GEOMETRY. 



403 



which form that angle, the angular point C being the 

 centre ; and it is estimated by the number of degrees 

 contained in that arc. An angle in a segment B A DC , 

 fig. 25, is that which is contained by two lines, BA, 

 CA, or BD, CD, drawn from any point A or D in 

 the arc of the segment, to the two extremities of that 

 arc B and C. An angle on a segment, or an arc, 

 is that which is contained by two lines, drawn from 

 any point in the opposite or supplemental part of the 

 circumference, to the extremities ot the arc, and 

 containing the arc between them. Thus the angle 

 BAC, fig. 25, is on the arc BC. 



29 



An angle ABC, fig. 26, at the circumference, is 

 that whose angular point or summit is anywhere in 

 the circumference. And an angle ADC at the cen- 

 tre, is that whose angular point is at the centre. A 

 right-lined figure, fig. 27, is inscribed in a circle, or 

 the circle circumscribes it, when ail the angular 

 points of the figure are in the circumference of the 

 circle. A right-lined figure, fig. 28, circumscribes 

 a circle, or the circle is inscribed in it, when all the 

 sides of the figure touch the circumference of the 

 circle. One right-lined figure, tig. 29, is inscribed in 

 another, or the latter circumscribes the former, when 

 all the angular points of the former are placed in the 

 sides of the latter. Identical figures, are such as are 

 both mutually equilateral and equiangular ; or that 

 have all the sides and all the angles of the one, 

 respectively equal to all the sides and all the angles 

 of the other, each to each ; so that, if the one figure 

 were applied to, or laid upon the other, all the sides 

 of the one would exactly fall upon and cover all the 

 sides of the other ; the two becoming as it were but 

 one and the same figure. Similar figures, are those 

 that have all the angles of the one equal to all the 

 angles of the other, each to each, and the sides about 

 the equal angles proportional. The perimeter of a 

 figure, is the sum of all its sides taken together. A 

 proposition, is something which is either proposed to 

 be done, or to be demonstrated, and is either a pro- 

 blem or a theorem. A problem is something pro- 

 posed to be done. A theorem is something proposed 

 to be demonstrated. A lemma, is something which 

 is premised, or demonstrated, in order to render wliat 

 follows more easy. A corollary, is a consequent 

 truth, gained immediately from some preceding 

 truth, or demonstration. A scholium, is a remark 

 or observation made upon something going before it. 



30 



31 



Properties of Lines. Let the line AB, fig. 30, 

 meet the line CD ; then will the two angles ABC, 

 ABD, taken together, be equal to two right angles. 

 Let the two lines AB, CD, fig. 31, intersect in the 

 point E ; then will the angle A EC be equal to the 

 angle BED, and the angle A ED equal to the angle 

 CEB. Let the line EF, fig. 32 . cut the two parallel 

 lines AB, CD; then will the angle AEF be equal to 

 the alternate angle EFD. 



34 



K 8 



35 



K B 



I 



Let the line EF, fig. 33, cutting the two lines 

 AB, CD, make the alternate angles AEF, DFE. 

 equal to each other; then will AB be parallel to 

 CD. Let the line EF, fig. 34, cut the two paralle. 

 lines AB, CD ; then will the outward angle EGB 

 be equal to the inward opposite angle GHD, on the 

 same side of the line EF ; and the two inward angles 

 BGH, GHD, taken together, will be equal to two 

 right angles. Let the fines AB, CD, fig. 35, be each 

 of them parallel to the line EF ; then shall the lines 

 AB, CD, be parallel to each other. 



36 37 38 39 40 



Let AD, fig. 36, be the one line, and AB the 

 other, divided into the parts AE, EF, FB ; then will 

 the rectangle contained by AD and AB, be equal to 

 the sum of the rectangles of AD and AE, and AD 

 and EF, and AD and FB : thus expressed, AD . AB 

 = AD . AE + AD . EF + AD . FB. If a right 

 line be divided into any two parts, the square of the 

 whole line is equal to both the rectangles of the 

 whole line and each of the parts. Let the line AB, 

 fig. 37, be the sum of any two lines AC, CB ; then 

 will the square of AB be equal to the squares of AC, 

 CB, together with twice the rectangle of AC . CB. 

 That is, AB" = AC" -f- CB* + 2 AC . CB. If a line 

 be divided into two equal parts, the square of the 

 whole line will be equal to four times the square of 

 half the line. Let AC, BC,fig. 38, be any two lines, 

 and AB their difference ; then will the square of AB 

 be less than the squares of AC, BC, by twice the 

 rectangle of AC and BC. Or, AB = AC" + BC 

 2 AC . BC. Let AB, AC, fig. 39, be any two 

 unequal lines ; then will the difference of the squares 

 of AB, AC, be equal to a rectangle under their sum 

 and difference. That is, AB 8 _ AC* =TB + AC . 

 AB AC. If AB, AC, AD, &c., fig. 40, be lines 

 drawn from the given point A, to the indefinite line 

 BE, of which AB is perpendicular; then shall the 

 perpendicular AB be less than AC, and AC less than 

 AD, &c. Let the four lines A, B, C, D, be propor- 

 tionals, or A : B : : C : D ; then will the rectangle 

 of A and D be equal to the rectangle of B and C ; 

 or the rectangle A X D = B x C. 



41 42 43 



\ 



Properties of Triangles. In the two triangles 

 ABC, DEF, fig. 41, if the side AC be equal to the 

 side DF, and the side BC equal to the side EF, and 

 the angle C equal to the angle F ; then will the two 

 triangles be identu'tl, or equal in all respects. Let 

 the two triangles ABC, DEF, fig. 41, have the angle 

 A equal to the angle D, the angle B equal to the 

 angle E, and the side AB equal to the side DE ; then 

 2c 2 



