Demonstration of the formulæ for the radius of curvature in terms of the Demonstration of formula for determining the figure and dimensions of the Formulas for latitude, altitude, and time of observation used in prime verti- cal transits, Description of the Pulkova and Washington instruments, Struve's formula, Example from Struve's observations,. Concluding note, PLANE PART I. TRIGONOMETRY. 1. THE term TRIGONOMETRY is compounded of two Greek words, gyvos, a triangle, and pergov, measure, signifying literally the measurement of triangles. It has for its object to determine the unknown parts of a triangle when a sufficient number of the parts is known. By parts or elements of a triangle are understood commonly the sides and angles, though trigonometry properly includes the measurement of the surface also. There will accordingly be six elements of every triangle, namely the three sides and the three angles. 2. It has been proved (Plane Geom., Theorems 1, 2, and 5), that when two triangles have three elements, one of which is a side, in the one, equal respectively to the corresponding elements in the other, the triangles are identical. One element must be a side, because if the three angles only were equal respectively in the two triangles they would be but similar (Plane Geom., Theorem 63); that is, alike in shape but not necessarily in size. Since all triangles which have three elements equal, are by consequence equal, it is said that three given elements determine a triangle; that is, with these three given elements, but one triangle can be formed. There is one exception to this principle, pointed out in Prob. 8, Plane Geom., where two sides and the angle opposite one of them are given, in which case two triangles can be constructed with the given elements. 3. Three elements of a plane triangle being given then (except they be the three angles), it ought to be possible to find the other three, since these are fixed by their dependence upon the three given. This may be accomplished with sufficient accuracy for many purposes, by means of constructions such as are exhibited at Problems 5 and 8 of Plane Geometry. We shall repeat one of these constructions, enunciating the problem somewhat differently. The two sides and included angle of a triangle being given, let it be required to find the remaining side and the other two angles. A B F Let A and B be the two given sides, and c the given included angle. Draw two lines DH and DG of indefinite length, making with each other an angle equal to the given angle c. Lay off on the first of these the given line a from D to E, and on the second the given line в from D to F. Join EF. The only possible triangle DEF will thus be formed with the three given elements, in which EF will be the required side, and E and F the required angles. Α D H E The finding the unknown elements of a triangle by means of those which are given is called its solution. 4 The method of solution just exhibited is rendered more practically useful by the employment of scales of equal parts and protractors. The most simple form of the scale of equal parts is shown in the annexed figure. It is a straight rule divided into any number of equal parts; in this example ten, and one of these again into ten, so that the smallest division is one hundredth of the whole length of the rule. The following is the manner of using it. Suppose that it is required to draw upon paper a line equal in length to 56. Place one foot of a pair of dividers at the line of division marked 5, and extend them till the other foot reaches exactly to the sixth smaller division mark on the right of 0; the feet of the dividers will then be at a distance of 56 apart. A To draw now the required line upon paper, let a be the point from which it is to be drawn. Placing one foot of the dividers at A, extended the distance 56 obtained from the scale, describe with the other an arc of a circle on the side towards which the line is to be drawn; then from A draw the line in the proper direction, terminating it at the arc before described, and it will be the line required. Another line of 42 being measured from the scale and laid down upon the paper, the two lines will be in the ratio of 56 to 42. If they are lines upon a map, and the first corresponds to a line of 56 feet upon the ground, the second will correspond to a line of 42 feet. If the first represent 56 yards, or chains, or miles, the second will represent 42 yards, or chains, or miles. And in general lines upon the same drawing which are measured in parts of the same scale must be understood to be expressed in units of the same kind. The sectoral scale of equal parts consists of a ruler of two arms moving on a hinge, each arm being divided into a number (usually 100) of equal parts. To set this scale to any size, say 40 parts to the inch, the arms must be separated by turning them round the hinge, till a pair of dividers opened to the distance of an inch will extend exactly from the division marked 40 on one arm, to that marked 40 on the other. If now any other distance be required upon a scale of 40 parts to the inch, as for instance the distance 65, the dividers must be opened till they will extend from the 65th division on the one arm, to the 65th division on the other. This kind of scale is constructed on the principle that lines drawn parallel to each other between the sides of an angle are proportional to the parts into which they divide the sides. (See Plane Geom., Theorems 61, 63, 65.) The diagonal scale of equal parts is constructed as seen in the diagram, |