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A circle is a plain figure contained by one line, which is called

the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

XVI.
And this point is called the center of the circle.

XVII. A diameter of a circle is a straight line drawn thro' the center, See N. and terminated both ways by the circumference.

XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

xix. « A fegment of a circle is the figure contained by a straight line « and the circumference it cuts off.”

XX.
Rectilineal figures are those which are contained by straight
lines.

XXI.
Trilateral figures, or triangles, by three straight lines.

XXII.
Quadrilateral, by four straight lines.

XXIII.
Multilateral figures, or Polygons, by more than four straight lines.

XXIV.
Of three sided figures, an equilàteral triangle is that which has
three equal fides.

XXV.
An isosceles triangle, is that which has only two sides equal.

Book I.

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XXIV

XXVI.
A scalene triangle, is that which has three unequal fides.

XXVII.
A right angled triangle, is that which has a right angle.

XXVIII.
An obtufe angled triangle, is that which has an obtuse angle.

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XXIX.
An acute angled triangle, is that which has three acute angles.

XXX.
Of four sided figures, a square is that which has all its sides

equal, and all its angles right angles.

XXXI.
An oblong is that which has all its angles right angles, but has
not all its fides equal.

XXXII.
A rhombus is that which has all its fides equal, but its angles

are not right angles.

77

See N.

XXXIII.
A rhomboid is that which has its opposite sides equal to ane ano-

ther, but all its fides are not equal, nor its angles right angles.

Book I.

XXXIV.
All other four sided 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.

POSTULAT E S.

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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 straight line.

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

distance from that center.

A X I O M S

Tanother

.

I.
HINGS which are equal to the same are equal to one
another.

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

III.
If equals be taken from equals, the remainders are equal. X

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 fame, are equal to one another.

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

Took 1.

IX.
The whole is greater than its part.

X.
Two straight 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 fame side of it taken together less « than two right angles, these straight lines being continually “ produced shall at length meet upon that fide on which are “ the angles which are less than two right angles. See the

notes on Prop.-29. of Book I.”

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Book I.
PROPOSITION I. PROBLEM.
O describe an equilateral triangle upon a given

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

it. From the center A, at the distance AB describe a the circle

a. 3d Postue

late. BCD. and from the center B, at the distance BA describe the circle D A . B E ACE; and from the point C in which the circles cut one another draw the straight lines b CA, CB to the points A, B. ABC shall be an equilateral triangle.

Because the point A is the center of the circle BCD, AC is equal e to AB. and because the point B is the center of the circle ACE,

C. 15th De.

finition. BC is equal to BA. but it has been proved that CA 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 d. ift Axie CA is equal to CB. wherefore CA, AB, BC are equal to one another. and the triangle ABC is therefore equilateral, and it is defcribed

upon the given straight line AB. Which was required to be done.

b. 2d Poft.

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PRO P. II. PRO B.
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 the

a. I. Poft. Straight line AB; and upon it de

K scribe the equilateral triangle DAB,

H and produce the straight lines DA, DB to E and F; from the center B, at the distance BC describe d the

d. 3. Pott circle CGH, and from the center D,

B В at the distance DG described the G

E circle GKL. AL Thall be equal to BC.

b. I. I.

c. 2. Poft,

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