IF

LEMM A.

F a straight line (EF), meeting two straight lines (LN, CD,) fituated in the fame plane, makes the alternate angles (p+n & o) unequal; thofe two ftraight lines (LN & CD,) being continually produced, will at length meet in (M), upon that fide on which is the leffer of the alternate angles (0).

Preparation.

For fince Vpn is fupofed > Vo.
1. There may be made in the greater

p+n, on the ftraight

P. 23. B. 1.
Pof. 2.

line EF, at the point G, an angle n = V o. 2. And AG may be produced at will to B.

BECAUS

DEMONSTRATION.

ECAUSE the two lines AB, CD, are cut by a third EF, so that

the alternaten & are to one another (Prep. 1.).

1. Those two lines AB, CD, are plle.

But the line LN cuts one of the two plles, viz. AB in G.

P. 27. B. 1.

2. Therefore, if produced fufficiently, it will cut alfo the other CD fome-
where in M, upon that side on which is the leffer of the alternate Vo*. C. N.

Which was to be demonftrated.

COROLLARY.

WHEN Vo<Vp+n, the two interior angles o+m are ne

ceffarily two ; fince the two angles + n & m are equal to two L. P. 13. B. 1. Confequently, when the two interior V, are two L; the lines LN, CD, which form those angles with EF, will meet fomewhere on the side of the line EF, where thofe angles are fituated, provided they are produced fufficiently.

Euclid regards as a felf evident principle that, a straight line (EF), which cuts one of two parallels as (AB) will neceffarily cut the other (CD), provided this cutting line (EF) be fufficiently produced. See the prep. of propofitions XXX, XXXVII, and of feveral others.

I

PROPOSITION XXIX. THEOREM XX.

Fa ftraight line (EF), falls upon two parallel straight lines (AB, CD), it makes the alternate angles (n & m) equal to one another; and the exterior angle (r) equal to the interior & oppofite upon the fame fide (m); and kewife the two interior angles upon the fame fides (p+m) equal to two right angles.

B

The Vm & n are unequal,

And one of them as Vm will be the other Vn.

ECAUSE the Vm is < n; if the Vp be added to both.
ECAUS

1. The Vm+p will be the

n+p.

But fince then & Vp are adjacent V, formed by the straight

C. N.

Ax. 4.

line EF which falls upon AB.

2. These n+pare to two L..

P.

13.

B. 1.

3. Confequently, the Vmp (lefs than the Vn+p) are alfo <two L. C. Ñ.

4.

From whence it follows, that the lines AB, CD, are not plle.
But the ftraight lines AB, CD, are plle. (Hyp.).

Cor. of lem,

5. Confequently, the Vm & n are not unequal.

6. They are therefore equal, or n= m.

P. 27. B. 1.
C. N.

Which was to be demonftrated. I.

7. These angles are to one another.

Moreover, Vr & Yn being vertically oppofite.

But Vm being: ton (Arg. 6.), & Vr being to the fame Vn,
(Arg. 7.).

8. It follows, that Vr is to V m.

P. 15. B. 1.

Ax. 1.

T

PROPOSITION XXX. THEOREM XXI.

HE ftraight lines (AB, EF), which are parallel to the fame ftraight line (CD), are parallel to one another.

Hypothefis.

AB, EF, are ftraight lines, plle to CD.

Preparation.

Thefis.

The ftraight lines AB, EF are plle to one another.

Draw the ftraight line GH, cutting the three lines AB, CD, EF.

BECAUSE

DEMONSTRATION.

ECAUSE the ftraight lines AB, CD, are two plles, (Hyp.) cut by the fame ftraight line GH. (Prep).

1. The alternate Vm & n are to one another.

Likewise since the ftraight lines CD, EF are two plles. (Hyp.) cut by the same straight line GH. (Prep).

P. 29. B. 1.

2. The exterior angle n is to its interior oppofite one on the fame fide p. P. 29. B, 1. But Vn being to m (Arg. 1.) & the fame Vn being alfo top (Arg. 2).

3. The Vm & p will be to one another.

But these Vm & p (Arg. 3.) are alternate V, formed by the two ftraight lines AB, EF, which are cut by the ftraight line GH. 4. Confequently, these straight lines AB, EF are plle.

Which was to be demonftrated

Ax. 1.

P. 27. B. 1.

To

PROPOSITION XXXI. PROBLEM X.

O draw a straight line (AB), thro' a given point (E), parallel to a given ftraight line (CD).

Given

The ftraight line CD and the point E.

Refolution.

Sought

The ftraight line AB, plle to CD, &paffing thro' the point E.

1. In the given straight line CD take any point F.

2. From the point F to the point E, draw the ftraight line FE.
3. At the point E in the straight line FE, make Vn to m.
4. And produce the fide EB to A.

DEMONSTRATION.

BECAUSE the alternate Vm &n, formed by the straight line

EF, which cuts the two lines AB, CD, are to one another (Ref. 3.). 1. The ftraight lines AB, CD, are plle.

Which was to be demonftrated,

PROPOSITION XXXII. THEOREM XXII. Fa fide as (AC) of any triangle (ABC) be produced, the exterior angle (cp) is equal to the fum of the two interior and oppofite angles (n + m); and the three interior angles (n + m + r) are equal to two right angles.

Thefis.

I. Vc+p is to m+n.
II. then+m+r are to a L.

Preparation.

Thro' the point C, draw the straight line CE, plle to the straight
line AB.

BECAUSE

DEMONSTRATION.

ECAUSE the ftraight lines AB, CE, are two plles (Prep.) c

by the fame ftraight line BC.

The alternate Vn & care to one another.

P. 31. B. 1.

cut

P. 29. B. 1.

Likewife because the straight line AB, CE, are two plles (Prep.) cut by the fame ftraight line AD.

2.

The exterior angle is to its interior oppofite one m, on the fame fide.

The being therefore to Vn (Arg. 1.), & \ p = \ m, (Arg. 2).

3. The V+p is to the V n & m taken together.

Which was to be demonftrated. I.

Since then the c+p is to V nm (Arg. 3); if the Vr be added to both fides.

Ax. 2.

4. The VC++ will be to the three Vn+m+r of the ▲ ABC. Ax, 2.
But thefe V+p+r are the adjacent V, formed by the line BC,
which meets AD at the fame point C.

5. Confequenly, the Wc+pare to two L.
Wherefore, the three Vnmr, which are to Vc+p+r,
(Arg. 4.) are alfo to two L.

Which was to be demonstrated. II.

P. 13. B. 1.

Ax. 1.