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That is, the difference between the true and apparent level is equal to the square of the distance between the places, divided by the diameter of the earth ; and consequently it is always proportional to the square of the dise

tance.

Now suppose, for example, we want to know the difference between the true and apparent level at the distance of an English mile, or 1760 yards : the square of this number is 3097600, which, being divided by the diameter of the earth 13953280, expressed in the same measure, gives o‘222 of a yard, nearly ; which being multiplied by 36, the number of inches in a yard, the product is 7.992, or nearly 8 inches, for the said difference.

Note 1. As the correction for reducing the apparent to the true level at the distance of i mile is very nearly 8 inches; therefore, as the square of 1 mile : 8 inches :: the square of any other distance : the correction for curvature,

NOTE 2. A table of corrections for given distances may be easily computed by means of logarithms. For the logarithm of the diameter of the earth 13953280, expressed in yards, is 7'14468, (more figures being unnecessary) which being constantly subtracted from double the logarithms of the distances, the differences will be the logo arithms of the corrections, expressed in decimal parts of a yard ; which, being multiplied by 36, the number of inches in a yard, gives the corrections in inches and decimals of an inch.

PROBLEM PROBLEM II.

To find the level of two places ; or the ascent or descent from

one to the other.

This is best done with a spirit level, having telescopic sights; which may be set horizontal by screws, that raise or lower the ends of the tube ; and station staves with

. sliding vanes.

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To take a level from A to 1:- let one assistant stand at A and another at C, and at a convenient place, as P, between A and C, place the level, and set it horizontal by means of the screws. Let the assistant at A hold the staff upright in his hand at A, while from P you look through the sights toward A, and direct him to slip the vane up or down till the white line at B be level with the sights; then note AB. Direct the other assistant to stand at C, and to hold the staff upright; then turning the instrument round at P, and looking toward C, direct him to raise or lower the vane till, through the sights, you can see the white line at D; then note the height above the ground, CD. And the difference of the heights, AB and CD, is the as.cent or descent with respect to the apparent level, if there : be no other station.

But, if there be many stations, make a table with two çolumns, one for the back stations, and the other for the fore stations. Now to proceed, set down the two heights AB, CD ; and let the assistant at A go to F with his staff, and remove the instrument to Q, and level it ; then direct

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the sights toward C, and let the assistant at C slip his vane up or down till the white line appear through the sights, as ar E ; and note CE. Turning the instrument about, look toward F till, through the sights, you see the white line at G, and note FG; in the mean time let the assistant at C remove to I, and placing the instrument at R, direct it backward toward F and forward toward K. And so proceed, from station to station, to the end. And set down the back heights AB, CE, FO, and the fore heights CD, FG, IK, in their proper columns. Then add the columns, and take the difference of the sums ; and if the fore heights exceed, the line is a descent ; but if the back heights exceed, it is an ascent. If they be equal, it is an apparent level.

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Hence, as the fore heights are greater, there is a descent, below the apparent level, of 3 feet 5 inches from A to I.

But to reduce this to the difference of true heights, the distances from each station of the level, both to the back and fore stations of the staves, must be measured, and the heights corrected, by the last Problem, when necessary ; then, the columns of corrected heights being added, the difference of their sums will be the difference of true heights.

NOTE

NOTE 1. The operation may also be performed by placing the Level first at one place, as A, and then successively at other convenient stations, taking fore observa. tions only, till a height at the other, as I, is found.

NOTE 2. If the distances of the instrument from the back station staff be every where equal to its distances from the corresponding forestation staff, there will be no. need of correction for curvature.

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NOTE *3. The velocity of running water depends on the fall. Where the fall is only 3 or 4 inches in a mile, the velocity is very small. Some canals have been cut with a fail from 4 to 6 inches,

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NAVIGATION.

NAVIGATION teaches to conduct a ship on the sea from one port or place to another.

In Navigation there are four principal things; two of which being given, the rest are thence determined.

1. Difference of latitude.

2. Difference of longitudc.
3. Distance, or length of the run.

4. Course, or rhumb line, on which the ship sails. The distance is measured by the Log line ; and the

' rhumb is shewn by the Compass.

PLANE CHART:

It is absolutely necessary to a traveller, that he should be acquainted with the situation of those places, to which he intends to go. And as the situations of places on the earth are known from their latitudes and longitudes, a

sufficiently

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