PRO P. VI. THE OR.

a right line be divided into equal parts, and another line add

as one fide of the rectangle, and the added line for the other fide, together with the fquare of half the line, are equal to the fquare of the half and added line, as one fide of the square.

Let the right line AB be bifected in C, and BD added to it, the rectangle under AD, DB, together with the fquare of BC, are equal to the fquare of CD.

Defcribe the fquare CEFD; conftruct the figure; and compleat the parallelogram under AC, CL.

BOOK II.

Then the parallelograms AL, CH, are equal; but CH is e- a 36. r. qual to HF; add BM to both; then CM is equal to BF; add b 43. 1. AL to both; then AM is equal to the gnomon CMG. To each add LG, that is, the fquare of CB ; then AM, LG, arec Cor. 4. equal to the gnomon CMG, and LG; that is, the rectangle under AD, DB, for DM is equal to DB, together with the fquare CB, are equal to the fquare of CD. Wherefore, &c.

I'

PRO P. VII. THEOR.

F a right line be any how cut, the fquare of the whole line, and one of the parts, is equal to twice the rectangle contained by the whole line, and faid part, together with the fquare of the other part.

Let the right line AB be any how cut in C, the fquares of AB, BC, are equal to twice the rectangle under AB, BC, and the fquare of AC.

b 43. 1. and AX. 2. 1.

Upon AB describe the fquare ADEBa, and conftruct the fi- a 46. I. gure; then, because the rectangle AF is equal to CE, and AF, CE, together, are equal to twice AF, that is, equal to the gnomon AFK, together with the square of CB, that is, CF; add c Cor. 4. HK to both; then twice AF, and HK, are equal to the gnomon AFK, and the fquares of AC, BC; that is, to the fquares of AB, BC. Wherefore, &c,

PROP

Book II.

b 36, I. C 29. I.

d

34. 1. and

def. 30. 1.

IF

PRO P. VIII. THE OR.

a right line be cut into two parts, four times the rectangle under the whole line, and one of the parts, together with the fquare of the other part, are equal to the fquare of the whole line, and the first part taken as the fide of the fquare.

Let the right line AB be any how cut in C, four times the rectangle under AB, BC, together with the fquare of AC, are equal to the fquare of AD; that is, AB produced to D, so that BD equal BC. Upon AD defcribe the fquare AEFD, and conftruct the double figure. Then, because BN, GR, are a Cor. 4. fquares, and CK, BN, are equal parallelograms ; but the fides CB, BK, are equal, and CBK is a right angle, for it is equal to BDN; therefore CK is a fquared. For the same reason, KO is a fquare; therefore CK, BN, GR, KO, are each squares; but they are conftitute upon equal right lines; therefore equal to one another, and, together, quadruple KC. But the rectangle AG is equal to MP, and PL to RF; but MP is equal to PLf; therefore the four rectangles are quadruple AG; and the four fquares and four rectangles quadruple the rectangle AK, that is, the rectangle under AB, BC; add the fquare XH, that is, the fquare of AC; then four times AK, that is, four times the rectangle under AB, BC, together with the fquare of AC, are equal to the fquare of AD. Wherefore, &c.

43. I.

a Cor. 32.

1.

b 47. I.

IF Fa right line be cut into two equal parts, and into two unequal parts, the fquares of the two unequal parts are double the fquare of the half line, and double the fquare of the intermediate part.

Let the right line AB be cut equally in C, and unequally in D, the fquares of AD, BD, are double the fquares of AC, CD.

For, through C draw CE, at right angles, to AB, and equal to AC, or CB; join EA, EB; through D draw DF parallel to to CE, and FG through F, parallel to AB; join AF.

Then, because AC is equal to CE, and the angle ACE a right angle, the angles AEC, EAC are each half right angles", and the fquares of AC, CE double the fquare of AC; but the fquare of AE is equal to the fquares of AC, CE; therefore,

double the fquare of AC. For the fame reafon, the angles CEB, Book II. EBC are each half right angles; but the angle EGF is a right angle; therefore GFE is half a right angle; therefore the fides 29. 1. EG, GF are equal 4; but the fquare of EF is equal to the fquares d.6. 1. of EG, GF, or double the fquare of GF or CD; but the fquares e 34. 1. of AE, EF are equal to the fquare of AF, for the angle AEF is a right angle; but the fquares of AE, EF are double the fquares of AC, CD; therefore the fquare of AF is double the fquares of AC, CD; but the angle DFB is half a right angle ; for it is equal to CEB; therefore, DFB, DBF are each half right angles; therefore FD, DB are equal; but the fquare of AF is equal to the fquares of AD, DF 5, or DB; therefore, the fquares of AD, DB are double the fquares of AC, CD. Wherefore, &c.

IF

PRO P. X. THE OR.

a right line be cut into two equal parts, and another right line added to it, the fquare of the whole and added line taken as one line, and the fquare of the added line, are double the Square of the half line, and double the fquare of the half and added line, taken as one line.

Let the right line AB be bifected in C, and BD added to it, the fquares of AD, DB are double the fquares of AC, CD.

For, from the point C, draw CE perpendicular to AB, and equal to AC or CB; join AE, EB; through E, draw EF parallel to AD; and through D, draw DF parallel to CE.

с

Becaufe AC is equal to CE, and the angle ACE a right angle, each of the angles AEC, EAC are half right angles; and therefore the fquare of AE is equal to the fquares of AC, CE, or double the fquare of AC'. For the fame reason, CEB, CBE a 47. x. are each half right angles; therefore AEB is a right angle; but the angles FEC, ECD are equal to two right angles, and ECD b 29. 1° is a right angle; therefore CEF is likewife a right angle; therefore CF is a rectangle; therefore, the angles DFE, FEC are equal e Def, 1. to two right angles ; therefore, DEF, FEB are lefs than two right angles; therefore, FD, EB will meet one another, which let be a. Cor. 17. in G. But CEF is a right angle, and CEB half a right angle; there- 1. fore, FEB is half a right angle, and EGF is likewife half a right angle; therefore, EF is equal to FG, and the fquare of EG equal e 6. 1. to the fquares of EF, FG 2, or double the fquare of EF, or CD; therefore, the fquares of AE, EG are double the fquares of AC, CD; but the fquare of AG is equal to the fquares of AE, EG,

and

d.

Book II and likewife equal to the fquares of AD, DG; for the angles AEG, and ADG are each right ones; therefore, the fquares of AD, DG, or DB its equal, are double the fquares of AD, CD. Wherefore, &c.

a 46. I.

b 6.
c Conft.

d 47. I.

a 12. 1,

b 47. I.

T°

PRO P. XI. PRO B.

O cut a given right line fo, that the rectangle contained under the whole line, and one of the parts, be equal to the Square of the other part.

Upon any given right line, as AB, defcribe the square ABDC; bifect AC in E; join EB, and produce EA to F; make EF equal to EB; upon AF, defcribe the fquare FGHA, and produce GH to K; then AB is fo cut in the point H, that the rectangle under AB, BH is equal to the fquare of AH.

For the rectangle under CF, FA, together with the fquare of AE, is equal to the fquare of EF; but EF is equal to EB, and the fquare of EB is equal to the fquares of BA, AEd; therefore, the rectangle under CF, FA, together with the square of AE, are equal to the fquares of BA, AE. Take the fquare of AE from both, there remains the rectangle under CF, FA, that is, the rectangle under CF, FG, that is, FK, equal to the fquare of AD. Take AK from both, there remains FH equal to HD; but FH is the fquare of AH, and HD the rectangle under AB, BH, for BD is equal to AB. Wherefore, &c.

IN

PRO P. XII. THE OR.

Nevery obtufe angled triangle, the fquare of the fide fubtending the obtufe angle, is greater than the fquares of the fides containing the obtufe angle, by twice a rectangle under one of the fides containing the obtufe angle, and that part of the fide produced, lying betwixt the obtufe angle, and perpendicular let fall from the oppofite angle.

Let BAC be the obtufe angle of the triangle ABC; produce the fide CA till it meet the peopendicular BD, let fall from the point B. The fquare of BC is greater than the fquares of BA, AC, by twice the rectangle under CA, AD.

For the fquare of BC but the fquare of DC is

is equal to the fquares of BD, DC; equal to the fquares of DA, AC, and

C

twice the rectangle under AD, AC; but the fquare of AB Book II. is equal to the fquares of BD, DA; therefore the fquare of BC is equal to the fquares of BA, AC, and twice the rectangle c 4 under DA, AC; therefore the fquare of BC is greater than the fquares of BA, AC, by twice the rectangle under DA, AC. Wherefore, &c.

b 47. I.

PRO P. XIII. THE OR.

Nevery acute angled triangle, the fquare of the fide fubtending the acute angle, is less than the fquares of the fide containing the acute angle, by twice a rectangle contained under one of the fides about the acute angle, and that part of the fide lying between the acute angle and the perpendicular let fall from the oppofite angle.

Let B be an acute angle in the triangle ABC; from the angle A let fall the perpendicular AD, cutting BC in D; a 12. r. the fquare of AC is lefs than the squares of AB, BC, by twice the rectangle under CB, BD.

C 7.

For the fquare of AC is equal to the fquares of AD, DC; b 47. 12 and the fquare of AB is equal to the fquares of AD, DB b; but the fquares of BC, BD, are equal to twice the rectangle under BC, BD, together with the fquare of DC; therefore the c fquares of AB, BC, are equal to the fquares of AD, DC, and twice the rectangle under CB, BD; but the fquare of AC is equal to the fquares of AD, DC; therefore the fquare of AC is lefs than the fquares of AB, BC, by twice the rectangle under CB, BD. Therefore, &c.

PRO P. XIV. P R O B.

O make a fquare equal to a given right lined figure.

Make the rectangle BCDE equal to a given right lined figure A; If BE be equal to ED, then BCDE is a fquare; a 45. 1. and what was required is done. If not, produce BE to F; make EF equal to ED, and bifect BF in G ; with the center b 10. 1. G, and diftance GB, defcribe a femicircle BHF; produce DE to H, and join GH.

E

Ther