Book I.

XXXIV.
All other four fided figures besides these, are called Trapeziums.

XXXV.
Parallel straight lines, are such as are in the same plane, and

which, being produced ever so far both ways, do not meet.

POST U L A T E S.

1 L

I.
ET it be granted that a straight line may be drawn from
any one point to any other point.

II.
That a terminated straight line may be produced to any length
in a flraight line.

III.
And that a circle may be described from any centre, at any

distance from that centre.

T

III.

A X I 0 MS.

1.
VHINGS which are equal to the fame are equal to one an-
other.

II.
If equals be added to equals, the wholes are equal.
If equals be taken from equals, the remainders are equal.

IV.
If equals be added to unequals, the wholes are unequal.

V.
If equals be taken from unequals, the remainders are unequal.

VI.
Things which are double of the same, are equal to one another.

VII.
Things which are halves of the same, are equal to one another.

VIII.
Magnitudes which coincide with one another, that is, which
exactly fill the same space, are equal to one another.

IX.

Book 1.

IX.
The whole is greater than its part.

X.
Two ftraight lines cannot inclose a space.

XI.
All right angles are equal to one another.

XII.
“ If a straight line meets two straight lines, so as to make the

“ two interior angles on the same fide of it taken together “ lefs than two right angles, these straight lines being con“ tinually produced, shall at length meet upon that lide on " which are the angles which are less than two right angles, " See the notes on Prop. 29. of Book I.”.

PROPO.

PROPOSITION I PROBLEM.
I.

Book I.
describe an equilateral triangle upon a given fi-

nite straight line. Let AB be the given straight line; it is required to describe an equilateral triangle upon it.

From the centre A, at the distance AB, describe the circle

a. 3. PostuBCD, and from the centre B, at

late. the distance BA, describe the circle ACE ; and from the point ID All В Е C, in which the circles cut one

Isi ancther, draw the straight lines o CA, CB to the points A, B; ABC

b.zd. Polt. ihall be an equilateral triangle.

Because the point A is the centre of the circle BCD, AC is equal to AB, and because the point B is the centre of the c!sth De circle ACE, BC is equal to BA: But it has been proved that CA

finition,
is equal to AB; therefore CA, CB are each of them equal to
AB, but things which are equal to the same are equal to one
another d; therefore CA is equal to CB; wherefore CA, AB, d. ift Asis
BC are equal to one another; and the triangle ABC is there-om.
fore equilateral, and it is described upon the given straight line
AB. Which was required to be done.

PROP. II. PROB.
ROM a given point to draw a straight line equal to

a given straight line.
Let A be the given point, and BC the given straight line; it is
required to draw from the point A a straight line equal to BC.
From the point A. to B draw a

K

a. 1. Post. the straight line AB; and upon it describe b the equilateral triangle

H DAB, and produce the straight

C. 2. Pont. lines DA, DB, to E and F; from D the centre B, at the distance BC, described the circle CGH, and B

d. 3. Poft from the centre D, at the distance

G

E DG, describe the circle GKL. AL thall be equal to BC.

F

b. I. I.

Becaulc

Book I. Because the point B is the centre of the circle CGH, BC is

equal e to BG ; and because D is the centre of the circle GKL, c. 15. Def. DL is equal to DG, and DA, DB, parts of them, are equal; f. 3. Ax. therefore the remainder AL is equal to the remainder f BG:

But it has been shewn, that BC is equal to BG; wherefore AL and BC are each of them equal to BG, and things that are equal to the same are equal to one another; therefore the straight line AL is equal to BC. Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC. Which was to be done,

PRO P. III. PROB.
"ROM the greater of two given straight lines to cut

off a part equal to the less.
Let AB and C be the two gi-
ven straight lines, whereof AB is
the greater. It is required to cut
off from AB, the greater, a part
equal to C the less.
From the point A draw a the

Е В
straight line AD equal to C;

and from the centre A, and at b. 3. Poft. the distance AD, describe o the

circle DEF; and because A is
the center of the circle DCF, AE shall be equal to AD, but the
straight line C is likewise equal to AD; whence AE and C are

each of them equal to AD, wherefore the straight line AE is 61. Ax. equal to C, and from AB, the greater of two straight lines,

a part AE has been cut off equal to C the less. Which was to
be done.

PROP. IV. THEOREM.
IF two triangles have two sides of the one equal to two

sides of the other, each to each; and have likewise the angles contained by those fides equal to one another; they shall likewise have their bales, or third fides, equal ; and the two triangles fhail be equal ; and their other angles shall be equal, cach to each, viz. those to which thc equal sides are opposite.

Le ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz.

АВ

Book I.

AB to DE, and AC to DF;
and the angle BAC equal. A

D
to the angle EDF, the base
BC shall be equal to the
base EF; and the triangle
ABC to the triangle DEF;
and the other angles, to
which the equal fides are
opposite, shall be equal each
to each, viz. the angle ABC
to the angle DEF, and the B

CE

F angle ACB to DFE.

For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE ; the point B shall coincide with the point E, because AB is equal to DE and AB coinciding with DE, AC shall coincide with DF, because the angle BAC is equal to the angle EDF; wherefore also the point C shall coincide with the point F, because the straight line AC is equal to DF: But the point B coincides with the point E ; wherefore the base BC shall coincide with the base EF, because the point B coinciding with E, and C with F, if the base BC does not coincide with the base EF, two straight lines would inclose a space, which is impossible. Therefore the base BC shall coincide with the base EF, and be equal to it. Wherefore the whole triangle ABC shall coincide with the whole triangle DEF, and be equal to it; and the other angles of the one thall coincide with the remaining angles of the other, and be equal to them, viz. the angle iBC to the angle DÉF, and the angle ACB to DFE. Therefore, if two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles contained by chose fides equal to one another, their bases shall likewife be equal, and the triangles be equal, and their other angles to which the equal fides are opposite shall be equal, each to each. Which was to be demonstrated.

a 10. Axi

THE

THE angles at the base of an Isosceles triangle are

equal to one another; and, if the equal sides be produced, the angles upon the other side of the base Thall be equal. Let ABC be an Isosceles triangle, of which the fide AB is e B

qual