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PROP. XIV. PROB.

To describe a square that shall be equal to a given rectilineal figure.

Let A be the given rectilineal figure; it is required to describe a square that shall be equal to A.

Describe the rectangular parallelogram BCDE

equal to the rectilineal figure A.

If then the sides of it BE, ED are equal to one another,

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BOOK II.

a xlv. 1.

to F,

and make EF equal to ED

and bisect BF in G;

and from the centre G, at the distance GB, or GF,

describe the semicircle BHF,

and produce DE to H,

and join GH:

Therefore because the straight line BF is divided

into two equal parts in the point G, and into two unequal at E,

the rectangle BE, EF, together with the square of EG,

is equal to the square of GF:

But GF is equal to GH:

therefore the rectangle BE, EF, together with the square

of EG,

is equal to the square of GH:

b v.2.

But the squares of HE, EG are equal to the square of vii. 1. GH:

Therefore the rectangle BE, EF,

I

BOOK II. together with the square of EG,

is equal to the squares of HE, EG:
Take away the square of EG,

which is common to both;

and the remaining rectangle BE, EF is equal to the square of EH:

But the rectangle contained by BE, EF is the parallelogram

BD,

because EF is equal to ED;

therefore BD is equal to the square of EH;

but BD is equal to the rectilineal figure A;

therefore the rectilineal figure A is equal to the square of EH.
Wherefore a square has been made equal to
the given rectilineal figure A;

viz. the square described upon EH.
Which was to be done.

[In constructing this figure it will be necessary to divide the rectilineal figure A into triangles, and construct a rectangle equal to one of them, and then apply to one of the sides of that rectangle another rectangle which shall be equal to another of the triangles, and so on till a rectangle has been made equal to the whole rectilineal figure. Then proceed as directed in the Proposition. The learner should here draw correctly the figure which he was directed to omit at the first reading of i. 45.]

DEDUCTIONS FROM BOOK II.

THERE is not much of interest or importance in the Deductions usually appended to this book. The learner may exercise himself with the following, or proceed immediately to the Third Book.

(1.) To divide a straight line so that the rectangle of the parts shall be equal to

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A

(2.) If a straight line be drawn from the vertical angle of a
triangle bisecting the base, the squares of the two sides of the
triangle are together
double of the squares
of the bisecting line,
and of half the base.
The square of AC is
greater than the squares
of AD, DC, by twice

the rectangle AD,

DE (II. 12.).

E

B

Or, in other words, is equal to the squares of AD, DC, together with twice the rectangle AD, DE;

that is, is equal to the squares of AD, DC,

together with twice the rectangle BD, DE, for AD is
equal to BD.

BOOK II.

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The square of BC is equal to the squares of BD, DC,

less twice the rectangle BD, DE (II. 13.).

Add these equals together, and the Proposition is proved.

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(4.) The squares described upon the sides of a parallelogram

are together

equal to the

squares upon

its diameters.

Remember

that the diago

nals of a paral- A

lelogram bisect each other.

B

D

Then apply Ex. 2. to both the triangles ABC, ADC.
Then use Ex. 3.

C

(5.) In any triangle, if a perpendicular be drawn from one
of the angles to the opposite
side, the difference of the
squares described upon the
sides is equal to the dif-
ference of the squares on the
segment of the base.

The square on AC is equal to
the squares on AO, OC.

A

B

The square on BC is equal to the squares on BO, OC.
Subtract these equals from each other.

(6.) Prove that if a straight line be divided into two parts, the rhombus described upon the whole line shall be equal to those which are described upon the parts, and have their angles equal to its angles, together with twice the

parallelogram whose sides are equal to

the parts, and whose angles are equal to

E

B

K

H

BOOK II.

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In other respects construct the figure as in Prop V., and

the proof is nearly the same with that in Prop. V.

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