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MATHEMATICS. PLANE GEOMETRY. 



[BOOK it. PROP. r. v. 



CHAPTER II. 

 ELEMENTS OF EUCLID. BOOK IL 



|iKKIMT!iiNs. 



11 I. 



T> B C. 



Every right-angled parallelogram, or rectangle, is said 

 to be contained by any two of the sides which contain 

 one of its angles : tliat is, by any two adjacent sides. 



Thus the rectangle A C to said to be contained by the adjacent sides 

 All, BC, or by AD, DC, and is often called, for brerity. "the 

 reeumgie A B liC;" or "the rectangle A D DC." And wtirn the 

 adjacent side* are made equal to two detached lines, each to each, 

 ilu common to refer to the rectangle as contained by the lines to 

 which the adjacent sides hare been made equal. Thus the rectangle 

 B 11. in Proposition 1. following, U referred to u the rectangle 

 A- I)C, because 11 O = A. 



II. 



In every parallelogram, either 

 of the parallelograms about, a x 

 diagonal, together with the two 

 complements, is called a gnomon. 

 Thus the parallelogram H G, to- 

 ;_;. -tin -r with the complements AF, 

 I a gnomon ; it is expressed 

 by the tftree letters A G K, or 

 E H C, at the opposite vertices of 

 the parallelograms, which make the gnomon. 



PROPOSITION I. THEOREM. 



If there be two straight lines (A and B C), one of which 

 (B C) is i/iVi'f/. / into any number of parts, the rectangle 

 contained by the two straight lines is equal to the rect- 

 angles contained by the undivided line and the several 

 parts (B D, D E, ic. ) of the divided line. 



From B draw BF at right angles toBC,* 

 3 i. and make B G A.t Though 

 G draw G H parallel to B C, and 

 through D, E, C, draw D K, E L, C H 

 31 I. parallel to BG.* Then the 

 rectangle BH = BK + DL+ K II ; 

 but B H is contained by A and B C, 

 + Const, for BG = A,t and 1! K is 



contained by A and B D, for B G = A ; 



also, D L is contained by A and D E, 

 MI. for D K = BG* = A ; and, in like manner, 

 K 1 1 in contained by A and E C, /. the rectangle A' B C 

 A-BD + A-DE + A- EC, however many divisions 

 there may be in B C ; . ' . if there be two straight lines, 

 tic. Q. E. D. 



PROPOSITION II. THEOREM. 



If a straight line (A B) be divided into any two parts (A C, 

 C 11) the rectangles contained by the whole and each of the 

 parts, are together equal to the square of the whole line : 

 that is, A B-A C + A B B C - A B. 



46 I. Upon A B describe the square A E,* and 

 through C draw C F parallel to A D or 

 31 i. B E.f Then A E = A F + 

 Const. C E ; but A E is A B 8 ,* and 

 A F - A D-AC - A B-A C, because 

 AD-AB; also CE-ABBC, for 

 BE-AB, .-. AB-AC + ABBC = 

 A I! J ; .-. if a straight line. <tc. 

 Q. E.D. 



Norm. Th proposition might hare been made a corollary to the 

 preceding, since it U only that particular caw of the former in 

 which Ihr two proposed lines (A, IK.') are equal. It Isslwinln n.u> 

 that the restriction of the number of parts to itco U unnecessary. 



PROPOSITION IIL THSORBM. 



JT/ a >lr,,,.ihl line (A B) be divided into any tiro parts 

 (A C, C B), the rectangle contained by the whole and one 

 of the parti is equal tv the rectangle contained by the 



L H 



two parts, together irith the square of the aforesaid 

 part : that is, A B B C - A C C B + li C' 2 . 



, c I'pon B C describe the square 



.|.il. CE ;* prolong K D t<i F, 

 and through A draw A F parallel 

 + 311. to CD or BE.f Then 

 A E = A 1) + C E ; but A E is 

 the rectangle A B B C, for B C = 

 B E ; and A D is the rectangle 

 " : A.O-OB, for CD= CB; also 

 A 1! 1!C = AC-CB + BC 2 ; .-. if a 



D 



CE is BC, .'. 



strait/lit line, <tc. 



Q. E. D. 



/ 



PROPOSITION IV. THEOREM. 



If a straight line (A B) be divided into any two parts 

 (A C, C B) the square of the whole line is equal tu the 

 squares of the two parts, together with twice the rect- 

 angle contained by the parts : that is, A B 8 = A C 2 + 

 CB' + twice AC-CB 



461. Upon AB describe the square AE:* draw 

 B D ; through C draw C G F parallel to A D or B E, and 



+ 31 1. through G draw H K parallel to A B or D E ;t 

 then C F, A D being parallels, the angle B (i ' 



291. ADB:* but ADB = ABD, because A K 



+ 51. AD.f AE being a square. .-. C G B = C I: < i, 



6 i. .;. CB = CG;* but CB=GK, andCG=l! K, 

 ' CK is ,,. It jg likewise rectangular, for 



CG, BK being parallels, the angles 

 B KBC, GCB are = two ri-ht :ui- 

 + 29 1. gles :t but KBC is a right 

 K angle, .-. GCB is a right in 

 .-. CGK, GKB, opposite to these, 

 3l. are right angles,* .'. C K is 

 lular ; and since it is also equi- 

 lateral, it is a square the square of 

 C B. For a similar reason H F is 

 the square of H G, or of A C, /. H F, C K are 

 AC 2 , Cfi 2 . And because the complement AG = 

 + 4S I. the complement G E,t and AG is the rectangle 

 AC-CB, for CG = CB, .-. GE = ACCB, /. A G + 

 GE=twice AC-CB ; and HF + CK=AC 2 + ('!,. 

 /. H F + C K + A G -f G E = A ('-' + C K' + twice 

 AC-CB; that is, A B*= A C 8 + C B 8 -f twice A C r I !. 

 . . if a straight line, <tc. Q E D. 



COR. From this it is manifest, that parallelograms 

 about the diagonal of a square are likewise squares. 



PROPOSITION V. THEOREM. 



If a straight line (A B) be divided into two equal parts (in 

 C), and oho into two unequal parts (in ])), the rectangle 

 contained by the unequal parts, together with the */ 

 f (C 1 )) the line between the points of section, is 

 to the square of half the line : that is, A D -I) B + f U 2 

 = CB 2 . 



46 1. Upon C B describe the square C F :* draw 



C D B B E ; through D draw 



D H G parallel to C E 

 t 31 1. or B F,t and 

 throii','h A draw A K 

 parallel to C L or B M. 

 .iipleinent CH = 

 43 i. H F :* to each 

 of these add D M, .-. < M 

 - D F : but C M = 



A L,t since A C = C B, .-. A L = D F ; t,. eaeh 

 <.f these add U H, /. A H - D F + C H : but A H is the 

 4 n. cor. rectangle A DP U, for 1MI---DH:* and 

 DF + CH is theBuomoii CM G,.-. CMC - ADDB: 



K 



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