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Constant log. =
0.978604 Log. co-sine . 9.988724 Log.co-sine. 9.999735
11 degrees. Nat. co-sine=981627
Nat, number 9.269=Log. 0.967063
Arch = ..11: 0:10" = Nat. co-s.=981617.731
Difference = .0: 0:10; which, therefore, is the change of altitude in one minute from noon.
Let the latitude be 40 degrees, and the declination of a celestial object 8 degrees, of a contrary name to that of the latitude; required the variation or change of altitude in one minute from noon?
Constant log. =
0.978604 9.884254 9.995753
48 degrees. Nat. co-sine=669131
Nat. num. = 7.221 Log.=0.858611
Arch= . . 48: 0; 2"-Nat.co-sine=669123.779
0:0! 27., which, therefore, is the change of altitude in
It is to be observed, however, that, with the view of introducing every possible degree of accuracy into the present Tables, the natural and log. co-sines, &c., employed in their construction, have had their respective numbers extended to seven places of decimals.
Note.-The difference between the meridional altitude of a celestial object and its altitude at a given interval from noon, is found, by actual observation, to be very nearly proportional to the square of that interval, under certain limitations, as pointed out in page 138; and hence the rule, in page 139, for computing the meridional altitude of a celestial object
The Miles and Parts of a Mile in a Degree of Longitude at every
Degree of Latitude. This Table consists of seven compartments : the first column in each compartment contains the degrees of latitude, and the second column the miles and parts of a mile in a degree of longitude corresponding thereto. In taking out the numbers from this Table, proportion is to be made, as usual, for the minutes of latitude; this proportion is subtractive from the miles, &c., answering to the given degree of latitude.
Example. Required the number of miles contained in a degree of longitude in latitude 37:48! ?
Miles in a degree of longitude, in latitude 37 degrees =
64 x 48' Difference to 1 degree of latitude = .64; now
Miles in a degree of long. in latitude 37 degs. 48 min., as required=47.41
Remarks,-Since the difference of longitude between two places on the earth is measured by an arch of the equator intercepted between the meridians of those places; and since the meridians gradually approach each other from the equator to the poles, where they meet, it hence follows that the number of miles contained in a degree of longitude will decrease in proportion to the increase of the latitude; the ratio of decrease being as radius to the co-sine of the latitude. Now, since a degree of longitude at the equator contains 60 miles, we have the following rule for computing the present Table ; viz.,
As radius is to the co-sine of the latitude of any given parallel, so is the measure of a degree of longitude at the equator to the measure of a degree in the given parallel of latitude.
Example. Required the number of miles contained in a degree of longitude in the parallel of latitude 37 degrees?
As radius . . 90 degrees
Log. sine = .
1.680500; Hence the measure of a degree of longitude in the given parallel of latitude, is 47.92 miles.
Proportional Miles for constructing Marine or Sea Charts.
In this Table the parallels of latitude are ranged in the upper horizontal column, beginning at 0!, and numbered 10., 20., 30., &c., to 89:; the horizontal column immediately under the parallels of latitude contains the number of miles of longitude corresponding to each parallel's distance from the equator ; under which, in the horizontal column marked“ Difference of the Parallels, &c.," stands the number of miles of longitude contained between the parallel under which it is placed and that immediately preceding it.
The left-hand vertical column contains the intermediate or odd degrees of latitude, from 0 to 10: ; opposite to which, and under the respective parallels of latitude, will be found the number of miles of longitude corresponding to each degree of latitude in those parallels : these are intended to facilitate, and render more accurate, the subdivision of the different parallels of latitude into degrees and minutes.
To make a Chart of the World, in which the Parallels of Latitude and
Longitude are to consist of 10 Degrees each.
Draw a straight, or meridian, line along the right hand, or east margin of the paper intended to receive the projection; bisect that line, and from the point of bisection draw a straight line perpendicular to the former, which continue to the left-hand or west margin of the paper, and it will represent
From any diagonal scale of convenient size take 600 miles in the compasses (the number of miles of the equator contained in 10 degrees of longitude), and lay it off from the point of bisection along the equator, and it will graduate it into 36 equal parts of 10 degrees each; through which let straight lines be drawn at right angles to the equator, and parallel to that drawn along the right-hand margin, and they will represent the meridians or parallels of longitude. Take, from the same scale, 60 miles in the compasses, and it will subdivide each of those 36 divisions, or parallels of longitude, into ten equal parts consisting of one degree each ; and then will the equator be divided into 360 degrees of 60 miles each.
On the meridian lines drawn along the right and left-hand margins of the
paper, let the parallels of latitude be laid down, as thus :--For the first parallel, or 10 degrees from the equator, take 603. 1 miles in the compasses (found in the horizontal column immediately under the parallels of latitude, and marked." Ditto in miles of the Equator, &c."); place one foot on the
equator, and where the other falls upon the right and left-hand marginal lines, when turned northward and southward, there make points ; through which let straight lines be drawn parallel to the equator, and they will represent the parallels of latitude at 10 degrees north and south of the equator: in the same manner, for 20 degrees, lay off 1225. I miles; for 30 degrees, 1888.4 miles; for 40 degrees, 2622.6 miles, and so on.
But since the common compasses are generally too small for taking off such high numbers, it will be found more convenient to lay down the parallels of latitude by the numbers contained in the third horizontal column, or that marked “Difference of the Parallels, &c.” Thus, for 10 degrees, take 603. I miles in the compasses; place one foot on the equator, and with the other make points north and south thereof on the east and west marginal lines, through which let straight lines be drawn, and they will represent the parallels of latitude at 10 degrees north and south of the equator. From these parallels respectively, lay off 622.0 miles, by placing one foot of the compasses on the respective parallels and the other on the east and west marginal lines ; through the points thus made by the compasses draw straight lines, and they will represent the parallels of latitude at 20 degrees north and south of the equator. From the parallels, thus obtained, lay off 663.3 miles, and the parallel of 30 degrees will be determined: thence lay off 734.2 miles, and it will show the parallel of 40 degrees; and so on for the succeeding parallels.
The numbers for subdividing those parallels will be found in the vertical columns under each respectively, and are to be applied as follows ; thus, to graduate the parallel between 50 and 60 degrees : take 94.3 miles in the compasses, and lay it off from 50 degrees towards 60 degrees, and it will give the parallel of 51 degrees; from which lay off 96.4 miles, and it will show the parallel of 52 degrees; from this lay off 98.6 miles, and the parallel of 53 degrees will be obtained; and so on of the rest. In the same manner let the other parallels of latitude be subdivided; then let the parallels of latitude be numbered along the east and west marginal columns, from the equator towards the poles, according to the number of degrees contained in that arc of the meridian which is intercepted between them and the equator, as 10., 20., 30., 40:, &c. &c.; and let the parallels of longitude be numbered at the top and bottom, and also along the equator; these are to be reckoned east and west of the first meridian, as 10., 20., 30., 40:, &c., to 180., both ways; and since the first meridian is entirely arbitrary, it may be assumed as passing through any particular place on the earth, such as Greenwich Observatory: then will the chart be ready for receiving the latitudes and longitudes of all the principal places on the earth, and which are to be placed thereon by the following rule; viz.,
Lay a ruler over the given longitude found at the top and bottom of the
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.
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.