480 NOTES. 



dually increase from the polar radius C N, fig. 30, which is least, to the equa- 

 torial radius C Q, which is greatest. There is also an increase in the lengths 

 of the arcs corresponding to the same number of degrees from the equator to 

 the poles; for, the angle NCr being equal to qCd, the elliptical arc Nr is 

 less than qd. 



NOTE 123, pp. 52, 296. Cosine of latitude. The angles mCa, mCb, fig. 4, 

 being the latitudes of the points a, b, &c., the cosines are Cq, Cr, &c. 



NOTE 124, p. 53. An arc of the meridian. Let N Q S q, fig. 30, be the meri- 

 dian, and mn the arc to be measured. Then, if Z'm, Zn, be verticals, or 

 lines perpendicular to the surface of the earth, at the extremities of the arc 

 mn they will meet in p. Qan, Qbm, are the latitudes of the points m and 

 n, and their difference is the angle mpn. Since the latitudes are equal to the 

 height of the pole of the equinoctial above the horizon of the places m and n, 

 the angle mpn may be found by observation. When the distance mn is 

 measured in feet or fathoms, and divided by the number of degrees and parts 

 of a degree contained in the angle mpn, the length of an arc of one degree 

 is obtained. 



NOTE 125, p. 53. A series of triangles. Let MM', fig. 31, be the meridian 

 Fig. 31. 



of any place. A line AB is measured with rods, on level ground, of any 

 number of fathoms, C being some point seen from both ends of it. As two of 

 the angles of the triangle ABC can be measured, the lengths of the sides 

 AC, BC, can be computed; and if the angle wiAB, which the base AB 

 makes with the meridian, be measured, the length of the sides Em, Am, may 

 be obtained by computation, so that Am, a small part of the meridian, is 

 determined. Again, if D be a point visible from the extremities of the known 

 line BC, two of the angles of the triangle BCD may be measured, and the 

 length of the sides CD, BD, computed. Then, if the angle Emm' be mea- 

 sured, all the angles and the side Em of the triangle Emm' are known, 

 whence the length of the line mm' may be computed, so that the portion 

 A m' of the meridian is determined, and in the same manner it may be pro- 

 longed indefinitely. 



NOTE 126, pp. 54, 56. The square of the sine of the latitude. Q bm, fig. 30, 

 being the latitude of m, em is' the sine and be the cosine. Then the number 

 expressing the length of em, multiplied by itself, is the square of the sine of 

 the latitude ; and the number expressing the length of be, multiplied by itself, 

 is the square of the cosine of the latitude. 



