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The Propositions of the third and fourth Book

Prop I. Prob. To find the Center of a given Circle.

Prop VI Theor
If two circles touch one another internally,
they shall not have the same center.

Prop X Theor
One circumference of a circle cannot cut
another in more than two points.

Prop XV Theor
The Diameter is the greatest straight line in a circle;
and of all others, that which is nearer to the Center is
always greater than one more remote: and the greater
is nearer to the Center than the less.

Prop XX Theor
The angle at the Center of a Circle is double of
the angle at the Circumference, upon the same base,
that is, upon the same part of the Circumference.

Prop XXV Prob.
A Segment of a Circle being given to describe
<the circle> of which it is the Segment.

---page break---

Prop II Theor
If any two points be taken in the circumference
of a Circle, the straight line which joins them
shall fall within the circle.

Prop VII Theor
If any point be taken in the diameter of a circle, which
is not the center, of all the straight lines which can be
drawn from it to the circumference, the greatest is that
in which the Center is, and the other part of that
diameter is the least: and of any others, that which is nearer
to the line which passes thro' the Center is always greater
than one more remote, and from the same point there can be
drawn only two straight lines that are equal to one another
upon each side of the line.

Prop XI Theor
If two Circles touch each other internally, the straight
line which joins their centers being produced shall
pass thro' the point of contact.

Prop XVI Theor
The straight line drawn at right angles to the Diameter of a circle
from the extremity of it, falls without the Circle; & no straight
line can be drawn, between that straight line and the
circumference from the extremity, so as not to cut the circle:
or, which is the same thing, no straight line can make
so great an acute angle with the diameter at its
extremity, or so small an angle wh with the straight
line which is at right angles to it, as not to cut the circle.

Prop XXI Theor
The angles in the same Segment of a circle are
equal to one another.

Prop XXVI Theor
In equal circles, equal angles stand upon equal
circumferences, whether they be at the centers
or circumferences.

---page break---

Prop III Theor
If a straight line be draw thr
a circle, bisect a straight line
does not pass thro' the Center
at right angles, & if it cuts
it shall bisect it.

If any point be taken without a Circle
passes thro the Center, of those which
thro' the Center; and of the rest, tha
more remote: but of those which
point without the circle and the
less than the more remote, and
the circumference, one upon ea

Prop XII Theor
If two Circles touch each other ea
line which joins their Centers shall pass

Prop XVII Prob.
To draw a straight line from a
without or I the Circumferen
touch a given circle.

Prop XXII Theor
The opposite angles of any qu
are described in a circle, are to
two right angles.

Prop XXVII Theor
In equal circles, the angles
equal circumferences are equal
whether they be at the center

Identifier: | JB/135/073/002
"JB/" can not be assigned to a declared number type with value 135.


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the propositions of the third and fourth books of euclid as expressed by simson





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sir samuel bentham


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