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a right angle, the

two

GB, HC; and through A drawb AL parallel to BD or CE, and Book I.

join AD, FC; then, because each of the angles BAC, BAG is

b 31. 1. c 30. def.

G

ftraight lines AC, AG up

on the opposite sides of AB,

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make with it at the point Ar
the adjacent angles equal to
two right angles; therefore
CA is in the same straight

A

K

d 14. 1.

lined with AG; for the fame
reason, AB and AH are in
the same straight line; and
because the angle DBC is e-
qual to the angle FBA, each
of them being a right angle,
add to each the angle ABC,

B

C

D

E

and the whole angle DBA is

L

equal to the whole FBC; and because the two fides AB, BD c 2. Ax.

are equal to the two FB, BC, each to each, and the angle

DBA equal to the angle FBC; therefore the base AD is e

qual to the base FC, and the triangle ABD to the triangle f 4.1.
FBC: Now the parallelogram BL is double of the triangle g 41. 1.
ABD, because they are upon the same base BD, and between
the same parallels, BD, AL; and the square GB is double of
the triangle FBC, because these also are upon the fame base
FB, and between the fame parallels FB, GC. But the doubles
of equals are equal to one another: Therefore the parallelo- h 6. Ax.
gram BL is equal to the square GB: And in the fame manner,
by joining AE, BK, it is demonftrated that the parallelogramı
CL is equal to the square HC: Therefore the whole square
BDEC is equal to the two squares GB, HC; and the square
BDEC is defcribed upon the straight line BC, and the squares
GB, HC upon BA, AC: Wherefore the square upon the fide
BC is equal to the squares upon the fides BA, AC. Therefore,
in any right angled triangle, &c. Q. E. D.

I angle,

PROP. XLVIII. THEOR.

F the square described upon one of the fides of a tribe equal to the squares described upon the other two fides of it; the angle contained by these two fides is a right angle.

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f

Book I.

If the square described upon BC, one of the sides of the tri angle ABC, be equal to the squares upon the other fides BA AC; the angle BAC is a right angle.

a II. I.

From the point A draw AD at right angles to AC, and make AD equal to BA, and join DC: Then, because DA i

equal to AB, the square of DA is equal

b 47. I.

to the square of AB: To each of these
add the square of AC; therefore the squares
of DA, AC, are equal to the squares of
BA, AC: But the square of DC is equal A
▸ to the squares of DA, AC, because DAC

D

c 8. g.

is a right angle; and the square of BC, by
hypothesis, is equal to the squares of BA,
AC; therefore the square of DC is equal
to the square of BC; and therefore alfo B
the fide DC is equal to the fide BC. And
because the side DA is equal to AB, and AC common to the
two triangles DAC, BAC, the two DA, AC are equal to the
two BA, AC; and the base DC is equal to the base BC; there-
fore the angle DAC is equal to the angle BAC: But DAC
is a right angle; therefore also BAC is a right angle. There-
fore, if the square, &c. Q. E. D.

C

THE

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E

BOOK II.

DEFINITIONS.

Ι.

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

right angles.

II.

In every parallelogram, any of the parallelograms about a dia

meter, together with the

E

two complements, is called A

D

a Gnomon. Thus the pa

'rallelogram HG, toge'ther with the comple'ments AF, FC, is the gno

F

mon, which is more brief H

K

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F there be two straight 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 several parts of the divided line.

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1

Book II.

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 straight lines A, BC is equal

to the rectangle contained by A, B

DEC

BD, together with that contain-
ed by A, DE, and that contained

by A, EC.

a II. I

G

A

b 3. I.

C 31. 1.

d 34. 1.

From the point B draw BF
at right angles to BC, and make
BG equal to A; and through
G draw GH parallel to BC;
and through D, E,
C draw DKF

EL, CH parallel to BG; then the

KLH

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 by A, DE, because DK, that is, d 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 several rectangles contained by A, BD, and by A, DE; and alfo by A, EC. Wherefore, if there be two straight lines, &c. QE, D.

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a 46. 1.

b 31. 1.

IF a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole | line.

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N. B. To avoid repeating the word contained too frequently, the rectangle contained by two straight lines AB, AG is sometimes simply called the rectangle AB, AC.

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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 CЕ 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.

PROP. III. THEOR.

IF a straight 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 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.

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square of BC; therefore the rectangle

E

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

PROP. IV. THEOR.

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.

D 3

Upon

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