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HYPOTHESES.

If the same straight line is
perpendicular to each of
two planes.

If two straight lines be cut
by parallel planes.
If two straight lines meeting
one another be parallel to
two other straight lines
which meet one another,
but are not in the same
plane with the first two.
If a plane be perpendicular
to another plane, and a
straight line be drawn from
a point in one of the planes
perpendicular to the other
plane.

If two parallel planes be cut
by another plane.
If every two of three plane
angles be greater than the
third, and if the straight
lines which contain them
be all equal.

CONSEQUENCES.

They are parallel to one an-
other.

They shall be cut in the same
ratio.

The plane which passes
through these is parallel to
the plane passing through
the others.

This straight line shall fall
on the common section of
the planes.

Their common sections with
it are parallels.

A triangle may be made of
the straight lines that join
the extremities of those
equal straight lines.

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Z. Of Solid Figures.

HYPOTHESES.

If three straight lines be pro-
portionals.

If four straight lines be pro-
portionals.

If four straight lines be con-
tinual proportionals.

If solid parallelopipeds are
similar.

If solid parallelopipeds have
the same altitude;
And are upon equal bases
If prisms are of equal alti-
tudes;

CONSEQUENCES.

The solid parallelopiped de-
scribed from all three, as its
sides, is equal to the equi-
lateral parallelopiped de-
scribed from the mean pro-
portional, one of the solid
angles of which is contained
by three plane angles equal,
each to each, to the three
plane angles containing one
of the solid angles of the
figure.

The similar solid parallel-
opipeds similarly described
from them shall also be
proportionals.

As the first is to the fourth,
so is the solid parallelopiped
described from the first
to the similar solid simi-
larly described from the
second.

They are to one another in the
triplicate ratio of their ho-
mologous sides.

They are to one another as
their bases.

They are equal to one another.
They are to one another as
their bases.

And are upon triangular bases Idem.
If solid parallelopipeds are
equal.

If there be two triangular
prisms of the same altitude,
the base of one of which is
a parallelogram, and the
base of the other a trian-
gle: if the parallelogram
be double of the triangle.
If a solid be contained by six
planes, two and two of
which are parallel.
If solid figures are contained
by the same number of
equal and similar planes
alike situated, and having
none of their solid angles
contained by more than
three plane angles.

{Their bases and altitudes are
reciprocally proportional.

The prisms shall be equal to
one another.

The opposite planes are simi-
lar and equal parallel-

ograms.

They are equal and similar
to one another.

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CONSEQUENCES.

They have to one another the ratio which is the same with the ratio compounded of the ratios of their sides.

They are equal to one another.

It divides the whole into two solids, the base of one of which shall be to the base of the other, as the one solid is to the other.

It shall be cut into two equal parts.

The common section of the planes passing through the points of division, and the diameter of the solid parallelopiped, cut each other into two equal parts. (It may be divided into three pyramids that have triangular bases, and are equal to one another. (It may be divided into two

equal and similar pyramids having triangular bases, and which are similar to the whole pyramid; and into two equal prisms which together are greater than half of the whole pyramid. They are to one another in the triplicate ratio of that of their homologous sides.

They are to one another as their bases.

Their bases and altitudes are reciprocally proportional.

They are equal to one another.

It is the third part of a prism which has the same base and altitude,

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PROBLEMS.

A. Relating to Straight Lines.

To draw a straight line perpendicular to a plane, from a given point above it.

To draw a straight line at right angles to a given plane, from a point given in that plane.

From a given finite straight line to cut off any required part.

To divide a given straight line similarly to a given divided
straight line.

To cut a given straight line in extreme and mean ratio.
To divide a given straight line into two parts, such that
parallelograms of equal altitude may be constructed
upon them, one equal to a given rectilineal figure, and
the other similar to a given parallelogram; the rectili-
neal figure not being greater than the parallelogram
constructed on half the given line, and similar to the
given parallelogram.

To find a mean proportional between two given straight
lines.

To find a third proportional to two given straight lines.
To find a fourth proportional to three given straight lines.
To produce a given straight line so that a parallelogram
similar to a given one being constructed on the produced
part, another parallelogram of equal altitude constructed
on the whole line produced, may be equal to a given
rectilineal figure.

B. Relating to Rectilineal Angles.

To divide a given right angle into five equal parts.

C. Relating to Triangles.

To construct an isosceles triangle, in which each of the angles at the base shall be double of the angle opposite to the same.

F. Relating to Inscribed Figures:

In a given circle to inscribe a straight line equal to a given straight line, which is not greater than the diameter of the circle.

To inscribe a circle in a given triangle.

To circumscribe a circle about a given triangle.

In a given circle to inscribe a triangle equiangular to a given triangle.

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