:

chart, and with a pair of compasses take the latitude from the east or west marginal columns; which being applied to the edge of the ruler, placing one foot on the equator or on the parallel that the latitude was counted from, the other foot turned north or south according to the name of the latitude, will point out or fall upon the true position of the given latitude and longitude.

From what has been thus laid down, the manner of constructing a chart for any particular place or coast must appear obvious.

Note. Since this Table is merely an extract from the Table of Meridional parts, the reader is referred to page 113 for the method of computing the different numbers contained therein.

TABLE LV.

To find the Distance of Terrestrial Objects at Sea.

If an observer be elevated to any height above the level of the earth or sea, he can not only discern the distant surrounding objects much plainer than he could when standing on its surface, but also discover objects which are still more remote by increasing his elevation. Now, although the great irregularity of the surface of the land cannot be subjected to any definite rule for determining the distance at which objects may be seen from different elevations; yet, at sea, where there is generally an uniform curvature of the water, on account of the spherical figure of the earth, the distance at which objects may be seen on its surface may be readily obtained by means of the present Table; in which the distance answering to the height of the eye, or to that of a given remote object, is expressed in nautical miles and hundredth parts of a mile; allowance having been made for terrestrial refraction, in the ratio of the one-twelfth of the intercepted arch.

Note. The distance between two objects whose heights are given, is found by adding together the tabular distances corresponding to those heights. And, when the given height exceeds the limits of the Table, an aliquot part thereof is to be taken; as one fourth, one ninth, or one sixteenth, &c.; then, the distance corresponding thereto in the Table, being multiplied by the square root of such aliquot part, viz., by 2, 3, or 4, &c., according as it may be, will give the required distance.

Example 1.

The look-out man at the mast-head of a man-of-war, at an elevation of 160 feet above the level of the sea, saw the top of a light-house in the horizon whose height was known to be 290 feet; required the ship's distance therefrom?

which, therefore, is the ship's distance from the light-house.

Example 2.

The Peak of Teneriffe is about 15300 feet above the level of the sea; at what distance can it be seen by an observer at the mast-head of a ship, supposing his eye to be 170 feet above the level of the water?

One ninth of 15300 is 1700, answering to which is 47.50 miles; this being multiplied by 3 (the square root of one ninth) gives 142.50 miles. Distance ans. to 170 feet (height of the eye) is

15.03 do.

Remark 1.-Since the distances given in this Table are expressed in nautical miles, whereof 60 are contained in one degree, and there being 69.1 English miles in the same portion of the sphere; if, therefore, the distance be required in English miles, it is to be found as follows; viz.,

As 60, is to 69.1; so is the tabular distance to the corresponding distance in English miles; which may be reduced to a logarithmic expression, as thus:

To the log. of the given tabular distance, add the constant logarithm 0.061327,* and the sum will be the log. of the given distance in English miles.

Example.

Let it be required to reduce 157.53 nautical miles into English miles?

Given distance in nautical miles = 157.53, log.
Constant log.

= 2. 197364

.0.061327

Distance reduced to English miles 181.42 Log. = 2.258691

=

=

The log.of 69.1 1.239478, less the log. of 60 = 1.778151 is 0. 061327; which, therefore, is the constant logarithm.

The converse of this (that is, to reduce English miles into nautical miles,) must appear obvious.

Remark 2.-This Table was computed by the following rule; viz.,

To the earth's diameter in feet, add the height of the eye above the level of the sea, and multiply the sum by that height; then, the square root of the product being divided by 6080 (the number of feet in a nautical mile), will give the distance at which an object may be seen in the visible horizon, independent of terrestrial refraction. This rule may be adapted to logarithms, as thus :

Let the earth's diameter in feet be augmented by the height of the eye; then, to the log. thereof add the log. of the height of the eye; from half the sum of these two logs. subtract the constant log. 3. 783904,* and the remainder will be the log. of the distance in nautical miles, which is to be increased by a twelfth part, of itself, on account of the terrestrial refraction.

Example.

At what distance can an object be seen, in the visible horizon, by an observer whose eye is elevated 290 feet above the level of the sea?

Distance uncorrected by refraction 18.11=Log. = 1.257907
Add one-12th part on acc, of refrac. 1.51

Note. For the principles of this rule, see how the distance of the visible horizon, expressed by the line O T, is determined in page 5.

This is the log. of 6080, the number of feet in a nautical mile.

TABLE LVI.

To reduce the French Centesimal Division of the Circle into the English Sexagesimal Division; or, to reduce French Degrees, &c., into English Degrees, &c., and conversely.

This Table is intended to facilitate the reduction of French degrees of the circle into English degrees, and conversely. The Table is divided into two parts: the first or upper part exhibits the number of English degrees and parts of a degree contained in any given number of French degrees and parts of a degree; and the second or lower part exhibits the number of French degrees, &c., contained in any given number of English degrees, &c.

Note. In the general use of this Table, when any given number of French degrees exceeds the limits of the first part, take out for 100 degrees first, and then for as many more as will make up the given number; and, when any given number of English degrees exceeds the limits of the second part, take out for 90 degrees first, and then for as many more as will make up the given number.

Example 1.

If the distance between the moon and a fixed star, according to the French division of the circle, be 128:93.96", required the distance agreeably to the English division of the circle?

Distance reduced to English degs., as required 116: 2:44".30

Example 2.

If the distance between the moon and sun, according to the English division of the circle, be 116:53'47", required the distance agreeably to the French division of the circle?

Distance reduced to French degs, as required=129:88:48".76

Remark 1.-This Table was computed in conformity with the following considerations and principles; viz.,

The French writers on trigonometry have recently adopted the centesimal division of the circle, as originally proposed by our excellent countryman Mr. Henry Briggs, about the year 1600. In this division, the circle is divided into 400 equal parts or degrees, and the quadrant into 100 equal parts or degrees; each degree being divided into 100 equal parts or minutes, and each minute into 100 equal parts or seconds: these degrees, &c. &c., are written in the usual manner and with the customary signs, as thus; 128:93.96".

Hence, the French degree is evidently less than the English, in the ratio of 100 to 90; a French minute is less than an English minute, in the ratio of 100 x 100' to 90: × 60; and a French second is less than an English second, in the ratio of 100: × 100 × 100% to 90 × 60 × 60′′: now, the converse of this being obvious, we have the following general rule for converting French degrees into English, and the contrary.

As 100, the number of degrees in the French quadrant, is to 90, the number of degrees in the English quadrant; so is any given number of French degrees to the corresponding number of English degrees.

As 10000, the number of minutes in the French quadrant, is to 5400, the number of minutes in the English quadrant ; so is any given number of French minutes to the corresponding number of English minutes. And,

As 1000000, the number of seconds in the French quadrant, is to 324000, the number of seconds in the English quadrant; so is any given number of French seconds to the corresponding number of English seconds.

English degrees, minutes, and seconds, are reduced into French by a converse proportion; viz.,

As 90, is to 100; so is any given number of English degrees to the corresponding number of French degrees.

As 5400, is to 10000; so is any given number of English minutes to the corresponding number of French minutes. And,

As 324000, is to 1000000; so is any given number of English seconds to the corresponding number of French seconds.

Remark 2.-French degrees and parts of a degree may be turned into English, independently of the Table, by the following rule; viz.,

Let the French degrees be esteemed as a whole number, to which annex the minutes and seconds as decimals; then one-tenth of this mixed number, deducted from itself, will give the corresponding English degrees, &c.