UPON

PONa given straight line (AB) to defcribe a fquare (AD).

Given

The ftraight line AB.

Sought

A fquare made upon the straight line AB.

Refolution

1. At the point A, erect upon the straight line AB the perpendi-
cular AK.

=

P. 11. B.I..

P. 3. B. 1.

2. From the ftraight line AK cut off a part AC to AB.
3. Thro' the point C, draw the ftraight line CO plle to
4. And thro' the point B, draw the straight line BD plle to AB, P. 31. B. 1.

which will cut CO fomewhere in D.

DEMONSTRATION.

BECAUSE in the figure AD the oppofite fides AB, CD, & AC, BD,

1. The figure AD is a Pgr.

are plle (Ref. 3 & 4.).

2. Confequently, the oppofite fides AB, CD, & AC, BD, are to one

D. 35. B. 1.

But AC is to AB (Ref. 2.).

P. 34. B. 1.

another.

3. Confequently, the four fides AB, CD, AC, BD, are = to one ano

ther.

Again, because the ftraight lines AB, CD, are plle (Ref. 3.).

4. The interior oppofite VA & ACD, are to two L.

But the A being a (Ref. 1.).

5. It is evident, that V ACD is alfo a L. Moreover, because AD is a Pgr (Arg. 1.).

6. The oppofite are to one another.

Ax. 1.

P. 29. B. 1.

C. N.

P. 34. B. 1.

7. Wherefore, the V BDC & B oppofite to the right VA & ACD, are alfo L.

The figure AD being therefore an equilateral Pgr (Arg. 3.), & rectangular (Arg. 7.).

8. It follows, that this figure AD described upon the ftraight line AB,

is a fquare.

D. 30. B. 1.

Which was to be done.

COROLLARY I

EVERY parallelogram, that has two equal fides AB, AC, including a right

angle, is a square; for drawing thro' the points C & B the ftraight lines CD, BD, parallel to the two fides AB, AC, the Square AD will be defcribed (D. 30. B. 1.).

Evi

COROLLARY II.

VERY parallelogram that has one right angle, has all its angles right angles. For fince the oppofite angles A & BDC, are equal (P. 34. B. 1.), & the angle A is a right angle, the angle BDC will be also a right angle: moreover, the lines AB, CD, & AC, BD, being parallels; the interior angles A & ACD, likewife A & B, are equal to two right angles (P. 29. B. 1.); but the angle A being a right angle, it is manifeft that the angles ACD & B, are also right angles.

COROLLARY III.

THE squares described on equal straight lines, are equal to one another, & re

ciprocally, equal fquares are defcribed on equal ftraight lines.

I

PROPOSITION XLVII. THEOREM XXXIII.

IN any right angled triangle (ABC); the fquare which is defcribed upon

the fide (AC) fubtending the right angle, is equal to the squares made upon the fides (AB, BC,) including the right angle.

Hypothefis.

Thefis.

The AABC is Rgle, or V ABC is a L. The of the fide AC is

1. On the three fides
AG, AM, CD.

2, Thro' the point B,

3. From the point B

AB, together with the

Preparation.

AC, AB, BC, describe (Fig. 1.) the

draw the ftraight line BH plle to CG.

to the of of BC.

P. 46. B. 1.

P. 31. B. 1.

to the point F, draw the straight in BF. Pof. 1. 4. From the point C to the point N, draw the straight line CN.

BECAUS

DEMONSTRATION,

ECAUSE the figure AM is a (Prep. 1.).

1. The V ABM is a L.

But ABC being alfo a L (Hyp.).

2. The two adjacent V ABM, ABC, are to two L.

The ftraight lines MB, BC, which meet on the oppofite fides of the

line AB at the point B, making with this ftraight line AB the fum of

the adjacent VABM, ABC, to two (Arg. 2.).

3. Thefe ftraight lines MB, BC, are in one and the fame straight line MC, P. 14. B. 1. which is plle to NA.

In like manner it may be demonftrated.

4. That AB, BD, are in one & the fame ftraight line AD, which is

plle to CE.

Moreover, becaufe AG, AM, are (Prep. 1.).

5. The V FAC, NAB, are to one another, (being right angles) & the fides AF, AC, & AB, AN, are alfo to one another.

Therefore, if to thofe equal FAC, NAB, V CAB be added.

P. 28. B. 1.

D. 30. B. 1.

Since then in the ▲ AFB, ACN, the fides AF, AB, & AC, AN, are

Ax. 2.

But the AFB & the Pgr AH, are upon the fame bafe AF & between the fame plles AF, BH, (Prep. 2.).

8. From whence it follows, that the Pgr AH is double of the ▲ AFB. P. 41. B. 1, AM being upon the fame bate AN,

Likewife, the AACN & the

and between the fame plles AN, MC, (Arg. 3.).

9. The AM is double of the AACN. The ▲ AFB, ACN, being therefore

to one another (Arg. 7.). and the Pgr AH & the AM their doubles (Arg. 8 & 9.).

10. It follows, that the Pgr AH is to the AM.

In the fame manner, by drawing (Fig. 2.) the lines BG, AE, it is demonftrated, that the Pgr CH is to the CD.

AG.

11. But the Pgr AH, together with the Pgr CH, form the
12. Wherefore, this AG is to the fum of the AM & CD.
But fince the AG is the made upon the fide AC, & the AM
and CD the upon the fides which include the L ABC.
13. The made upon the fide AC is to the

taken together.

made upon AB & BC

Which was to be demonstrated.

A

H

B

PROPOSITION XLVIII.

C

THEOREM XXXIV.

F the fquare defcribed upon one of the fides (CA) of a triangle (CBA) be equal to the fquares defcribed upon the other two fides of it (AB, BC,); the angle (ABC) included by these two fides (AB, BC,), is a right angle.

Hypothefis.

of AB,
Preparation.

Thefis.

The ABC included by the fides AB, BC, is L.

1. At the point B, in the straight line BA, erect the perpendi

cular BH.

2. Make BH = BC.

P. 11. B. 1.

P. 3.

3. From the point H to the point A, draw the ftraight line HA. Pof. 1.

B. 1.

SP. 45. B. 1.
Cor. 3.

to the of AB & BC. (Prep. 1.).

3. It follows, that the of HA is to the of AB & BH.
Since then the of CA is to the of AB & BC (Hyp. 1.), the
of HA to the of AB & BH (Arg. 3.), & the

=

Ax. 2.

P. 47. B. 1.

of AB & BH,

Ax. 1.

are = to the ☐ of AB & BC, (Arg. 2.).

But in the A CBA, HBA, the fide CA is to the fide
(Arg. 5.), AB is common to the two ▲, & the base BC is to the
bafe BH (Prep. 2.).

6. Wherefore, the V ABC, ABH, included by the equal fides AB, BC,
and AB, BH, are to one another.

But the ABH is a L (Prep. 1.).

P. 8. B. 1.

Which was to be demonftrated,

7. Confequently, the V ABC will be alfo a L.