and AF is the rectangle contained by BA, AC; for it is con- Book II. tained by DA, AC, of which AD'is equal to AB; and CE M is contained by AB, BC, for BE is equal to AB; therefore the rectangle contained by AB, AC, together with the rectangle AB, BC, is equal to the square of AB. If therefore a straight line, &c. Q. E. D. IF rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the foresaid part. Let the straight line AB be divided into any two parts in the point C; the rectangle AB,BC is equal to the rectangle AC, CB, together with the square of BC. Upon BC describe a the square a 46. I. CDĖB, and produce ED to F, and A C B b 31, 1. through A draw 6 AF parallel to CD or BE ; then the rectangle AE is equal to the rectangles AD, CE; and AE is the rectangle contained by AB, BC, for it is contained by AB, BE, of which BE is equal to BC ; and AD is contained by AC, CB, for CD is equal to CB; and DB is the F D E square of BC; therefore the rectangle AB, BC is equal to the rectangle AC, CB together with the square of BC. If therefore a straight line, &c. Q. E. D. IF a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. Let the straight line AB be divided into any two parts in C; the square of AB is equal to the squares of AC, CB and to twice the rectangle contained by AC, CB.. Upon D3 Upon AB describe a the square ADEB, and join BD, and thro' C draw 6 CGF parallel to AD or BE, and thro' G draw HK parallel to AB or DE: And because CF is parallel to AD, and BD falls upon them, the exterior angle BGC is equal to the interior and opposite angle ADB; but ADB is equald to the angle ABD, because BA is equal to AD, being sides of a square; wherefore the angle CGB A С B is equal to the angle GBC; and therefore the side BC is equal to the fide CG: But CB is equal f also to GK, and CG to BK; wherefore H K the figure CGKB is equilateral : It is likewise rectangular; for CG is parallel to BK, and CB meets them the angles KBC, GCB are therefore equal to two right angles; and KBC D F E is a right angle; wherefore GCB is a right angle; and therefore also the angles CGK, GKB opposite to these are right angles, and CGKB is rectangular : But it is also equilateral, as was demonstrated ; wherefore it is a square, and it is upon the side CB: For the same reason HF also is a square, and it is upon the fide HG which is equal to AC: Therefore HF, CK are the squares of AC, CB ; and because the complement AG is equal s to the complement GE, and that AG is the rectangle contained by AC, CB, for GC is equal to CB; therefore GE is also equal to the rectangle AC, CB ; wheresore AG, GE are equal to twice the rectangle AC, CB: And HF, CK are the squares of AC, CB ; wherefore the four figures HF, CK, AG, GE are equal to the squares of AC, CB and to twice the rectangle AC, CB: But HF, CK, AG, GE make up the whole figure ADEB, which is the square of AB: Therefore the square of AB is equal to the squares of AC, CB and twice the rectangle AC, CB." Wherefore, if a straight line, &c. Q_ E. D. Cor. From the demonftration, it is manifeft, that the parallelograms about the diameter of a square are likewise squares. % 43.1. PRO P. Book 11. I PRO P. V. THE O R. a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts at the point D; the rectangle AD, DB, together with the square of CD, is equal to the square of CB. Upon CB describe the square CEFB, join BE, and thro'a 46. I. D draw 0 DHG parallel to CE or BF ; and thro' H draw 6 31 1. KLM parallel to CB or EF; and also thro' A draw AK pasallel to CL or BM: And because the complement CH is equal to the complement HF, to each of these add DM; C 43. I therefore the whole CM is equal to the whole DF; but CM is equal to AL, А С D B d 36.7 because AC is equal to L H CB ; therefore allo AL is K M equal to DF. To each of these add CH, and the wbole AH is equal to DF and CH : But AH is the rectangle contained by E G F AD, DB, for DH is equal 'to DB; and DF together with CH is the gnomon CMG; Cor. 4. therefore the gnomon CMG is equal to the rectangle AD, DB: To each of these add LG, which is equal to the square of CD; therefore the gnomon CMG together, with LG, is equal to the rectangle AD, DB, together with the square of CD: But the gnomon CMG and LG make up the whole figure CEFB, which is the square of CB : Therefore the rectangle AD, DB, together with the square of CD, is equal to the square of CB. Wherefore, if a straight line, &c. Q. Ė. D. From this proposition it is manifeft, that the difference of the squares of two unequal lines AC, CD, is equal to the rectangle contained by their sum and difference. Book II. PRO P. VI. T H E O R. IF a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half of the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to the point D; the rectangle AD, DB, together with the square of CB, is equal to the square of CD. a 46.1. Upon CD describe a the square CEFD, join DE, and thro' b 31.1. B drawb BHG parallel to CE or DF, and thro' H draw KLM parallel to AD or EF, and also thro' A draw AK parallel to CL A C B D C 36. 1. AL' is equal to CH; but d 43. I. CH is equald to HF; there L H fore alfa AL is equal to M M : K CM; therefore the whole AM is equal to the gnomon CMG: And AM is the rectangle contained by E F AD, DB, for DM is equal c Cor. 4. 20 C to DB : Therefore the gnomon CMG is equal to the rect angle AD, DB: Add to each of these LG, which is equal to the square of CB; therefore the rectangle AD, DB, together with the square of CB, is equal to the gnomon CMG and the figure LG : But the gnomon CMG and LG make up the whole figure CEFD, which is the square of CD; therefore the rectangle AD, DB, together with the square of CB, is equal to the square of CD. Wherefore, if a straight line, &c. Q. E. D. IF a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the the point C; the squares of AB, BC are equal to twice the Book II. rectangle AB, BC together with the square of AC. Upon AB describe a the square- ADEB, and construct the a 46. I. figure as in the preceeding propositions : And because AG is equal o to GE, add to each of them CK; the whole AK is b 43. 1. therefore equal to the whole CE; therefore. AK, CE are double of A с AK : But AK, CE are the gnomon B AKF together with the square CK; therefore the gnomon AKF, toge-H G K ther with the square CK, is double of AK: But twice the rectangle AB, BC is double of AK, for BK is e c Cor. 4. 2. qual to BC: Therefore the gnomon AKF, together with the square CK, is equal to twice the rectangle D E AB, BC: To each of these equals add HF, which is equal to the square of AC; therefore the gnomon AKF, together with the squares CK, HF, is equal to twice the rectangle AB, BC and the square of AC : But the gnomon AKF, together with the squares CK, HF, make up the whole figure ADEB and CK, which are the Tquares of AB and BC: Therefore the squares of AB and BC are equal to twice the rectangle AB, BC, together with the square of AC. Wherefore, if a {traight line, &c. Q. E. D. IF a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part. Let the straight line AB be divided into any two parts in the point C; four times the rectangle AB, BC, together with the square of AC, is equal to the square of the straight line made up of AB and BC together. Produce AB to D, fo that BD be equal to CB, and upon AD describe the square AEFD; and construct two figures such as in the preceeding. Because CB is equal to BD, and that CB is equal to GK, and BD to KN; therefore GK is a 24.1. equal |