mentioned, so as to keep the sun's image always in the proper part of the horizon-glass. When the sun is near the horizon, the line going from the eye to the sun will not be far removed from the axis of vision; and consequently the principal motion of the quadrant will be performed on the axis of vision, and the part of motion made on the vertical axis will be but small. On the contrary, when the sun is near the zenith, the line going to the sun is not far removed from a vertical line, and consequently the principal motion of the quadrant will be performed on a vertical axis, by the observer's turning himself about, and the part of the motion made on the axis of vision will be but small. In intermediate altitudes of the sun, the motions of the quadrant on the axis of vision, and on the vertical axis, will be more equally divided.

Observations taken with the quadrant are liable to errors, arising from the bending and elasticity of the index, and the resistance it meets with in turning round its centre; whence the extremity of the index, on being pushed along the arch, 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 seems to have been apprehensive that his instrument would be liable to errors from this cause; and, in order to avoid them, gives particular directions that the index be made broad at the end next the centre, and that 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 far, perhaps, as to render it nearly insensible, where the index is made strong, and the proper medium between the two extremes of a shake at the centre on 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 finding the reality of the error to which he is liable from the Latitude and Longitude at Sea.

Of finding the reality of the error to which he is liable from this cause, the observer, if he is provided with a quadrant furnished with a screw for moving the index gradually, may thus satisfy himself. After finishing the observation, lay the quadrant on a table, and note the angle; then cautiously loosen the screw which fastens the index, and it will immediately, if the quadrant is not remarkably well constructed, be seen to start from its former situation, more or less according to the perfection of the joint and the strength of the index. This starting, which is owing 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 0, 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, 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 0 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 obviated. 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 take Altitudes by the Fore Observation.

1. Of the Sun.

Turn down either of the coloured glasses before the horizon-glass, according to the brightness of the sun; 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 appear in the horizon-glass; then give the quadrant a slow vibratory motion about the axis of vision; move the index until the lower or upper limb of the sun is in contact with the horizon at the lowest part of the arch described by this motion, and the degrees and minutes shown by the index on the limb will be the altitude of the sun.

2. Of the Moon.

Put the index to 0, turn down the green glass, place the eye at the lower hole in the sight-vane, and observe the moon in the silvered part of the horizon-glass; move the index gradually, and follow the moon's reflected image until the enlightened limb is in contact with the horizon at the lower part of the arch described by the vibratory motion as before, and the index will show the altitude of the observed limb of the moon. If the observation is made in the day-time, the coloured glass is unnecessary.

3. Of a Star or Planet.

The index being put to 0, direct the sight to the star through the lower hole in the sight-vane and transparent part of the horizon-glass; move the plane of the quadrant a very 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 of the star will appear to descend; continue moving the index gradually until the star is in contact with the horizon at the lowest part of the arch described, and the degrees and minutes shown by the index on the limb will be the altitude of the star.

Put the stem of the coloured glasses into the perforation between the horizon-glasses; turn down either according to the brightness of the sun, and hold the quadrant vertically; then direct the sight through the hole in the back sight-vane, and the transparent slit in the horizon-glass to that part of the horizon which is opposite to the sun; now move the index till the sun is in the silvered part of the glass, and by giving the quadrant a vibratory motion, the axis of which is that of vision, the image of the sun will describe an arch the convex side of which is upwards; bring the limb of the sun, when in the upper part of this arch, in contact with the horizon, and the index will show the altitude of the other limb of the sun.

The altitude of the moon is observed in the same manner as that of the sun, with this difference only, that the use of the coloured glass is unnecessary unless the moon is very bright; and that the enlightened limb, whether it be the upper or lower, is to be brought in contact with the horizon.

Look directly to the star through the vane and transparent slit in the horizon-glass; move the index until the opposite horizon, with respect to the star, is seen in the silvered part of the glass, and make the contact perfect, as formerly. If the altitude of the star is known nearly, the index may be set to that altitude, the sight directed to the opposite horizon, and the observation made as before.

The observation necessary for ascertaining the latitude of a place, is that of the meridional altitude of a known celestial object; or two altitudes when the object is out of the meridian. The latitude is deduced with more certainty and with less trouble from the first of these methods than from the second; and the sun, for various reasons, is the object most proper for this purpose at sea. It, however, frequently happens that, by the interposition of clouds, the sun is obscured at noon, and by this means the meridian altitude is lost. In this case, therefore, the method by double altitudes becomes necessary. The latitude may be deduced from three altitudes of an unknown object, or from double altitudes, the apparent times of observation being given.

The altitude of the limb of an object observed at sea requires four separate corrections in order to obtain the true altitude of its centre; these are for semidiameter, dip, refraction, and parallax. (See ASTRONOMY, and the respective articles.) The first and last of these corrections vanish when the observed object is a fixed star.

When the altitude of the lower limb of any object is observed, its semidiameter is to be added thereto in order to obtain the central altitude; but if the upper limb be observed, the semidiameter is to be subtracted. The dip is to be subtracted from, or added to, the observed altitude, according as the fore or back observation is used. The refraction is always to be subtracted from, and the parallax added to, the observed altitude.

The dip in the preceding table answers to an entirely open and unobstructed horizon. It, however, frequently happens that the sun is over the land at the time of observation, and the ship nearer to the land than the visible horizon would be if unconfined. In this case, the dip will be different from what it would otherwise have been, and is to be taken from the subjoined table, in which the height is expressed at the top, and the distance from the land in the side column in nautical miles. Seamen, in general, can estimate the distance of any object from the ship with sufficient exactness for this purpose, especially when that distance is not greater than six miles, which is the greatest distance of the visible horizon from an observer on the deck of any ship.

NAVIGATION.

TABLE VI.—To reduce the Sun's Declination to any other Meridian, and to any given Time under that Meridian.

PROB. I. To reduce the sun's declination to any given meridian.

RULE. Find the number in the table answering to the longitude in the table nearest to that given, and to the nearest day of the month. Now, if the longitude is west, and the declination increasing, that is, from the 20th of March to the 22d of June, and from the 22d of September to the 22d of December, the above number is to be added to the declination; during the other part of the year, or while the declination is decreasing, this number is to be subtracted. In east longitude the contrary rule is to be applied.

Ex. 1. Required the sun's declination at noon 16th April 1836, in longitude 81° W.

Ex. 2. Required the sun's declination at noon 22d March 1836, in longitude 151° E.

PROB. II. Given the sun's meridian altitude, to find the latitude of the place of observation.

RULE. The sun's semidiameter is to be added to or subtracted from the observed altitude, according as the lower or upper limb is observed; the dip answering to the height from Table IV. or V. is to be subtracted if the fore observation is used; otherwise, it is to be added; and the refraction answering to the altitude from Table vol. IV. p. 100, art. ASTRONOMY, is to be subtracted; hence the true altitude of the sun's centre will be obtained. Call the altitude south or north, according as the sun is south or north at the time of observation, which subtracted from 90°, will give the zenith distance of a contrary denomination.

Reduce the sun's declination to the meridian of the place of observation, by Prob. I.; then the sum or difference of the zenith distance and declination, according as they are of the same or of a contrary denomination, will be the latitude of the place of observation, of the same name with the greater quantity.

Ex. 1. October 19, 1836, in longitude 32° E., the meridian altitude of the sun's lower limb was 48° 53' S.; height of the eye 18 feet. Required the latitude.

PROB. III. Given the meridian altitude of a fixed star, to find the latitude of the place of observation.

RULE. Correct the altitude of the star by dip and refraction, and find the zenith distance of the star as formerly; take the declination of the star from the Nautical Almanac, and reduce it to the time of observation. Now, the sum or difference of the zenith distance and declination of the star, according as they are of the same or of a contrary name, will be the latitude of the place of observation.

Ex. 1. December 1, 1836, the meridian altitude of Sirius was 59° 50' S., height of the eye 14 feet. Required the latitude.

PROB. IV. Given the meridian altitude of a planet, to find the latitude of the place of observation.

RULE. Compute the true altitude of the planet as directed in last problem (which is sufficiently accurate for altitudes taken at sea); take its declination from the Nautical Almanac, and reduce it to the time and meridian of the place of observation; then the sum or difference of the zenith distance and declination of the planet will be the latitude, as before.

Ex. 1. August 7, 1836, the meridian altitude of Saturn was 68° 42' N., and height of the eye 15 feet. Required the latitude.

PROB. V. Given the meridian altitude of the moon, to find the latitude of the place of observation.

RULE. Take the proportional part of the daily variation of the moon's passing the meridian at Greenwich, answering to the ship's longitude; which being applied to the time of passage given in the Nautical Almanac, will give the time of the moon's passage over the meridian of the ship.

Reduce this time to the meridian of Greenwich; and by means of the Nautical Almanac find the moon's declination, horizontal parallax, and semidiameter at the reduced time.

Apply the semidiameter, dip, and refraction to the observed altitude of the limb, and the apparent altitude of the moon's centre will be obtained; to which add the moon's parallax in altitude (found from Problem VII. of next chapter), and the sum will be the true altitude of the moon's centre; which subtracted from 90°, the remainder is the zenith distance, and the sum or difference of the zenith distance and declination, according as they are of the same or of a contrary name, will be the latitude of the place of observation.

Ex. 1. December 19, 1836, in longitude 30° W., the meridian altitude of the moon's lower limb was 81° 15' N., height of the eye 12 feet. Required the latitude.

Remark. If the object be on the meridian below the pole at the time of observation, then the sum of the true altitude and the complement of the declination is the latitude of the same name as the declination or altitude.

Ex. 1. July 2, 1836, in longitude 15° W. the altitude of the sun's lower limb at midnight was 8° 58', height of the eye 18 feet. Required the latitude.

PROB. VI. Given the latitude by account, the declination and two observed altitudes of the sun, and the interval of time between them, to find the true latitude.

RULE. To the log. secant of the latitude by account, add the log. secant of the sun's declination; the sum, rejecting 20 from the index, is the logarithm ratio. To this add the log. of the difference of the natural sines of the two altitudes, and the log. of the half elapsed time from its proper column.

Find this sum in column of middle time, and take out the time answering thereto; the difference between which and the half elapsed time will be the time from noon when the greater altitude was observed.

Take the log. answering to this time from column of rising, from which subtract the log. ratio, the remainder is the logarithm of a natural number; which being added to the natural sine of the greater altitude, the sum is the natural cosine of the meridian zenith distance; from which and the sun's declination the latitude is obtained as formerly.

If the latitude thus found differs considerably from that by account, the operation is to be repeated, using the computed latitude in place of that by account.¹

Ex. In latitude 49° 48' N. by account, the sun's declination being 9° 37' S. at 0^h 32^m P. M. per watch, the altitude of the sun's lower limb was 28° 32', and at 2^h 41^m it was 19° 25'; the height of the eye 12 feet. Required the true latitude.

As the latitude by computation differs 1° 27' from that by account, the operation must be repeated.

As this latitude differs only 5' from that used in the computation, it may therefore be depended on as the true latitude.

PROB. VII. Given the latitude by account, the sun's declination, two observed altitudes, the elapsed time, and the course and distance run between the observations; to find the ship's latitude at the time of observation of the greater altitude.

RULE. Find the angle contained between the ship's course and the sun's bearing at the time of observation of the least altitude, with which enter the Traverse Table as a course, and the difference of latitude answering to the distance made good will be the reduction of altitude.

Now, if the least altitude be observed in the forenoon, the reduction of altitude is to be applied thereto by addition or subtraction, according as the angle between the ship's course and the sun's bearing is less or more than eight points. If the least altitude be observed in the afternoon, the contrary rule is to be used.

The difference of longitude in time between the observations is to be applied to the elapsed time by addition or subtraction, according as it is east or west. This is, however, in many cases so inconsiderable as to be neglected.

With the corrected altitudes and interval, the latitude by account and sun's declination at the time of observation of the greatest altitude, the computation is to be performed by the last problem.

Remark. If the sun come very near the zenith, the sines of the altitude will vary so little as to make it uncertain which ought to be taken as that belonging to the natural sine of the meridian altitude. In this case the following method will be found preferable.

To the log. rising of the time from noon found as before, add the log. secant of half the sum of the estimated meridian altitude, and greatest observed altitude; from which subtract the log. ratio, its index being increased by 10, and the remainder will be the log. sine of an arch; which added to the greatest altitude, will give the sun's meridian altitude.

¹ This method is only an approximation, and ought to be used under certain restrictions; namely, The observations must be taken between nine o'clock in the forenoon and three in the afternoon. If both observations be 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. If one observation be in the forenoon and the other in the afternoon, the interval must not exceed four hours and a half; and, in all cases, the nearer the greater altitude is to noon the better.

If the sun's meridian zenith distance be less than the latitude, the limitations are still more contracted. 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 three hours and a half. The observations must be taken still nearer to noon if the latitude exceed the zenith distance in a greater proportion. See Mackay's Treatises on the Longitude and Navigation, &c.; Requisite Tables, 3d edit.; Mendoza Rios's Tables; Norie's and Riddle's Treatises on Navigation; &c.

Of finding CHAP. II.—METHOD OF FINDING THE LONGITUDE AT SEA
the Longitude at Sea by Lunar Observations.

SECT. I.—Introduction.

The observations necessary to determine the longitude by this method are, 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 Aquila, Fomalhaut, and α 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 xiiith—xviith 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 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 watch is not necessary; but if that object be near the meridian, the watch 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 estimate 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.—Of the Sextant.

This instrument is constructed for the express purpose of measuring with accuracy the angular distance between

the sun and moon, or between the moon and a planet or fixed star, in order to ascertain the longitude of a place by lunar observations. It is, therefore, made with more care than the quadrant, and has some additional appendages that are wanting in that instrument.

Fig. 40 represents the sextant, so framed as not to be

Fig. 40.

liable to bend. The arch AA is divided into 120 degrees; each degree is divided into three parts; each of these parts, therefore, contains twenty minutes, which are again subdivided by the vernier into every half minute or thirty seconds. The vernier is numbered at every fifth of the longer divisions, from the right towards the left, with 5, 10, 15, and 20; the first division to the right being the beginning of the scale.

In order to observe with accuracy, and make the images come precisely in contact, an adjusting screw B is added to the index, which may thereby be moved with greater accuracy than it can be by the hand; but this screw does not act until the index is fixed by the finger-screw C. Care should be taken not to force the adjusting screw when it arrives at either extremity of its adjustment. When the index is to be moved any considerable quantity, the screw C at the back of the sextant must be loosened; but when the index is brought nearly to the division required, this back screw should be tightened, and then the index may be moved gradually by the adjusting screw.

There are four tinged glasses D, each of which is set in 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 requires.

There are three more such glasses placed behind the horizon-glass at E, to weaken the rays of the sun or moon when they are 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 frame of the index-glass I is firmly fixed by a strong cock to the centre plate of the index. The horizon-glass F is fixed in a frame that turns on the axes or 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

if finding is on one side of the socket through which the stern of the finger-screw passes.

The sextant is furnished with a plain tube without any glasses; and to render the objects still more distinct, it has two telescopes, one representing the objects erect, or in their natural position: the longer one shows them inverted; it has a large field of view, and other advantages, and a little use will soon accustom the observer to the inverted position, and the instrument will be as readily managed by it as by the plain tube alone. By a telescope the contact of the images is more perfectly distinguished; and by the place of the images in the field of the telescope, it is easy to perceive whether the sextant is held in the proper place 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 thereto by two screws: by turning one or other 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 that slides in a socket, and, by means of a screw at the back of the quadrant, may be raised or lowered so as to move the centre of the telescope to point to that part of the horizon-glass which shall be judged the most fit for observation. Tinged glasses are provided to screw on the eye end of either of the telescopes or the plain tube.

Adjustments of the Sextant.

The adjustments of a sextant are, to set the mirrors perpendicular to its plane and parallel to each other when the index is at zero, and to set the axis of the telescope parallel to the plane of the instrument. The three first of these adjustments are performed nearly in the same manner as directed in the section on the quadrant; as, however, the sextant is provided with a set of coloured glasses placed behind the horizon-glass, the index error may be more accurately determined by measuring the sun's diameter twice, with the index placed alternately before and behind the beginning of the divisions; 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 0 is less or greater than the diameter measured with the index to the right of the beginning of the divisions. It will be more accurate to measure the sun's horizontal diameter, as the vertical diameter is often affected with refraction.

Adjustment IV.—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 be selected, as two stars of the first magnitude, whose distance is not less than $90^\{\backslash circ\}$ or $100^\{\backslash circ\}$; make the contact of these objects as perfect as possible at the wire nearest the plane of the instrument; fix the index in this position; move the sextant till 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; but if the objects are apparently separated, or do partly cover each other, 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.

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 division shown by the index of the limb, corrected by the index error, will be the angular distance between the wires. Other and better methods will readily occur to the observer at land.

Use of the Sextant.

When the distance between the moon and the sun, a planet or a star, is to be observed, the sextant must be held so that its plane may pass through the eye of the observer and both objects; and the reflected image of the most luminous of the two is to be brought in contact with the other seen directly. To effect this, therefore, it is evident, that when the brightest object is to the right of the other, the face of the sextant must be held upwards; but 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 by the right hand.

Sometimes a sitting posture will be found very 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 is provided with a ball and socket, and a staff, one of whose ends is attached thereto, and the other rests in a belt fastened round the body of the observer, the greater part of the weight of the instrument will by this means be supported by his body.

To observe the Distance between the Moon and any Celestial Object.

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 either 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 enlightened limb of the moon; now fasten the index, and by a gentle motion of the instrument make the image of the sun move alternately past the moon; and, when in that position where the limbs are nearest each other, make the coincidence of the limbs perfect by means of the adjusting 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.

Of finding the Longitude at Sea by Lunar Observations.

2. 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 if to the left the face is to be held from the observer; 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; fasten the index, and turn the adjusting screw till the coincidence of the star and enlightened limb of the moon is perfect; and the degrees and parts of a degree shown by 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. III.—Of the Circular Instrument of Reflection.

This instrument was proposed with a view to correct the errors to which the sextant is liable, particularly the error arising from the inaccuracy of the divisions on the limb. It consists of the following parts, a circular ring or limb, two moveable indices, two mirrors, a telescope, coloured glasses, &c.

The limb of this instrument is a complete circle of metal, and is connected with a perforated central plate by six radii; it is divided into 720 degrees, each degree being divided into three equal parts, and the division carried to minutes by means of the index scale as usual.

The two indices are moveable about the same axis, which passes exactly through the centre of the instrument:—the first index carries the central mirror, and the other the telescope and horizon-glass, each index being provided with an adjusting screw for regulating its motion, and a scale for showing the divisions on the limb.

The central mirror is placed on the first index, immediately above the centre of the instrument, and its plane makes an angle of about 30° with the middle line of the index. The four screws in its pedestal for making its plane perpendicular to that of the instrument have square heads, and are therefore easily turned either way by a key for that purpose.

The horizon-glass is placed on the second index near the limb, so that as few as possible may be intercepted of the rays proceeding from the reflected object when to the left. The perpendicular position of this glass is rectified in the same manner as that of the horizon-glass of a sextant, to which it is similar. It has another motion, whereby its plane may be disposed so as to make a proper angle with the axis of the telescope, and a line joining its centre and that of the central mirror.

The telescope is attached to the other end of the index. It is an achromatic astronomical one, and therefore inverts objects; it has two parallel wires in the common focus of the glasses, whose angular distance is between two and three degrees, and which, at the time of observation, must be placed parallel to the plane of the instrument. This is easily done, by making the mark on the eye-piece coincide with that on the tube. The telescope is moveable

by two screws in a vertical direction with regard to the plane of the instrument, but is not capable of receiving lateral motion.

There are two sets of coloured glasses, each set containing four, and differing in shade from each other. The glasses of the larger set, which belongs to the central mirror, should have each about half the degree of shade with which the correspondent glass of the set belonging to the horizon-mirror is tinged. These glasses are kept tight in their places by small pressing screws, and make an angle of about 85° with the plane of the instrument, by which means the image from the coloured glass is not reflected to the telescope. When the angle to be measured is between 5° and 34°, one of the glasses of the largest set is to be placed before the horizon-glass.

The handle is of wood, and is screwed to the back of the instrument, immediately under the centre, with which it is to be held at the time of observation.

Fig. 41 is a plan of the instrument, wherein the limb is represented by the divided circular plate; A is the central

Fig. 41.

mirror; aa, the places which receive the stems aa of the glass; EF the first or central index, with its scale and adjusting screw; MN the second or horizon-index; GH the telescope; IK the screws for moving it towards or from the plane of the instrument; C the plane of the coloured glass; and D its place in certain observations.

Fig. 42 is a section of the instrument, wherein the se-

Fig. 42.

veral parts are referred to by the same letters as in fig. 41. The circular reflecting instrument was greatly improved by Messrs Mendoza Rios, Troughton, Dollond, &c.

Adjustments of the Circular Instrument.

1. To set the horizon-glass so that none of the rays from the central mirror shall be reflected to the telescope from the horizon-mirror, without passing through the coloured glass belonging to this last mirror.—Place the coloured glass before the horizon-mirror; direct the telescope to the silvered part of that mirror, and make it nearly parallel to the plane of the instrument; move the first index; and if the rays from the central mirror to the horizon-glass, and from thence to the telescope, have all the same degree of shade with that of the coloured glass used, the horizon-glass is in its proper position; otherwise, the pedestal of the glass must be turned until the uncoloured images disappear.

Of finding the Longitude at Sea by Lunar Observations.

II. Place the two adjusting tools on the limb, about 350° of the instrument distant, one on each side of the division on the left, answering to the plane of the central mirror produced; then, the eye being placed at the upper edge of the nearest tool, move the central index till one half only of the reflected image of this tool is seen in the central mirror towards the left, and move the other tool till its half to the right is hid by the same edge of the mirror; then, if the upper edges of both tools are apparently in the same straight line, the central mirror is perpendicular to the plane of the instrument; if not, bring them into this position by the screws in the pedestal of the mirror.

III. To set the horizon mirror perpendicular to the plane of the instrument.—The central mirror being previously adjusted, direct the sight through the telescope to any well-defined distant object; then, if, by moving the central index, the reflected image passes exactly over the direct object, the mirror is perpendicular; if not, its position must be rectified by means of the screws in the pedestal of the glass.

A planet, or star of the first magnitude, will be found a very proper object for this purpose.

IV. To make the line of collimation parallel to the plane of the instrument.—Lay the instrument horizontally on a table; place the two adjusting tools on the limb, towards the extremities of one of the diameters of the instrument; and at about fifteen or twenty feet distant let a well-defined mark be placed, so as to be in the same straight line with the tops of the tools; then raise or lower the telescope till the plane, passing through its axis and the tops of the tools, is parallel to the plane of the instrument, and direct it to the fixed object; turn either or both of the screws of the telescope till the mark is apparently in the middle between the wires; then is the telescope adjusted; and the difference, if any, between the divisions pointed out by the indices of the screws will be the error of the indices. Hence this adjustment may in future be easily made.

In this process, the eye-tube must be so placed as to obtain distinct vision.

V. To find that division to which the second index being placed, the mirrors will be parallel, the central index being at zero.—Having placed the first index exactly to 0, direct the telescope to the horizon-mirror, so that its field may be bisected by the line of separation of the silvered and transparent parts of that mirror; hold the instrument vertically, and move the second index until the direct and reflected horizons agree; and the division shown by the index will be that required.

This adjustment may be performed by measuring the sun's diameter in contrary directions, or by making the reflected and direct images of a star or planet to coincide.

Use of the Circular Instrument.

To observe the Distance between the Sun and Moon.

I. The sun being to the right of the moon.

Set a proper coloured glass before the central mirror if the distance between the objects is less than 35°; but if above that quantity, place a coloured glass before the horizon-mirror; make the mirrors parallel, the first index being at 0, and hold the instrument so that its plane may be directed to the objects, with its face downwards, or from the observer; direct the sight through the telescope to the moon; move the second index, according to the order of the divisions on the limb, till the nearest limbs of the sun and moon are almost in contact; fasten that index, and make the coincidence of the limbs perfect by the adjusting screw belonging thereto; then invert the instrument, and move the central index towards the second by a quantity equal to twice the arch passed over by that index; direct the plane of the instrument to the objects;

look directly to the moon, and the sun will be seen in the field of the telescope; fasten the central index, and make the contact of the same two limbs exact by means of the adjusting screw: Then half the angle shown by the central index will be the distance between the nearest limbs of the sun and moon.

II. The sun being to the left of the moon.

Hold the instrument with its face upwards, so that its plane may pass through both objects; direct the telescope to the moon, and make its limb coincide with the nearest limb of the sun's reflected image, by moving the second index; now put the instrument in an opposite position; direct its plane to the objects, and the sight to the moon, the central index being previously moved towards the second by a quantity equal to twice the measured distance; and make the same two limbs that were before observed coincide exactly, by turning the adjusting screw of the first index; then half the angle shown by the first index will be the angular distance between the observed limbs of the sun and moon.

To observe the Angular Distance between the Moon and a Fixed Star or Planet.

I. The star being to the right of the moon.

In this case the star is to be considered as the direct object; and the enlightened limb of the moon's reflected image is to be brought in contact with the star or planet, both by a direct and inverted position of the instrument, exactly in the same manner as described in the last article. If the moon's image is very bright, the lightest tinged glass is to be used.

II. The star being to the left of the moon.

Proceed in the same manner as directed for observing the distance between the sun and moon, the sun being to the right of the moon, using the lightest tinged glass if necessary.

SECT. IV.—Of the Method of determining the Longitude from Observation.

PROB. I. To convert degrees or parts of the equator into time.

RULE. Multiply the degrees and parts of a degree by 4, beginning at the lowest denomination, and the product will be the corresponding time; observing that minutes multiplied by 4 produce seconds of time, and degrees multiplied by 4 give minutes.

Ex. Let 26° 45' be reduced to time.

1^h 47^m 0^s = time required.

PROB. II. To convert time into degrees.

RULE. Multiply the given time by 10, to which add the half of the product. The sum will be the corresponding degrees.

Ex. Let 3^h 4^m 25^s be reduced to degrees.

Corresponding deg. = 46 7 0

PROB. III. Given the time under any known meridian, to find the corresponding time at Greenwich.

RULE. Let the given time be reckoned from the preceding noon, to which the longitude of the place in time is to be applied by addition or subtraction, according as it

Of finding is east or west; and the sum or difference will be the longitude responding time at Greenwich.

Ex. What time at Greenwich answers to 6^h 15^m at a ship in longitude 76° 45' W.?

PROB. IV. To reduce the time at Greenwich to that under any given meridian.

RULE. Reckon the given time from the preceding noon, to which add the longitude in time if east, but subtract it if west; and the sum or remainder will be the corresponding time under the given meridian.

Ex. What is the expected time of the beginning of the lunar eclipse of February 25, 1793, at a ship in longitude 109° 48' E.?

PROB. V. Given the latitude of a place, the altitude and declination of the sun, to find the apparent time, and the error of the watch.

RULE. If the latitude and declination are of different names, let their sum be taken; otherwise, their difference. From the natural cosine of this sum or difference, subtract the natural sine of the corrected altitude, and find the logarithm of the remainder; to which add the log. secants of the latitude and declination: the sum will be the log. rising of the horary distance of the object from the meridian, and hence the apparent time will be known. The equation of time being then applied as directed in the Nautical Almanac, the mean time of observation is obtained.

Example. September 15, 1792, in latitude 33° 56' S. and longitude 18° 22' E., the mean of the times per watch was 8^h 12^m 10^s A. M., and that of the altitudes of the sun's lower limb 24° 48'; height of the eye 24 feet. Required the error of the watch.

PROB. VI. Given the latitude of a place, and the altitude of a known fixed star, to find the mean time of observation and error of the watch.

RULE. Correct the observed altitude of the star, and reduce its right ascension and declination to the time of observation.

With the latitude of the place, the true altitude and declination of the star, compute its horary distance from the meridian by last problem; which being added to or subtracted from its right ascension, according as it was observed in the western or eastern hemisphere, the sum or remainder will be the right ascension of the meridian.

From the right ascension of the meridian subtract the sidereal time of mean noon, reduced to the meridian of the place of observation, and obtained from the Nautical Almanac, the remainder will be the approximate time of observation; from which subtract the reduction of sidereal to mean time, answering thereto, the result will be the mean time of observation; and hence the error of the watch will be known.

Ex. December 12, 1836, in latitude 37° 46' N., and longitude 21° 15' E., the altitude of Arcturus east of the meridian was 34° 6' 4", the height of the eye 10 feet. Required the apparent time of observation.

PROB. VII. Given the apparent altitude of the sun, moon, a planet or star, to find the true altitude.

The altitude obtained from observation corrected for dip and for the semidiameter of the body, if the limb has been observed, is the apparent altitude of the centre.

From the apparent altitude subtract the refraction (Table vol. IV. p. 100, art. ASTRONOMY); and to the remainder add the parallax in altitude; the sum is the true altitude required.

The parallax in altitude is found by adding to the log. cosine of the apparent altitude corrected for refraction, the logarithm of the horizontal parallax; the sum is the logarithm of the parallax in altitude.

The horizontal parallax of the object observed will be found in the Nautical Almanac. The horizontal parallax of the sun may always be supposed 9". The parallax of a star is insensible.

Ex. The observed altitude of the moon's upper limb was observed 32° 17', the height of the eye being eighteen feet, the moon's horizontal parallax being 57' 58", and semidiameter 16'. Required the apparent and true altitudes.

PROB. VIII. Given the apparent distance between the moon and the sun or a fixed star or planet, and the apparent and true altitudes of these bodies, to find the true distance.

Example. Let the apparent altitude of the moon's centre be 48° 22', that of the sun's 27° 43', the apparent central distance 81° 23' 40", and the moon's horizontal parallax 58' 45". Required the true distance.

This is nearly the method of Borda, similar to Problem IV. and XX.

PROB. IX. To find the time at Greenwich answering to a given distance between the moon and the sun, or one of the stars or planets used in the Nautical Almanac.

RULE. If the given distance is found in the Nautical Almanac opposite to the given day of the month, or to that which immediately precedes or follows it, the time is found at the top of the page. But if this distance is not found exactly in the ephemeris, subtract the prop. log. of the difference between the distances which immediately precede and follow the given distance (which prop. log. is given in the Almanac) from the prop. log. of the difference between the given and preceding distances; the remainder will be the prop. log. of the excess of the time corresponding to the given distance, above that answering to the preceding distance; and hence the mean time at Greenwich is known.

Example. September 19, 1836, the true distance between the centres of the sun and moon was 110° 3' 5". Required the mean time at Greenwich.

RULE. To the logarithmic difference answering to the moon's apparent altitude and horizontal parallax, add the logarithmic sines of half the sum, and half the difference of the apparent distance and difference of the apparent altitudes; half the sum will be the logarithmic cosine of an arch: now add the logarithmic sines of the sum and difference of this arch, and half the difference of the true altitudes, and half the sum will be the logarithmic cosine of half the true distance.

Remarks. The apparent distance is found by adding to or subtracting from the observed distance of the limbs the sum of the semidiameters of the bodies (the moon's being increased by the augmentation for her altitude), according as the nearer or more remote limbs have been observed. If a star or the centre of a planet has been observed, no allowance is to be made for the semidiameter of the object. In computing the true altitude from the apparent by the preceding problem, the refraction and parallax should be taken to seconds.

XIX. PRACTICAL ASTRONOMY, which may be consulted,

PROB. X. The latitude of a place and its longitude by account being given, together with the distance between, and the altitude of the moon and the sun, or one of the stars or planets in the Nautical Almanac; to find the true longitude of the place of observation.

RULE. Reduce the estimate time of observation to the meridian of Greenwich by Problem III., and to this time take, from the Nautical Almanac, the moon's horizontal

Of finding the longitude at sea by Lunar Observations. Increase the semidiameter by the augmentation answering to the moon's altitude. Find the apparent and true altitudes of each object's centre, and the apparent central distance; with which compute the true distance by Problem VIII., and find the mean time at Greenwich answering thereto by the last problem.

If the sun or star be at a proper distance from the me-

ridian at the time of observation of the distance, compute the mean time at the ship. If not, the error of the watch may be found from observations taken either before or after that of the distance.

The difference between the mean times of observation at the ship and Greenwich will be the longitude of the ship in time, which is east or west according as the time at the ship is later or earlier than the Greenwich time.

Ex. 1. On the 8th of April 1835, at 2^h 21^m 28^s P. M. mean time, in latitude 21° 33' N. longitude 155° 44' E. by account, the distance between the sun and moon's nearest limbs was observed to be 111° 16' 7", the observed altitude of the sun's lower limb was 53° 8' 53", that of the moon's lower limb was 14° 38' 32"; the height of the eye 20 feet; the barometer being at 29.20 inches, and Fahrenheit's thermometer at 75°. Required the true longitude.

Remark. In the preceding example, the true distance and mean time at the ship have been computed by methods different from those before given, which would, however, have given the same results.

CHAP. 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 on 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 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 stopped, either 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.

PROB. To find the longitude of a ship at sea by a chronometer.

Let several altitudes of the sun, or of any fixed star or planet, be observed, and find the true mean altitude; with which, the ship's latitude, and object's declination, compute the mean time of observation.

To the mean of the times of observation, as shown by the chronometer, apply its 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, by observation, is later or earlier than the Greenwich time.

Ex. May 19, 1804, in latitude 42° 15' N., five altitudes of the sun's lower limb were observed in the afternoon, the mean being 43° 45', and the mean of the times of observation, as given by a chronometer, 7^h 0^m 56^s, the chronometer's error having been settled at the Royal Observatory at Greenwich, March 16, at noon, 1^m 18^s fast for mean time, and daily gain 7^s 83^s, height of the eye twenty-six feet. Required the longitude of the place of observation.

The true altitude of the sun's centre is found to be 43° 55', with which, the latitude, and sun's declination 19° 51' N., the sun's meridian distance is found to be 3^h 12^m 34^s, and the equator of time being 5^m 51^s subtractive, the mean time at the place of observation is 3^h 8^m 43^s.

For various other methods of determining the longitude of a place, the reader is referred to Mackay's Treatise on the Longitude, Inman's, Riddle's, and Norie's Treatises on Navigation; Mendoza Rios's and Thomson's Tables, &c.

CHAP. IV.—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 articles AZIMUTH, COMPASS, and VARIATION; and for the method of communicating magnetism to compass needles, see MAGNETISM.

PROB. I. Given the latitude of a place, and the sun's magnetic amplitude, to find the variation of the compass.

RULE. To the log. secant of the latitude add the log. sine of the sun's declination, the sun will be the log. cosine of the true amplitude; to be reckoned from the north or south, according as the declination is north or south.

The difference between the true and observed amplitudes, reckoned from the same point, and if of the same name, is the variation; but if of a different name, their sum is the variation.

If the observation be made before noon, the variation will be east or west, according as the observed amplitude is nearer to or more remote from the north than the true amplitude. The contrary rule holds good in observations taken after noon.

A journal is a regular and exact register of all the various transactions that happen aboard a ship, whether at sea or land, and more particularly that which concerns a ship's way, from whence her place at noon or any other time may be justly ascertained.

That part of the account which is kept at sea is called sea work; and the remarks taken down while the ship is in port are called harbour work.

At sea, the day begins at noon, and ends at the noon of the following day; the first twelve hours, or those contained between noon and midnight, are denoted by P. M., signifying after mid-day; and the other twelve hours, or those from midnight to noon, are denoted by A. M., signifying before mid-day. A day's work marked Wednesday, March 6, began on Tuesday at noon, and ended on Wednesday at noon. The days of the week are usually represented by astronomical characters. Thus ☉ represents Sunday; ☽, Monday; ☾, Tuesday; ☿, Wednesday; ♃, Thursday; ♄, Friday; and ♅, Saturday.

When a ship is bound to a port so situated that she will be out of sight of land, the bearing and distance of the port must be found. This may be done by Mercator's or Middle Latitude Sailing; but the most expeditious method is 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 course per compass, by allowing the variation to the right or left of the true course, according as it is west or east.

At the time of leaving the land, the bearing of some known place is to be observed, and its distance is usually found by estimation. As perhaps the distance thus found will be liable to some error, particularly in hazy or foggy weather, or when that distance is considerable, it will therefore be proper to use the following method for this purpose.

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 to be again observed. Now, having one side of a plain triangle, namely, the distance sailed, and all the angles, the other distances may be found by Prob. I. of Oblique Sailing.

The method of finding the course and distance sailed in a given time is by the compass, the log-line, and half-minute glass. These have been already described. In the royal navy, and in ships in the service of the East India Company, the log is hove once every hour; but in most other trading vessels only every two hours.

The several courses and distances sailed in the course of twenty-four hours, or between noon and noon, and whatever remarks are thought worthy of notice, are set down with chalk on a board painted black, called the log-board, which is usually divided into six columns; the first column on the left hand contains the hours from noon to noon; the second and third the knots and parts of a knot sailed every hour, or every two hours, according as the log is marked; the fourth column contains the courses steered; the fifth, the winds; and in the sixth the various remarks and phenomena are written. The log-board is transcribed every day at noon into the log-book, which is ruled and divided after the same manner.

The courses steered must be corrected by the variation of the compass and leeway. If the variation is west, it must be allowed to the left hand of the course steered; but if east, to the right hand, in order to obtain the true course. The leeway is to be allowed to the right or left of the course steered, according as the ship is on the larboard or starboard tack. The method of finding the va-

Ship's Journal. riation, which should be determined daily if possible, is given in the preceding chapter; and the leeway may be understood from what follows.

When a ship is close hauled, that part of the wind which acts upon the hull and rigging, together with a considerable part of the force which is exerted on the sails, tends to drive her to the leeward. But since the bow of a ship exposes less surface to the water than her side, the resistance will be less in the first case than in the second; the velocity in the direction of her head will therefore in most cases be greater than the velocity in the direction of her side; and the ship's real course will be between the two directions. The angle formed between the line of her apparent course and the line she really describes through the water is called the angle of leeway, or simply the leeway.

There are many circumstances which prevent the laying down rules for the allowance of leeway. The construction of different vessels, their trim with regard to the nature and quantity of their cargo, the position and magnitude of the sail set, and the velocity of the ship, together with the swell of the sea, are all susceptible of great variation, and very much affect the leeway. The following rules, are, however, usually given for this purpose.

1. 1. When a ship is close hauled, has all her sails set, the water smooth, with a light breeze of wind, she is then supposed to make little or no leeway.
2. 2. Allow one point when the top-gallant sails are hauled.
3. 3. Allow two points when under close reefed top-sails.
4. 4. Allow two points and a half when one top-sail is hauled.
5. 5. Allow three points and a half when both top-sails are hauled.
6. 6. Allow four points when the fore course is hauled.
7. 7. Allow five points when under the main-sail only.
8. 8. Allow six points when under balanced mizen.
9. 9. Allow seven points when under bare poles.

These allowances may be of some use to work up the day's work of a journal which has been neglected; but a prudent navigator will never be guilty of this neglect. 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 taffarel, with its diameter at right angles to the ship's length, and dividing its circumference into points and quarters. The angle contained between the semidiameter which points right aft, and that which points in the direction of the wake, is the leeway. But the best and most rational way of bringing the leeway into the day's log is to have a compass or semicircle on the taffarel, 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, which, as a necessary article, ought to be entered into a separate column against the hourly distance on the log-board.

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 which the vessel would otherwise acquire, to the endangering her masts, and straining her timbers, &c. When a ship is brought to, the tiller is put close over to the leeward, which brings her head round to the wind. The wind having then very little power on the sails, the ship loses her way through the water; which ceasing to act on the rudder, 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 vibratory motion, coming up to the wind and falling off from it again alternately. Now the middle point between those upon which she comes up and falls off is taken for her apparent course; and the leeway and variation is to be allowed from thence, to find the true course.

The setting and drift of currents, and the heave of the sea, are to be marked down. These are to be corrected by variation only.

The computation made from the several courses, corrected as above, and their corresponding distances, is called a day's work; and the ship's place, as deduced therefrom, is called her place by account, or dead reckoning.

It is almost constantly found that the latitude by account does not agree with that by observation. From an attentive consideration of the nature and form of the common log, that its place is alterable by the weight of the line, by currents, and other causes, and also the errors to which the course is liable, from the very often wrong position of the compass in the binnacle, the variation not being well ascertained, an exact agreement of the latitudes cannot be expected.

When the difference of longitude is to be found by dead reckoning, if then 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 be 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 wrong. This last correction will, according to different authors, place the ship upon opposite sides of her meridian by account. As these corrections are, therefore, no better than guessing, they should be absolutely rejected.

If the latitudes are not found to agree, the navigator ought to 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 on 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 is not to receive any farther alteration.

It will be proper, however, for the navigator to determine the longitude of the ship from 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.

General Rules for working a Day's Work.

Correct the several courses for variation and leeway; place them, and the corresponding distances, in a table prepared for that purpose. From whence, by Traverse Sailing, find the difference of latitude and departure made good; hence the corresponding course and distance, and the ship's present latitude, will be known.

Find the middle latitude at the top or bottom of the traverse table, and the distance, answering to the departure found in a latitude column, will be the difference of longitude; or, the departure answering to the course made good, and the meridional difference of latitude in a latitude column, is the difference of longitude; the sum or difference of which, and the longitude left, according as they are of the same or of a contrary name, will be the ship's present longitude of the same name with the greater.

Compute the difference of latitude between the ship and the intended port, or any other place whose bearing and distance may be required: find also the meridional difference of latitude and the difference of longitude. Now the course answering the meridional difference of latitude found in a latitude column, and the difference of longitude in a departure column, will be the bearing of the place, and the distance answering to the difference of latitude will be the distance of the ship from the proposed place. If these numbers exceed the limits of the table, it will be necessary to take aliquot parts of them; and the distance is to be multiplied by the number by which the difference of latitude is divided.

It will sometimes be necessary to keep an account of the meridian distance, especially in the Baltic or Mediterranean trade, where charts are used in which the longitude is not marked. The meridian distance on the first day is that day's departure; and any other day it is equal to the

sum or difference of the preceding day's meridian distance and the day's departure, according as they are of the same or of a contrary denomination.

It will be found very satisfactory to lay down the ship's place on a chart at the noon of each day, and her situation with respect to the place bound to and the nearest land will be obvious. The bearing and distance of the intended or any other port, and other requisites, may be easily found by the chart, as already explained; and indeed every day's work may be performed on the chart, and thus the use of tables superseded.

Specimens of a ship's journal may be found in the works on navigation already mentioned. The reader is also referred to these works for the traverse table, and tables of meridional parts, log. rising, middle time, half elapsed time, logarithmic difference, &c. which have been employed in the solution of the various problems.

END OF VOLUME FIFTEENTH.

the difference of latitude between the ship and the place where she may place whose bearing and distance may be required; and also the meridional difference of latitude, and the difference of longitude. Now the course according to the meridional difference of latitude, found in a latitude column, and the difference of longitude in a longitude column, will be the bearing of the place, and the distance according to the difference of latitude will be the distance of the ship from the proposed place. If these numbers exceed the limits of the table, it will be necessary to take separate parts of them, and the distance is to be multiplied by the number by which the difference of latitude is divided.

It will sometimes be necessary to keep an account of the meridian distance, especially in the Indian or Mediterranean seas, where the ship may be in which the longitude is not required. The meridian distance on the first day is the difference of latitude, and on any other day it is equal to the

sum of differences of the preceding day's meridian distance and the day's departure, according as they are of the same or of a contrary denomination.

It will be a very satisfactory to find down the ship's place on a chart at the noon of each day, and her situation with respect to the place bound to and the nearest land. The bearing and distance of the intended port, and other remarks, may be easily as already explained, and indeed every part of the work may be performed on the chart, and thus the

Specimens of a navigation are furnished to these meridional parts, logarithmic differences in the solution of the

ship's journal may be found in the works mentioned. The reader is also referred to the traverse table, and tables of time, middle time, half elapsed time, which have been employed in the solution of the problems.

END OF VOLUME SIXTEENTH.

Fig. 27.

Fig. 28.

Fig. 29.

Fig. 30.

Fig. 31.

Fig. 32.

Fig. 33.

Fig. 34.

Fig. 35.

Fig. 36.

Fig. 37.

Fig. 38.

Fig. 39.

Fig. 40.

Fig. 41.

Fig. 42.

Fig. 43.

Fig. 44.

Fig. 45.

Fig. 46.

PINGAL'S CAVE, STALPEA

FINGAL'S CAVE, STAFFA.

FINGAL'S CAVE, STAFFA.

FINGAL'S CAVE, STAFFA.

This plate contains 13 numbered illustrations of mollusks:

• 1. Helix hirsuta: A large, detailed drawing of a snail with a prominent, ribbed shell and a long, segmented body.
• 2. Helix septa: A small, circular whorl showing concentric growth rings.
• 3, 5. Auricula: A large, elongated whorl with a distinct siphon.
• 4. Helix septa: A small, circular whorl with radial ribs.
• 6, 8. Mya: A large, elongated whorl with a distinct siphon.
• 7. Aspicula rotunda: A small, circular whorl with radial ribs.
• 9. Aspicula rotunda: A small, circular whorl with radial ribs.
• 10. Tridacna Mangu: A long, slender, worm-like organism.
• 11. Tridacna hibernica: A long, slender, worm-like organism.
• 12. Limnaea odora: A small, circular whorl with radial ribs.
• 13. Sagittaria truncata: A small, circular whorl with radial ribs.
• 14. Strombus: A small, circular whorl with radial ribs.
• 15. Strombus: A small, circular whorl with radial ribs.
• 16. Strombus: A small, circular whorl with radial ribs.
• 17. Strombus: A small, circular whorl with radial ribs.
• 18. Strombus: A small, circular whorl with radial ribs.

1.

2.

3.

5.

4.

7.

5.

6.

8.

6.

8.

9.

10.

Ja-te a ma-no - - ho i tahu-a-ta auch - - - - - ta ma-hu - - ma ch, &c.

Tutti. Soli. Tutti.

Rhythm of 2 measures.

Half Cadence.

Repetition of the same rhythm.

Half Cadence.

Rhythm of 2 measures.

the same.

the same.

Half Cadence.

Half Cadence.

Half Cadence.

the same.

Rhythm of 4 measures.

Half Cad.

Interrupted Cad.

Rhythm of 2 measures.

the same.

Perfect Cad.

5 measures.

the same.

Half Cad.

Half Cad.

Rhythm of 2 measures.

the same.

Half Cad.

Half Cad.

4 measures.

end of the period.

added period.

Rhythm of 2 measures.

Perfect Cad.

the same.

Rhythm of 2 measures.

the same.

Interrupted Cad.

Interrupted Cad.

Perfect Cad.

The musical score illustrates various rhythmic patterns and cadences. The notation is in 2/4 time with a key signature of one flat (B-flat). The score is divided into several staves, each demonstrating a specific rhythmic concept:

• Staff 1: Shows a "Rhythm of 5 measures." The melody consists of a series of eighth and sixteenth notes. It ends with a "Half Cadence."
• Staff 2: Shows a "Repetition of the same rhythm." The melody is identical to the first staff, ending with a "Half Cadence."
• Staff 3: Shows a "Rhythm of 2 measures." The melody consists of quarter notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 4: Shows a "Rhythm of 4 measures." The melody consists of quarter notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 5: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 6: Shows a "Rhythm of 5 measures." The melody consists of quarter notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 7: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 8: Shows a "Rhythm of 4 measures." The melody consists of quarter notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 9: Shows a "Rhythm of 3 measures." The melody consists of quarter notes. It ends with a "Perfect Cadence." The next two measures are labeled "end of the period." and "added period." and end with another "Perfect Cadence."
• Staff 10: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 11: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 12: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 13: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 14: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 15: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 16: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 17: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 18: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 19: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 20: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 21: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 22: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 23: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 24: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 25: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 26: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 27: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 28: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 29: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 30: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 31: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 32: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 33: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 34: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 35: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 36: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 37: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 38: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 39: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 40: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 41: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 42: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 43: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 44: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 45: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 46: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 47: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 48: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 49: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 50: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 51: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 52: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 53: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 54: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 55: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 56: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 57: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 58: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 59: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 60: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 61: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 62: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 63: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 64: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 65: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 66: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 67: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 68: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 69: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 70: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 71: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 72: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 73: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 74: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 75: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 76: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 77: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 78: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 79: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 80: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 81: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 82: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 83: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 84: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 85: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 86: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 87: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 88: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 89: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 90: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 91: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 92: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 93: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 94: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 95: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 96: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 97: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 98: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 99: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 100: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 101: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 102: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 103: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 104: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 105: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 106: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 107: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 108: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 109: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 110: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 111: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 112: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 113: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 114: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 115: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 116: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 117: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 118: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 119: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 120: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 121: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 122: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 123: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 124: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 125: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 126: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 127: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 128: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 129: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 130: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 131: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 132: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 133: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 134: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 135: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 136: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 137: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 138: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 139: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 140: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 141: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 142: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 143: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 144: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 145: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 146: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 147: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 148: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 149: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 150: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 151: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 152: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 153: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 154: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 155: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 156: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 157: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 158: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 159: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 160: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 161: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 162: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 163: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 164: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 165: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 166: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 167: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 168: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 169: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 170: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 171: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 172: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 173: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 174: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 175: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 176: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 177: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 178: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 179: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 180: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 181: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 182: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 183: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 184: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 185: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 186: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 187: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 188: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 189: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 190: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 191: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 192: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 193: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 194: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 195: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 196: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 197: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 198: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 199: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 200: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 201: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 202: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 203: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• Staff 204: Shows a "Rhythm of 2 measures." The melody consists of eighth notes. It ends with a "Half Cadence." The next two measures are labeled "the same." and end with another "Half Cadence."
• <

A.

B.

C. N^o 23. N^o 24. N^o 25.

D. N^o 27.

N^o 28A. Haydn. N^o 29. N^o 30A. Haydn. N^o 30B.

Mozart. N^o 32. N^o 33. N^o 34.

N^o 35. N^o 37. Spontini.

N^o 38A. Mozart. N^o 39. Perglezi. N^o 40.

N^o 42.

N^o 43. N^o 44.

TELEMAZIO DI HARMONIE

1747. N° 49. Beethoven. N° 49.

1748.

1749. N° 50. J. S. Bach. N° 50.

1750. N° 51. Mozart.

1751. Mozart. N° 52. Haydn. N° 53.

1752. Haydn. N° 54. C. P. E. Bach. N° 55. D. F. N° 56. D. F.

1753. D. F. N° 57. Beethoven. N° 58. Haydn. N° 59. S. M. Lasso. Haydn.

1754. N° 60. Haydn. N° 61. Adagio. Beethoven. N° 62. Cherubini.

1755. Adagio. C. P. E. Bach. O. Purcell.

1756. Opus. 1244.

1757.

No. 47. No. 48. Beethoven. No. 49.
No. 50.
No. 51. J. S. Bach. No. 52.
No. 53. Mozart.
No. 54. Mozart. No. 55. Haydn. No. 56.
No. 57. Haydn. No. 58. C. P. E. Bach. No. 59. D^♭. No. 60. D^♭.
No. 61. D^♭. No. 62. Beethoven. No. 63.
No. 64. S^♭ kassa. Haydn.
No. 65. No. 66. Haydn. No. 67. Adagio. Beethoven. No. 68. Cherubini.
No. 69. Adagio. C. P. E. Bach. No. 70. Purcell.
No. 71. Certon. 1544.