COR. A polygon may be converted into an equivalent rectangle, having one

of its sides of a given length.

ELEMENTS

OF

GEOMETRY.

BOOK II.

DEFINITIONS.

1. EVERY right angled parallelogram, or rectangle, is said to be contained by any two of the straight lines which are about one of the right angles. "Thus the right angled parallelogram AC is called the rectangle contain"ed by AD and DC, or by AD and AB, &c. For the sake of brevity, "instead of the rectangle contained by AD and DC, we shall simply say "the rectangle AD. DC, placing a point between the two sides of the "rectangle."

A. In Geometry, the product of two lines means the same thing as their rectangle, and this expression has passed into Arithmetic and Algebra, where it serves to designate the product of two unequal numbers or quantities, the expression square being employed to designate the product of a quantity multiplied by itself.

The arithmetical squares of

1, 2, 3, &c. are 1, 4, 9, &c.
So likewise the square de-
scribed on the double of
a line is evidently four
times the square described
on a single one; on a triple
line nine times that on a
single one, &c.

2 In every parallelogram, any of the
parallelograms about a diameter, to-
gether with the two complements, is
called a Gnomon. "Thus the paral-

66

66

lelogram HG, together with the "complements AF, FC, is the gnomon of the parallelogram AC. This 66 gnomon may also, for the sake of brevity, be called the gnomon AGK

⚫or EHC."

PROP. I. THEOR.

If 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.

Let A and BC be two straight lines; and let BC be divided into any parts in the points D, E; the rectangle A.BC is equal to the several rectangles A.BD, A.DE, A.EC.

From the point B draw (Prop. 11.1.) BF at right angles to BC, and make BG equal (Prop. 3. 1.) to A; and through G draw (Prop. 31. 1.) GH parallel to BC; and through D, E, C, draw DK, EL, CH parallel to BG; then BH, BK, DL, and EH are rectangles, and BH= BK+DL+EH.

B

G

But BHBG.BC= A.BC, because
BG=A: Also BK = BG.BD=A.BD,
because BG=A; and DL-DK.DE= F
A.DE, because (34. 1.) DK=BG=A.

DEC

KL H

A

In like manner, EH=A.EC. Therefore A.BC=A.BD+A.DE+A.EC; that is, the rectangle A.BC is equal to the several rectangles A.BD, A.DE, A.EC.

SCHOLIUM.

The properties of the sections of lines, demonstrated in this Book, are easily derived from Algebra. In this proposition, for instance, let the segments of BC be denoted by b, c, and d; then, A(b+c+d)=Ab+Ac+Ad.

PROP. II. THEOR.

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.

Let the straight line AB be divided into any two parts in the point C; the rectangle AB.BC, together with the rectangle AB.AC, is equal to the square of AB; or AB.AC+AB.BC=AB2.

On AB describe (Prop. 46. 1.) the square ADEB, and through C draw CF (Prop. 31. 1.) parallel to AD or BE; then AF+CE=AE. But AF-AD.AC=AB.AC, because AD AB; CE=BE.BC=AB.BC; and AE=AB2. Therefore AB.AC+AB.BC=AB2.

SCHOLIUM.

A

D

C B

FE

This property is evident from Algebra: let AB be denoted by a, and the segments AC, CB, by b and d, respectively; then, a=b+d; therefore, multiplying both members of this equality by a, we shall have a2ab+ad

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 aforesaid part.

Let the straight line AB be divided into two parts, in the point C; the rectangle AB.BC is equal to the rectangle AC.BC, together with BC2.

Upon BC describe (Prop. 46. 1.) the square CDEB, and produce ED to F, and through A draw (Prop. 31. 1.) AF parallel to CD or BE; then AE=AD +CE.

But AE AB.BE = AB.BC, because BE-BC. So also AD=AC. CD=AC.CB; and CE=BC2; therefore AB.BC=AC.CB+BC2.

A

C

B

F D

E

SCHOLIUM.

In this proposition let AB be denoted by a, and the segments AC and CB, by b and c; then a=b+c: therefore, multiplying both members of this equality by c, we shall have ac=bc+c2.

PROP. IV. THEOR.

If 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, that is, AB2=AC2+CB2+2AC.CB.

Upon AB describe (Prop. 46. 1.) the square ADEB, and join BD, and through C draw (Prop. 31. 1.) CGF parallel to AD or BE, and through G draw HK parallel to AB or DE. And because CF is parallel to AD, and BD falls upon them, the exterior angle BGC

B

C

ing sides of a square; wherefore the angle H
CGB is equal to the angle GBC; and there-
fore the side BC is equal (6. 1.) to the side
CG; but CB is equal (34. 1.) also to GK and
CG to BK; wherefore the figure CGKB is
equilateral. It is likewise rectangular; for
the angle CBK being a right angle, the other
angles of the parallelogram CGKB are also right angles
Wherefore CGKB is a square, and it is upon the side CB.

D

F E

(Cor. 46. 1.)

For the same

reason HF also is a square, and it is upon the side HG, which is equal to AC: therefore HF, CK are the squares of AC, CB. And because the complement AG is equal (43. 1.) to the complement GE; and because AG=AC.CG=AC.CB, therefore also GE=AC.CB, and AG÷GE= 2AC.CB. Now, HF-AC2 and CK=CB2; therefore, HF+CK+AG +·GE=AC2+CB2+2AC.CB.

But HF+CK+AG+GE=the figure AE, or AB2; therefore AB2=

AC2+CB2+2AC.CB.

COR. From the demonstration, it is manifest that the parallelograms about the diameter of a square are likewise squares.

SCHOLIUM.

This property is derived from the square of a binomial. For, let the two parts into which this line is divided be denoted by a and b; then, (a+b)2 =a2+2ab+b2.

PROP. V. THEOR.

If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D; the rectangle AD.DB, together with the square of CD, is equal to the square of CB, or AD.DB+CD2= CB2.

A

C

D B

L

IC M

Upon CB describe (Prop. 46. 1.) the square CEFB, join BE, and through D draw (Prop. 31. 1.) DHG parallel to CE or BF; and through H draw KLM parallel to CB or EF; and also through A draw AK parallel to CL or BM: And because CH=HF, if DM be added to both, CM=DF. But AL=(36. 1.) CM, therefore AL =DF, and adding CH to both, AH =gnomon CMG. But AH = AD. DH=AD.DB, because DH = DB (Cor. 4. 2.); therefore gnomon CMG =AD.DB. To each add LG-CD2, then, gnomon CMG+LG=AD.DB +CD2. But CMG+LG=BC2; therefore AD.DB+CD2=BC2.

E

G F

"COR. From this proposition it is manifest, that the difference of the squares of two unequal lines, AC, CD, is equal to the rectangle contain"ed by their sum and difference, or that AC2-CD2=(AC+ČD) (AC— · CD)."

SCHOLIUM.

In this proposition, let AC be denoted by a, and CD by b; then, AD= a+b, and DB=a-b; therefore, by Algebra, (a+b)×(a—b)—a2—b2; that is, the product of the sum and difference of two quantities, is equivalent to the difference of their squares