Posts in category 'solid-geometry'

G.524 solid angle

A:term:**solid angle** <solid-angle> is the inclination constituted by more than two:term:lines <G.2> which meet one another and are not in the same:term:surface <G.533>, towards all the lines. Otherwise: A:term:**solid angle** <solid-angle> is that which is contained by more than two:term:plane angles <G.22> which are not in the same:term:plane <G.7> and are constructed to one:term:point <G.1>.
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G.528 construct a solid angle out of three plane

Construct a solid angle out of three plane angles two of which, taken together in any manner, are greater than the remaining one: thus the three angles must be less than four right angles.
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G.545 there equal plane

If there be two equal plane angles, and on their vertices there be set up elevated segments containing equal angles with the original segments respectively, if on the elevated segments points be taken at random and perpendiculars be drawn from them to the planes in which the original angles are, and if from the points so arising in the planes segments be joined to the vertices of the original angles, they will contain, with the elevated segments, equal angles.
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G.546 segments proportional parallelepipedal

If three segments be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.
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G.547 segments proportional parallelepipedal

If four segments be proportional, the parallelepipedal solids on them which are similar and similarly described is proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the segments will themselves also be proportional.
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G.554 pyramid triangular base

Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.
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G.555 there pyramids height

If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two equal prisms, then, as the base of the one pyramid is to the base of the other pyramid, so will all the prisms in the one pyramid be to all the prisms, being equal in multitude, in the other pyramid.

G.561 cone

When, one side of those about the:term:right angle <G.33> in a:term:right-angled triangle <G.78> remaining fixed, the:term:triangle <G.12> is carried round and restored again to the same position from which it began to be moved, the:term:figure <G.9> so comprehended is a:term:**cone** <cone>. And, if the:term:segment <G.4> which remains fixed equals the remaining side about the right angle which is carried round, the cone is:term:**right-angled** <right-angled-cone>; if less,:term:**obtuse-angled** <obtuse-angled-cone>; and if greater,:term:**acute-angled** <acute-angled-cone>.
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G.570 similar cones cylinders

Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
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G.574 circles center inscribe

Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.
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G.577 spheres center inscribe

Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.
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G.586 equilateral equiangular pentagon

If in an equilateral and equiangular pentagon segments subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.
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G.588 equilateral pentagon inscribed

If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.
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G.589 circle diameter rational

If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational segment called minor.
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G.592 construct pyramid comprehend

To construct a :term:`pyramid <G.553>`, to comprehend it in a given :term:`sphere <G.575>`, and to prove that the :term:`square <G.63>` on the :term:`diameter of the sphere <G.576>` is one and a half times the square on the side of the pyramid.
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G.594 construct an octahedron and comprehend it in a

Construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double of the square on the side of the octahedron.
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G.595 construct a cube and comprehend it in a

Construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple of the square on the side of the cube.
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G.597 construct an icosahedron and comprehend it in a

Construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational segment called minor.
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G.600 construct a dodecahedron and comprehend it in a

Construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational segment called apotome.
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G.601 set out sides

To set out the sides of the five figures and to compare them with one another.