Book II.

THE

ELEMENTS

OF

EUCLID.

BOOK II.

DEFINITIONS,

I.

VERY right angled parallelogram is faid to be contained
by any two of the ftraight lines which contain one of the
right angles.

II.

In every parallelogram, any of the parallelograms about a diamęter, together with the two

F

H

K

pofite angles of the parallelograms which make the gnomon.'

PROP. I. THEOR.

F there be two ftraight lines, one of which is divided into any number of parts; the rectangle contained. by the two straight lines, is equal to the rectangles contained by the undivided line, and the feveral parts of the divided line,

Book II.

a. II. I.

b. 3. I.

C. 31. I.

Let A and BC be two straight lines; and let BC be divided into any parts in the points D, E; the rectangle contained by the ftraight lines A, BC is equal to

the rectangle contained by A, BD, B

and to that contained by A, DE;
and alfo to that contained by A,
EC.

From the point B draw

D. E. C

BFG

KL H

A

at right angles to BC, and make
BG equal b to A; and thro' GF
draw GH parallel to BC; and

thro' D, E, C draw DK, EL, CH parallel to BG. then the rectangle BH is equal to the rectangles BK, DL, EH; and BH is contained by A, BC, for it is contained by GB, BC, and GB is equal to A; and BK is contained by A, BD, for it is contained by GB, BD, of which GB is equal to A; and DL is contained d. 34. 1. by A, DE, because DK, that is a BG, is equal to A; and in like manner the rectangle EH is contained by A, EC. therefore the rectangle contained by A, BC is equal to the feveral rectangles contained by A, BD, and by A, DE, and alfo by A, EC. Wherefore if there be two straight lines, &c. Q. E. D.

a. 46. I.

b. 31. I.

IF

F a ftraight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the fquare of the whole line.

Let the straight line AB be divided into A
any two parts in the point C; the rectangle
contained by AB, BC together with the
rectangle * AB, AC fhall be equal to the
fquare of AB.

Upon AB defcribe the fquare ADEB,
and thro' G draw b CF parallel to AD or
BE. then AE is equal to the rectangles AF,
CE; and AE is the fquare of AB; and AFD

C B

FE

*N. B. To avoid repeating the word Contained too frequently, the rectangle contained by two straight lines AB, AC is fometimes fimply called the rectangle AB, AC.

is the rectangle contained by BA, AC; for it is contained by Book II. DA, AC, of which AD is equal to AB; and CE 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 fquare of AB. If therefore a straight line, &c. Q. E. D.

F a ftraight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the fquare of the forefaid 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 fquare of BC.

Upon BC defcribe the fquare A C CDEB, and produce ED to F, and thro' A draw b 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

F

D

2. 46. I.

B

b. 31. I.

E

AC, CB, for CD is equal to CB; and DB is the square of BC.

therefore the rectangle AB, BC is together with the fquare of BC. Q. E. D.

IF

equal to the rectangle AC, CB
If therefore a straight line, &c.

PROP. IV. THEOR.

F a ftraight line be divided into any two parts, the fquare of the whole line is equal to the fquares of the two parts, together with twice the rectangle contained by the parts.

Let the ftraight line AB be divided into any two parts in C; the fquare of AB is equal to the fquares of AC, CB and to twice the rectangle contained by AC, CB.

Book II.

a. 46. I.

b. 31. I. C. 29. I. d. 5. I.

e. 6, I. f. 34. I.

g. 43. I.

Upon AB defcribe the fquare ADEB, and join BD, and thro' C draw CGF parallel to AD or BE, and thro' G draw HK parallel to AB or DE. and becaufe CF is parallel to AD; and BD falls upon them, the exterior angle BGC is equal to the interior

e

d

A

C

B

Gh

K

H

D

FE

and oppofite angle ADB; but ADB is equal to the angle ABD,
because BA is equal to AD, being fides of a square; wherefore the
angle CGB is equal to the angle GBC,
and therefore the fide BC is equal to
the fide CG. but CB is equal alfo f to
GK, and CG to BK; wherefore the
figure CGKB is equilateral. it is like-
wife rectangular; for CG is parallel to
BK, and CB meets them, the angles
KBC, GCB are therefore equal to two
right angles; and KBC is a right an-
gle, wherefore GCB is a right angle; and therefore also the an-
gles of CGK, GKB oppofite to these are right angles, and CGKB
is rectangular. but it is alfo equilateral, as was demonstrated;
wherefore it is a fquare, and it is upon the fide CB. for the fame
reafon HF alfo is a fquare, 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 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; wherefore AG, GE are equal to twice the rectangle AC,
CB. and HF, CK are the fquares of AC, CB; wherefore the four
figures HF, CK, AG, GE are equal to the fquares of AC, CB
and to twice the rectangle AC, CB. but HF, CK, AG, GE make
up the whole figure ADEB which is the fquare of AB. therefore
the fquare of AB is equal to the fquares of AC, CB and twice
the rectangle AC, CB. Wherefore if a straight line, &c.
Q. E. D.

COR. From the demonstration it is manifeft, that the parallelograms about the diameter of a square are likewife fquares.

IF

PROP. V. THEOR.

a ftraight line be divided into two equal parts, and alfo into two unequal parts; the rectangle contained by the unequal parts, together with the fquare of the line between the points of fection, is equal to the fquare 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.

pa

Book II.

Upon CB defcribe a the square CEFB, join BE, and thro' D a. 46. I. draw b DHG parallel to CE or BF; and thro' H draw KLM b. 31. I. rallel to CB or EF; and also thro' A draw AK parallel to CL or BM. and because the complement CH is equal to the comple- • 43• I• ment HF, to each of these add DM, therefore the whole

C

C.

A

C

D B

e

e

AH is the rectangle contained by AD, DB, for DH is equal to e. Cor. 4. 2. DB; and DF together with CH is the gnomon CMG; therefore the gnomon CMG is equal to the rectangle AD, DB. to each of thefe add LG, which is equal to the fquare of CD, therefore the gnomon OMG 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 fquare of CB. therefore the rectangle AD, DB together with the fquare of CD is equal to the fquare of CB. Wherefore if a ftraight line, &c. Q. E. D.