The New American Practical Navigator; Being an Epitome of Navigation; Containing All the Tables Necessary to be Used with the Nauticl Almanac

or other object marked on the chart, so that any error which might arise in the course of the boat may be prevented. While proceeding in this direction, let one person take the soundings, while another observes, with a quadrant or sextant, the angular elevation of the top of the boat's mast above the horizontal line drawn from the eye of the observer, and a third person notes the observations in the minute-book, and the time of observation, in order to make the necessary reduction in the soundings, to reduce them to low water. Proceed in this manner from the sail-boat, till you get off the bank into deep water, or till the elevation of the mast is not much less than one degree; then row across the bank till the bearing of the mast is altered considerably, or till it appears in a range with another point of land, at a considerable angular distance from the point with which the mast ranged in the first observations; then row towards the boat, sounding and observing the angular elevation of the mast as before. Proceed in this manner, in sounding to and from the sail-boat, till you have procured a sufficient number of soundings in every direction. Then go on board the sail-boat, and shift her birth to another part of the bank, where soundings have not been taken, and proceed to sound as before. Continue sounding and shifting the situation of the boat, till the whole bank has been explored, and then the observations may be plotted off by the directions in the following example.

ADB

Let ABC (Plate VIII. fig. 1) be the mast of the sail-boat; D the situation of the eye of the person who observes the angular elevation of the mast. Draw the line BD parallel to the horizon, and join AD. Then the height AB must be measured* accurately, and, that being given and the observed angle ADB, the corresponding distance BD may be obtained by the usual rules of trigonometry, by saying, As radius: AB:: cotangent ADB: BD. Thus, if the height AB be 30 feet, and the angle ADB 1°, the distance BD will be 1719 feet (being 57.3 times as great as AB). The distances corresponding to 2o, 3o, &c., are given in the adjoined table, by examining which it will appear that the distance BD corresponding to any angle ADB (less than 30°) may be obtained nearly by dividing 1719 by the angle ADB in degrees. Thus, for 4 degrees, by this rule, the distance would be 1719 4293 nearly, as in the table. The greatest difference between the distances determined by the rule and by the table is 5 feet, corresponding to the angle 30°; for 171957, whereas by the table the distance is 52. In taking soundings by this method, it will be very rarely necessary to measure an angle so great as 30°; so that, for all practical purposes, the distance may be determined, in this example, to a sufficient degree of accuracy, by dividing 1719 by the observed angular elevation in degrees. On these principles we have the following rule for calculating the distance, corresponding to a mast of any given height, and to any observed angular elevation.

FEET.

1°

1719

859

572

429

343

170

2345028

RULE. Multiply the height of the mast above the eye of the observer by 57.3, and the product will be a constant quantity,† which, being divided by the observed angle of elevation, expressed in degrees and decimals of a degree, the quotient will be the sought distance nearly.

If the height of the mast be expressed in equal parts, taken from the scale by which the chart is plotted off, the distances found by the above rule will be expressed in the same equal parts; so that, if the distances thus expressed, corresponding to 1o, 2o, 3o, &c., be calculated and marked on a slip of paper (Plate VIII. fig. 2) from H to 1°, from H to 2°, and from H to 3°, &c., respectively, the slip H 1, thus marked, will be a very convenient scale for plotting off such distances.

For further illustration of this method, we have given an example in Plate VIII. fig. 4, in which C represents the place where the sail-boat is at anchor; A and B the

A mark may be made at B, and a vane placed at the top of the mast at A, to enable the observer to distinguish those objects when at a great distance. If the height of the observer above the horizon be small in comparison with the height of the mast, the angular distance ADE between the surface of the sea, near the boat, and the top of the boat's mast may be measured, instead of ADB; for, if the distances BC and CE remain the same in all observations, it will be immaterial which angle is measured; observing, however, that different scales must be used for plotting off the angles ADB and ADE. If AB represent the known vertical height of the summit of an island above the eye of an observer the distance from the island can be determined by measuring the angular elevation ÅDB, as is eviden from what has been said above.

This constant quantity may be determined without actually measuring the altitude AB, if the angular elevation can be measured at a place D, where the distance BD is known. Thus, in the example (Plate VIII. fig. 4), the distance AC being known, and the angular elevation of the mast at C being observed at Å in degrees and decimals of a degree, and multiplied by the distance AC, the product will be the constant quantity mentioned in the rule. This method may be used in determining the distance from an island by the method mentioned in the last note.

ADB BD

Let ABC (Plate VIII. fig. 1) be the mast of the sail-boat; D the situation of the eye of the person who observes the angular elevation of the mast. Draw the line BD parallel to the horizon, and join AD. Then the height AB must be measured accurately, and, that being given and the observed angle ADB, the corresponding distance BD may be obtained by the usual rules of trigonometry, by saying, As radius: AB:: cotangent ADB: BD. Thus, if the height AB be 30 feet, and the angle ADB 1°, the distance BD will be 1719 feet (being 57.3 times as great as AB). The distances corresponding to 2o, 3o, &c., are given in the adjoined table, by examining which it will appear that the distance BD corresponding to any angle ADB (less than 30°) may be obtained nearly by dividing 1719 by the angle ADB in degrees. Thus, for 4 degrees, by this rule, the distance would be 17194293 nearly, as in the table. The greatest difference between the distances determined by the rule and by the table is 5 feet, corresponding to the angle 30°; for 11957, whereas by the table the distance is 52. In taking soundings by this method, it will be very rarely necessary to measure an angle so great as 30°; so that, for all practical purposes, the distance may be determined, in this example, to a sufficient degree of accuracy, by dividing 1719 by the observed angular elevation in degrees. On these principles we have the following rule for calculating the distance, corresponding to a mast of any given height, and to any observed angular elevation.

FEET.

1°

1719

859

572

429

343

170

RULE. Multiply the height of the mast above the eye of the observer by 57.3, and the product will be a constant quantity, which, being divided by the observed angle of elevation, expressed in degrees and decimals of a degree, the quotient will be the sought distance nearly.

For further illustration of this method, we have given an example in Plate VIII. fig. 4, in which C represents the place where the sail-boat is at anchor; A and B the

A mark may be made at B, and a vane placed at the top of the mast at A, to enable the observer to distinguish those objects when at a great distance. If the height of the observer above the horizon be small in comparison with the height of the mast, the angular distance ADE between the surface of the sea, near the boat, and the top of the boat's mast may be measured, instead of ADB; for, if the distances BC and CE remain the same in all observations, it will be immaterial which angle is meas. ured; observing, however, that different scales must be used for plotting off the angles ADB and ADE. If AB represent the known vertical height of the summit of an island above the eye of an observer the distance from the island can be determined by measuring the angular elevation ÅDB, as is eviden from what has been said above.

This constant quantity may be determined without actually measuring the altitude AB, if the angular elevation can be measured at a place D, where the distance BD is known. Thus, in the example (Plate VIII. fig. 4), the distance AC being known, and the angular elevation of the mast at C being observed at Å in degrees and decimals of a degree, and multiplied by the distance AC, the product ill be the constant quantity mentioned in the rule. This method may be used in determining the distance from an island by the method mentioned in the last note.

points observed, in order to ascertain the position of the boat on the chart, by drawing thereon the lines AC, BC, in opposite directions to the bearings of the points A, B, observed from the boat,-the point of intersection C being evidently the place of the boat upon the chart. Suppose, now, that in the first set of observations, the mast of the sail-boat is made to range on the point A; in this case the course of the boat must be on the continuation of the line AC towards D: then the slip H I (Plate VIII. fig. 2) is to be laid upon the line CD (Plate VIII. fig. 4), with the point H upon C; and the angular elevation being found on the slip, the sounding corresponding (reduced to low water) is to be marked on the line CD, immediately under the mark on the slip. Thus, if the angle be 4°, the point corresponding will be G. Having plotted off the soundings taken in the direction CD, proceed in the same manner with the others, viz. those in the direction CE, found by keeping the boat's mast in a range with the church at H; those in the direction CF, found by keeping the boat's mast in a range with the point B; those in the direction CA, found by keeping the mast to bear E. N. E.; and so on with the other observations. When all the soundings are marked on the chart, dotted lines are to be made round the shoal soundings; and thus the true figure of the shoal part of the bank will be obtained.

This method I have frequently used in taking a survey of the part of the coast of Massachusetts Bay included between Manchester and Lynn. The height of the mast of the boat used on the occasion was about 30 feet; and it was found that distances less than a third of a mile could be obtained in this manner to a great degree of precision.

Second Method. This method of determining the place where soundings are taken, consists in keeping (while sailing in a boat and sounding) a particular point of land, or any other object, to bear always in the same direction, and measuring with a quadrant or sextant, held in a horizontal position, the angular distance between that object and another object making a considerable angle with the former; for by this means the situation of the boat at the time of sounding may be determined. Instead of bringing the object to bear upon a particular point of the compass, you may (when it can be done) bring the object in a range with another remarkable object, and by this means you will avoid the error which might arise from the use of a compass.

For an example of this method, suppose that a survey of the small islands A B, K (Plate VIII. fig. 3), and the large one CGH, has been taken and plotted off as in the figure. Then soundings may be taken, in the direction BCD, by bringing the small island B in a range with the southern part of the great island, and measuring the angle CDG formed by the extremes of the great island; or by keeping the small island A to range with the northern part of the great island, and measuring the angle HIK formed by the northern extreme of that island and the small island K; or by running in the direction KL, so as to keep the island K to bear W. S., and measuring the angle formed by that island and the northern extreme of the great island, &c.

The method I have generally used for plotting off such angles, is by means of a sector; and as that instrument is more easily procured than others better adapted to the purpose, I shall explain the method by showing how the angle CDG, measured as above, may be plotted off so as to determine the point D where that angular distance was observed. To do this, you must draw the line CD, and open the sector till the two legs form with each other an angle equal to the observed angle CDG; then slide one leg of the sector on the line CD till the other leg touches the northern extreme of the island at the point G, and the point directly under the centre of the joint of the sector will be the point of observation. As this point cannot be exactly marked, on account of the size of the joint of the instrument, you may mark with a pencil on the line CD the two points where the circumference of the joint touches that line, and note the sounding in the middle between those two marks.

If a quadrant of a circle be described on a piece of paper, with a radius equal in length to one of the legs of the sector, and then divided into 90°, the sector may, by means of that quadrant, be opened to any angle in a very expeditious manner.

This method of obtaining distances when sounding, I have frequently used with

success.

Third Method-with two observers. This method is founded upon the process explained in Problem VII. page 93. It consists in finding, at the same time, by means of two observers furnished with sextants, the horizontal angles ADC, BDC, (figure Problem VII. page 93) formed, at the point D of the shoal, by the right lines DA, DC, DB, drawn to three points of land or remarkable objects, A, C, B, whose positions are given on the chart, or have been ascertained by previous observations. In this way

and the corresponding soundings can, at the same time, be observed. As the process of projecting and computing such observations has already been explained in Problem VII., it will not be necessary to make any additional remarks in this place, except that great care must be taken in selecting the points to be observed, A, C, B, so as not to have the centres, F, G, of the two intersecting circles, ABD, BCD, near to each other; because, in that case, a slight error in either of the observed angles, ADC, BDC, may produce a very important error in the situation of the point D of the shoal, corresponding to the intersection of these circles; it being evident that the method would wholly fail if the point C were to be placed at E upon the circumference of the circle ABD, because the centres F, G, would then coincide, and there would be no single point of intersection D, since any point whatever of the circumference of the circle BCD would satisfy the observations. This difficulty is inherent in this method of observation, and no process of numerical calculation will help it; so that we may rest assured, that whenever it is difficult to find the precise point of intersection D, by a geometrical construction, the points A, C, B, have not been well selected; and the observations may lead to a very incorrect result, except the angles are taken with the utmost degree of accuracy.

To reduce soundings taken at any time of the tide to low water.

The soundings at low water are always to be marked on a chart; and if they are taken at any other time of the tide, a correction must be applied to reduce them to low water. This allowance may be made, if the whole vertical rise of the tide from low to high water be known, with the time of high and low water, as in the following example:

Suppose the vertical rise of the tide; from low to high water, to be 10 feet, the time of low water 5h. A. M., and the time of high water 11h. 30m. A. M.; required the allowance to be made on an observation taken at 8, A. M.

Draw the line AC (Plate VIII. fig. 5), and make it equal to the whole rise of the tide, 10 feet, taken from any scale of equal parts, and divide the line into equal parts, representing feet, at the points 1, 2, 3, &c. to 10, the mark 10 (corresponding to the whole rise of the tide) being at the point C; and through these points draw lines 11, 22, 33, &c., perpendicular to AC, to meet the circumference of a circle drawn on the diameter AC. Divide the semicircumference ABC of this circle into a number of equal parts representing the number of hours elapsed from low to high water * (which, in this case, is 64h.), the hour of low water being marked at A, and that of high water at C, the intermediate hours being marked in succession, as in the figure; then, any hour being found on the arc, the number of the line drawn perpendicular to AC, and passing through the hour, will represent nearly the number of feet to be subtracted from a sounding taken at that time, to reduce it to low water. Thus the number of feet corresponding to 8h. is between 4 and 5, because the mark 8h. falls between the lines marked 4 and 5; therefore the reduction is between 4 and 5 feet, on soundings taken at 8, A. M., to reduce them to low water, on the day of observation; and if, on that day, the tide does not ebb so much as on a spring tide, the reduction must be increased by the difference in the ebbing of the two tides. Thus, if, on the day of observation, the tide did not ebb so much by two feet as on a spring tide, the reduction corresponding to 8h. must be increased two feet, and will therefore be between 6 and 7 feet. Allowance may be made for this, by increasing the number of feet given in figure 5, by marking 2 feet at A, 3 feet at 1, 4 feet at 2, &c., as is evident.

To reduce a draught to a smaller scale.

With a black-lead pencil, draw, on the draught to be reduced, cross lines, forming exact squares; and on the clean paper for the copy draw the same number of squares, making their sides larger or smaller in proportion to the intended size of the scale, such as, 4, &c., the length of the other. Distinguish by a stronger mark every fifth or sixth row of squares in both, so that the several corresponding squares may be readily perceived; then, in each of the squares of the draught, draw, by the eye, a curve on the paper, similar to that in the square of the copying-draught, till the whole is copied, when the black-lead lines may be rubbed out with bread or India-rubber.

*This division of the semicircle may be made by means of a line of chords; the number of degrees corresponding to one hour being found by saying, As the whole elapsed time from low to high water (6 hours) is to 180°, so is one hour to the arc corresponding to 1 hour, 27° 42', which, being taken from a line of chords, and laid off from 5h., will reach to 6h., &c.

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The New American Practical Navigator; Being an Epitome of Navigation ...