demonstrate this proposition Euclid assumes it as an axiom, Book I. that " if a straight line meet two straight lines, so as to make "the interior angles on the same side of it less than two right

66

angles, these straight lines being continually produced, " will at length meet on the side on which the angles are " that are less than two right angles." This proposition, however, is not self-evident, and ought the less to be received without proof, because, as Proclus has observed, the converse of it is a proposition that confessedly requires to be demonstrated. For the converse of it is, that two straight lines which meet each other make the interior angles, with any third line, less than two right angles; or, in other words, that the two interior angles of any triangle are less than two right angles, which is the 17th of the First Book of the Elements; and it should seem that a proposition can never rightly be taken for an axiom, of which the converse requires a demonstration.,

It is of the utmost importance in explaining the elements of science, to connect truths by the shortest chain that can connect them; and till that is done, we can never consider them as being placed in their natural order. The reasoning in the first of the following propositions is so simple, that it seems hardly susceptible of abbreviation, and it has the advantage of immediately connecting together two truths so much alike, that one might conclude, even from the bare enumeration of them, that they are but different cases of the same general theorem, viz. That all the angles about a point, and all the exterior angles of any rectilineal figure, are constantly of the same magnitude, and equal to four right angles.

DEFINITION.

IF, while one extremity of a straight line remains fixed at A, the line itself turn about that point from the position AB to the position AC, it is said to describe the angle BAC, contained by the lines AB and AC.

C

A

B

Book I.

Cor. If a line turn about a point from the position AB, till it come into the position AB again, it describes angles which are together equal to four right angles. This is evident from the second corollary to the 15th proposition.

PROP. I.

All the exterior angles of any rectilineal figure are together equal to four right angles.

1. Let the rectilineal figure be the triangle ABC, of which the exterior angles are DČA, FAB, GBC; these angles are together equal to four right angles.

Let the line CD turn

F

B

A

E

D

about the point C till it co-
incide with CE, a part of
the side CA, and have de-
scribed the exterior angle
DCE or DCA. Let it
then be carried along the
line CA till it be in the
position AF, and, the point
A remaining fixed, let it
turn about A till it de-
scribe the angle FAB, and
coincide with a part of
the line AB. Let it next
be carried along AB till it
come into the position BG,
and, by turning about B, let it describe the angle GBC, so as
to coincide with a part of BC. Lastly, let it be carried along
BC, so as to coincide with CD, its first position. Then, be-
cause the line CD has turned about one of its extremities till
it has come into the position CD again, it has by the corol-
lary to the above definition described angles which are toge-
ther equal to four right angles. But the angles which it has
described are the three exterior angles of the triangle ABC;
therefore the exterior angles of the triangle ABC are equal to
four right angles.

G

2. If the rectilineal figure have any number of sides, the proposition is demonstrated just as in the case of a triangle.

1

Therefore all the exterior angles of any rectilineal figure are Book I. together equal to four right angles. Q. E. D.

Cor. 1. Hence, all the interior angles of any triangle are equal to two right angles. For all the angles of the triangle, both exterior and interior, are equal to six right angles, and the exterior being equal to four right angles, the interior are equal to two right angles.

Cor. 2. An exterior angle of any triangle is equal to the two interior and opposite, or the angle DCA is equal to the angles CAB, ABC. For the angles CAB, ABC, BCA are equal to two right angles; and the angles ACD, ACB are also (13. 1.) equal to two right angles; therefore the three angles CAB, ABC, BCA are equal to the two angles ACD, ACB; therefore if ACB be taken from both, the angle ACD is equal to the two angles CAВ, АВС.

COR. 3. The interior angles of any rectilineal figure are equal to twice as many right angles as the figure has sides, wanting four. For all the angles exterior and interior are equal to twice as many right angles as the figure has sides, and the exterior are equal to four right angles; therefore the interior are equal to twice as many right angles as the figure has sides, wanting four.

PROP. II.

Two straight lines, which make with a third line the inte rior angles on the same side of it less than two right angles, will meet on that side, if produced far enough.

Let the straight lines AB, CD make with AC the two angles BAC, DCA less than two right angles; AB and CD will meet if produced toward B and D.

In AB take AF = AC, and join CF; produce BA to H, and through C draw CE making the angle ACE equal to the angle CAH (23. 1.).

Because AC is equal to AF, the angles AFC, ACF are also equal (5. 1.). But the exterior angle HAC is equal to the two interior and opposite angles ACF, AFC, and therefore is double of either of them, as ACF; therefore ACE is

Book I. double of ACF, therefore the angle ACE is bisected by the line CF. If FG be taken equal to FC, and CG be drawn, it may be shown in the same manner that CG bisects the angle FCE; and so on continually. But if from a magnitude, as the angle ACE, there be taken its half, and from the remainder FCE its half FCG, and from the remainder GCE its half, &c. a remainder will at length be found less than the given angle DCE*. Let GCE be the angle whose half ECK

is less than DCE, then a straight line CK is found, which falls between CD and CE, but nevertheless meets the line AB in K. Therefore CD, if produced, must meet AB in a point between G and K. Therefore, if two straight lines, &c. Q. E. D.

PROP. III. 29. 1. Euclid.

If a straight line fall on two parallel straight lines it makes the alternate angles equal to each other, the exterior equal to the interior and opposite on the same side, and the two interior angles on the same side equal to two right angles.

Let the straight line EF fall on the parallel straight lines AB, CD; the alternate angles AGH, GHD are equal, the exterior angle EGB is equal to the interior and opposite GHD, and the two interior angles BGH, GHD are equal to two right angles.

For, if AGH be not equal to GHD, let it be greater; add Book I. BGH to both, then the angles AGH, HGB are greater than

the angles DHG, HGB.

E

less than two right angles, and therefore the lines AB,

CD will meet, by the last proposition, if produced to

C

H

D

ward B and D. But they do not meet, for they are parallel by hypothesis; there

F

fore the angles AGH, GHD are not unequal, that is, they are equal to each other. Now the angle AGH is equal to EGB (15. 1.), therefore EGB and GHD are equal. Lastly, to each of the equal angles EGB, GHD add the angle BGH, then the two angles EGB, BGH are equal to the two DHG, BGH. But EGB, BGH are equal to two right angles (13. 1.); therefore BGH, GHD are also equal to two right angles..

Therefore, if a straight line, &c. Q. E. D.

BOOK II.

THE demonstrations of this book are no otherwise Book II.

changed than by introducing into them some characters similar to those of algebra, which is always of great use where the reasoning turns on the addition or subtraction of rectangles. To Euclid's demonstrations others are sometimes added, serving to deduce the propositions from the fourth, without the assistance of a diagram.

PROP. A & B.

These theorems are added on account of their great use in geometry, and their close connection with the other propositions which are the subject of this book. Prop. A is an extension of the 9th and 10th.