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# JB/135/072/005

Completed

Book the Six

Prop I Theor.
Triangles and parallelograms of
the same altitude are to one
another as their bases —

Prop VII Theor.
If two triangles have one angle of the one equal to
one angle of the other, and the sides about two other
angles proportionals: then if each of the
remaining angles be either less or not less
than a right angle, or if one of them be a
right angle the triangle shall be equiangular
and have those angles equal about which the
sides are proportionals.

Prop XIV Theor.
Equal parallelograms which have
one angle of the one equal to one angle
of the other, have their Sides about the
equal angles reciprocally proportional,
and parallelograms that have one angle
of the one equal to one of the other
and their sides about their equal angle
reciprocally proportional, are equal to
one another.

Prop XXI Theor.
Rectilineal figures which are
similar to the same rectilineal
figure, are also similar to one
another.

Prop XXVIII. Prob.
To a given straight line to apply a
parallelogram equal to a given rectilineal
figure and deficient by a parallelogram
similar to a given parallelogram but the
given rectilineal figure to which the
parallelogram to be applied is to be equal
must not be greater than the parallelogram
applied to half of the given line,
having its defect similar to the defect of
that which is to be applied: that is to
the given parallelogram —

Prop C Theor.
If from an angle of
a triangle a strait line be drawn
perpendicular to the base: the rectangle
contained by the sides of the triangle is
equal to the rectangle contained by
the perpendicular and the diameter
of the circle described about the
triangle.

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Prop II Theor.
If a straight line be drawn parallel to
one of the Sides of a triangle, it shall
cut the other sides, or these produced,
proportionally. And if the sides, or the
sides produced be cut proportionally,
the straight lines which joins the
point of Section shall be parallel to
the remaining side of the triangle.

Prop VIII Theor.</lb> In a right angled triangle, if a perpendicular
be drawn from the right angle to
the base: the triangles on each side of it
are similar to the whole triangle &
to one another

Prop XV Theor.
Equal triangles which have one angle
of the one equal to one angle of the other,
have their Sides about the equal angles
reciprocally proportional: and triangles
which have one angle in the one
equal to one angle in the other and
their Sides about the equal angles
reciprocally proportional are equal to one another.

Prop XXII Theor.
If four straight lines be proportionals,
the similar rectilineal figures
described upon them shall also be
proportionals. And if the similar
rectilineal figures described upon 4
four strait lines be proportionals,
those straight lines shall be proportionals.

Prop XXIX Prob.
To a given straight line to apply a
parallelogram equal to a given
rectinineal figure exceeding by
a parallelogram similar to
another given —

Prop D Theor.
The rectangle contained by the
inscribed in a circle is equal
to both the rectangles contained
by its opposite sides —

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Prop III Theor.
If the angle of a triangle be divided into
two equal angles, by a straight line which
also cuts the base; the segments of the base
shall have the same ratio which the other
sides of the triangles have to one another
and if the segments of the base have the
same ratio which the other sides of the
triangle have to one another the straight
line drawn from the vertex to the point of
section divides the angle into two equal angles.

Prop IX Prob. From a given straight line to cut of
any part required.

Prop XVI Theor.
If four Straight lines be proportional,
the rectangle contained by the extremes
is equal to the rectangle contained by
the means, the four straight lines are
proportional.

Prop XXIII Theor.
Equiangular parallelograms
have to one another the ratio
which is compounded of the ratios
of their sides — <p>Prop XXX Prob. To cut a given straight line
in extreme & mean ratio

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Prop A Theor.
If the outer
sides be divided into two equal
by a straight line which also cuts
produced: the segments betw
dividing line and the extreme
base have the same ratio which
sides of the triangle have to one
if the segments of the bas produce
same ratio, which the other side
angle have the straight line dra
the vertex to the point of sect
the outward angle of the tria
two equal angles — <p>Prop X Theor.
To divide a given straight line
a given staight line, that is
which shall have the same ratio
other which the parts of the
given straight line have —

Prop XVII Theor.
If three straight lines be pro
the rectangle contained by the
is equal to the square of the
and the rectangle contained
extremes be equal to the Squ
mean, the 3 straight lines are prop

Prop XXIV Theor.
of any parallelogram, are
to the whole and to one

Prop XXXI Theor.
In right angled triangles
neal figures described upon
opposite to the right ang
equal to the similar, and
described figures upon the
containing the right ang

Identifier: | JB/135/072/005
"JB/" can not be assigned to a declared number type with value 135.

135

072

the propositions of the fifth and sixth books of euclid as expressed by simson those written with red ink are added by himself

005

private material

3

recto

sir samuel bentham

[[watermarks::[tall thin motif with prince of wales feathers] icv]]

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