406 



GEOMETRY GEORGE. 



If ABC or ADC, fig. 76, be a semicircle ; then 

 any angle U in that semicircle, is a right angle. 

 77 78 79 



If AB, fig. 77, be a tangent, and AC a chord, 

 and D any angle in the alternate segment ADC ; 

 then will the angle D be equal to the angle BAC 

 made by the tangent and chord of the arc A EC. 

 Let ABCJ), fig. 78, be any qjiadrilateral inscribed 

 in a circle ; then shall the sum of the two opposite 

 angles A and C, or B and D, be equal to two right 

 angles. If the side AB, fig. 79, of the quadrilate- 

 ral ABCD, inscribed in a circle, be produced to E ; 

 the outward angle DAE will be equal to the inward 

 opposite angle C. 



80 81 82 



Let the two chords AB, CD, fig. 80, be parallel ; 

 then will the arcs AC, BD, be equal ; or AC = BD. 

 Let the tangent ABC, fig. 81, be parallel to the 

 chord DF; then are the arcs BD, BF, equal; that 

 is, BD = BF. Let the two chords AB, CD, fig. 

 82, intersect at the point E; then the angle A EC, 

 or DEB, is measured by half the sum of the two 

 arcs, AC, DB. 



83 84 85 



A 



Let the angle E, fig. 83, be formed by two 

 secants EAB and BCD ; this angle is measured 

 by half the difference of the two arcs, AC, DB, 

 intersected by the two secants. Let EB, ED, 

 fig. 84, be two tangents to a circle at the points 

 A, C ; then the angle E is measured by half the 

 difference of the two arcs CFA, CG A. Let the two 

 lines AB, CD, fig. 85, meet each other in E ; then 

 the rectangle of AE, EB, will be equal to the rect- 

 angle of CE, ED. Or, AE, EB = CE, ED. 



(i 87 88 



When one of the lines in the second case, as DE, 

 fig. 86, by revolving about the point E, comes into 

 the position of the tangent EC or ED, the two 

 points C and D running into one ; then the rect- 

 angle of CE, ED, becomes the square of CE, 

 because CE and DE are then equal. Consequently, 



the rectangle of the parts of the secant, AE, K 13, is 

 equal to the square of the tangent, CE. Let CD, 

 fijj. 87, be the perpendicular, and CE the diameter 

 ot the circle about the triangle ABC ; then the rec- 

 tangle CA . CB is = the rectangle CD . CE. Let 

 CD, fig. 88, bisect the angle C ot the triangle ABC ; 

 then the square CD 8 + the rectangle AD . DB 

 is = the rectangle AC . CB. 

 89 



Let ABCD, fig. 89, be any quadrilateral in- 

 scribed in a circle, and AC, BD, its two diagonals ; 

 then the rectangle AC . BD is = the rectangle 

 AB . DC + the rectangle AD . BC. Similar 

 figures are to each other, as the squares of their 

 like sides. Similar figures inscribed in circles, have 

 their like sides, and also their whole perimeters, in 

 the same ratio as the diameters of the circles in which 

 they are inscribed. Similar figures inscribed in 

 circles, are to each other as the squares of the dia- 

 meters of those circles. The circumferences of all 

 circles are to each other as their diameters. The 

 areas or spaces of circles, are to each other as the 

 squares of their diameters, or of their radii. 



Such are the leading properties of lines and plane 

 figures. The more important details regarding 

 planes and solids, will be discussed under the arti- 

 cles, PLANES, Geometry of; and SOLIDS, Geometry of. 

 For the method of constructing geometrical figures, 

 see Mathematics, Practical, fyc. 



GEORGE, LAKE; a lake in New York, south of 

 lake Champlain, with which it communicates. It is 

 situated but a short day's ride from Saratoga springs, 

 from which an excursion to the lake is considered a 

 matter of course. Besides the interest which is 

 excited from the association of many important his- 

 torical events connected with the lake and its shores, 

 it is peculiarly interesting from its romantic scenery. 

 It generally varies from three fourths of a mile to 

 four miles in width. The whole length is thirty-six 

 miles. The waters are discharged into lake Cham- 

 plain at Ticonderoga, by an outlet which, in the 

 course of two miles, sinks 180 feet. Lake George is 

 remarkable for the transparency of its waters. They 

 are generally very deep, but at an ordinary depth the 

 clean gravelly bottom is distinctly visible. A great 

 variety of excellent fish are caught here. Salmon 

 trout abound, and weigh from twelve to twenty 

 pounds. The lake is interspersed with a great num- 

 ber of small islands, the principal of which, Diamond 

 island, once contained a small fortification. The 

 scenery on the shores is generally mountainous. With 

 the exception of some intervals checkered with 

 fruitful cultivation, the land recedes from the lake 

 with a gentle acclivity for a few rods, and then, with 

 a bolder ascent, to an elevation of from 500 to 1500 

 feet. The best view of the lake and its environs is 

 from the southern extremity, near the remains of old 

 fort George, whence the prospect embraces the vil- 

 lage of Caldwell, with numerous small islands. The 

 calm waters of the lake are seen, beautifully con- 

 trasted with the parallel ridges of craggy mountains, 

 through an extent of nearly fourteen miles. Near 

 the southern shore are the ruins of an old fortifica- 

 tion, called fort William Henry, taken by the mar- 

 quis de Montcalm, in 1757, with its garrison of 3000 

 men, nearly all of whom were massacred by the 

 Indian auxiliaries of the French. From this spot 



