Since the circumferences of circles are to one another as their radii, if the radius of the equator be taken to express a degree of the equator,h degree of any parallel will be ex pressed by the radius of that parallel. But the radius e ll (Fig. 2.) of any parallel e L, is the sine of e N the colati tude, or the cosine of E e, the latitude of that parallel to the radius ..EC. Hence, to construct the above Table, we have only to take the natural cosines of the different paral lels to radius 1, or the natural numbers corresponding to the logarithmic cosines, removing the decimal point ten places towards the left hand in each. Thus, let it be required to find a degree of longitude on the parallel of 25°. The natural cosine of 25° is 90,631 to radius 100,090, and making radius 1, the cosine becomes .90631, the length of the de gree required. Thus also the logarithmic cosine of 9.957276, and the number corresponding to this logarithm is 9,063,100,000, which is the length of the degree re quired, that of the equator being 10,000,000,000, or radius of the trigonometrical table. But as it would be inconve nient to operate with these numbers, they may both be di vidcd by 10,000,000,000, or the decimal point may be re moved ten places to the left hand in each, which will give I for the degree of the equator, and .90631, as in the pre ceding Table, for that of the parallel of 25°. This num ber may also be found at once from the logarithmic Tables, by subtracting 10 from the cosine, and finding the natural number corresponding to the remaining logarithm. Thus the cosine of 25° becomes 1957276, and the number cor responding in the Table of logarithms is .90631.
The application of the above 'f able for finding the length of a degree of longitude under any parallel, consists in sim ply multiplying the fraction opposite to the given latitude, by the length of a degree of the equator. Thus, to find the length of a degree on the parallel of 25°, that of the equa tor being 60 geographic miles, multiply 90,631 by 60, and the product 54.3786, or 54.38 nearly, gives the degree re quired in geographic miles. If the earth be considered as spherical, a degree of the equator may be assumed equal to the degree of the meridian bisected by the parallel of 45°, or 60739.473 fathoms, which gives for the geographi cal mile 6075.947 feet.
Before concluding this account of the dimensions of the globe, it may perhaps be of use to some of our readers, to point out a simple and expeditious method of finding the superficial contents of any given zone of the earth. By geo metry, the superficies of a sphere is equal to the product of the circumference, multiplied by the diameter, and that of a zone to the product of the circumference multiplied by that part of the diameter, intercepted between the planes of the two parallels containing the zone; that is, the area of the zone is to the area of the whole sphere, as the perpendi cular distance of the two parallels of the zone is to the dia meter. But the distance BD (Fig. 2.) between any two pa rallelsf g, e L, is the difference of the sines of IE e and f, the latitudes of e and f; therefore the area of the zone feLg:area of the globe : : diameter :: sin. /E. e—sin..E.; : radius. If, therefore, the radius of the sphere be taken to express the whole area ; half the differ ence of the natural sines, or of the natural numbers corre sponding to the logarithmic sines of any two latitudes, will express the area of the zone included between these.lati
tudes, the radius of the sphere being equal to the radius of the respective Tables. If the radius be reduced to unity, the area of the zone will be a decimal fraction. In the com mon logarithmic Tables, this is done by removing the deci mal point ten places towards the left hand, or the fraction may be found at once, thus : From the trigonometrical ta bles, take the sines of the latitudes, subtract ten from the index of each, and find the numbers corresponding to the remaining logarithms ; half the difference of these numbers will express the area of the zone, that of the sphere itself being unity.
ble being 100,000, and removing the decimal point five places towards the left, the radius becomes 1, and the area of the zone .004815.
Upon this principle, the following Table is constructed, exhibiting the area of every zone of 1° from the equator to the pole, that of the globe being unity.
To find from the preceding Table the area of a zone, less than 1° in breadth, take a proportional part of the zone of t° of which the other forms a part. Thus to find the area of a zone between 43° and 43° 35', take from the Table the area of the zone between 43° and 44°, which is .00633, and say 60' : 35' : .00633; area required nearly x .00633 60 =.0036925. The true area, as found from-the Table of sines, is .003705.
To find the area of a segment of a zone terminated at both extremities by meridians ; multiply the area of the whole zone, by the length of the segment, and divide by 360. Thus, to find the area of a segment of the zone be tween 43° and 35', terminated by two meridians 6° 20' distant from one another, multiply .0036925 by 6° 20', and 6° 20' divide by 360, that is the area of the segment.= or 360° To find the al ea of any particular country or district, di vide the country into SCf2,11telitS or zones, by parallels of la titude, and find the area of each segment separately ; the stun of these areas will be the area t cquired.
In sonic cases, this operation may be considerably ab breviated without affecting, in any great degree, the accu racy of the result. Let it be required, for example, to find the arca of Portugal. Instead or dividing the whole sur face into segments or zones of different lengths, according to the difference in the extent of the country from west to east, we may suppOse the whole to consist of one segment, ofa uniform length and breadth, viz. between 37° and 42° north latitude, and between 7° and 9° west longitude. By this arrangement, indeed, the eastern boundary cuts off a part of Tralos Mantes and Beira, and the western a part of Estremadura ; but, in lieu of these, the former includes a portion of Andalusia in Spain, and the latter a pat t of the Atlantic ocean. Supposing, therefore, these exchanges to be nearly equivalent, the area may be found thus : Take the sum of the areas or the zones between 37° and which is .03366, multiply by 2 the length of the seg ment, and divide by 360, that is, 6 Area = 360 180 x.03366= .0336 .000187.