Example 13. A farm is described in an old Deed, as bounded thus. Beginning at a pile of stones, and running thence twentyseven chains and seventy links South-Easterly sixty-six and a half degrees to a white-oak stump; thence eleven chains and sixteen links North-Easterly twen Fig. 197. 6 3 North-Westerly sixty-six degrees to a pine stump; 2 thence thirty-three chains and nineteen links South-Westerly twenty-two degrees to the place of beginning, containing ninety-two acres, be the same more or less. Required the exact content. (297) Mascheroni's Theorem. The surface of any polygon is equal to half the sum of the products of its sides (omitting any one side) taken two and two, into the sines of the angles which those sides make with each other. Thus, take any polygon, such as the fivesided one in the figure. Express the angle which the directions of any two sides, as AB, CD, make with each other, thus (ABCD). Then will A the content of that polygon be, as below; = [AB. BC. sin (AB ^ BC) + AB . CD . sin (AB ^ CD) + AB. DE. sin (AB A DE) + BC. CD. sin (BC CD) ^ + BC . DE . sin (BC ADE) + CD . DE . sin (CD A DE)] The demonstration consists merely in dividing the polygon into triangles by lines drawn from any angle, (as A); then expressing the area of each triangle by half the product of its base and the perpendicular let fall upon it from the above named angle; and finally separating the perpendicular into parts which can each be expressed by the product of some one side into the sine of the angle made by it with another side. The sum of these triangles equals the polygon. The expressions are simplified by dividing the proposed polygon into two parts by a diagonal, and computing the area of each part separately, making the diagonal the side omitted.* CHAPTER VII. N'N THE VARIATION OF THE MAGNETIC NEEDLE. (298) Definitions. The Magnetic Meridian is the Fig. 199. direction indicated by the Magnetic Needle. The True Meridian is a true North and South line, which, if produced, would pass through the poles of the earth. The Variation, or Declination, of the needle is the angle which one of these lines makes with the other.† In the figure, if NS represent the direction of the True Meridian, and N'S' the direction of the Magnetic Meridian at any place, then is the angle NAN' the Variation of the Needle at that place. OA (299) Direction of Needle. The directions of these two meridians do not generally coincide, but the needle in most places points to the East or to the West of the true North, more or less * The original Theorem is usually accredited to Lhuillier, of Geneva, who published it in 1789. But Mascheroni, the ingenious author of the "Geometry of the Compasses," had published it at Pavia, two years previously. The method is well developed in Prof. Whitlock's "Elements of Geometry." + " Declination" is the more correct term, and "Variation" should be reserved for the change in the Declination which will be considered in the next chapter; but custom has established the use of Variation in the sense of Declination. according to the locality. Observations of the amount and the direction of this variation have been made in nearly all parts of the world. In the United States the Variation in the Eastern States is Westerly, and in the Western States is Easterly, as will be given in detail, after the methods for determining the True Meridian, and consequently the Variation, at any place, have been explained. TO DETERMINE THE TRUE (300) By equal shadows of the Sun. any level surface, erect an upright staff, shown, in horizontal projection, at S. Two or three hours before noon, mark the extremity, A, of its shadow. Describe an arc of a circle with S, the foot of the staff, for centre, and SA, the distance to MERIDIAN. On the South side of N B the extremity of the shadow, for radius. About as many hours after noon as it had been before noon when the first mark was made, watch for the moment when the end of the shadow touches. the arc at another point, B. Bisect the arc AB at N. Draw SN, and it will be the true meridian, or North and South line required. For greater accuracy, describe several arcs before hand, mark the points in which each of them is touched by the shadow, bisect each, and adopt the average of all. The shadow will be better defined, if a piece of tin with a hole through it be placed at the top of the staff, as a bright spot will thus be substituted for the less definite shadow. Nor need the staff be vertical, if from its summit a plumb-line be dropped to the ground, and the point which this strikes be adopted as the centre of the arcs. This method is a very good approximation, though perfectly correct only at the time of the solstices; about June 21st and December 22d. It was employed by the Romans in laying out cities. To get the Variation, set the compass at one end of the True Meridian line thus obtained, sight to the other end of it, and take |