## The Elements of Euclid |

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Side 227

To describe in the greater of two

polyhedron , the superficies of which shall not meet the lesser

be two

a ...

To describe in the greater of two

**spheres**which have the some centre , a solidpolyhedron , the superficies of which shall not meet the lesser

**sphere**. * Let therebe two

**spheres**about the same centre A ; it is required to describe in the greatera ...

Side 231

And that the other planes between the quadrants BX , KX fall without the lesser

plane of the quadrilateral SOPT , and join 10 ; and , as was demonstrated of the

plane ...

And that the other planes between the quadrants BX , KX fall without the lesser

**sphere**, is thus demonstrated ; from the point A draw Al perpendicular to theplane of the quadrilateral SOPT , and join 10 ; and , as was demonstrated of the

plane ...

Side 232

divided into the same number of pyramids , and in the same order , the pyramids

shall be similar to one another each to each ; because they have the solid angles

...

**sphere**BCDE has to the diameter of the other**sphere**; for if these two solids bedivided into the same number of pyramids , and in the same order , the pyramids

shall be similar to one another each to each ; because they have the solid angles

...

Side 233

the

solid polyhedron in the

which BC has to EF . But the

ratio of ...

the

**sphere**DEF ; therefore the solid polyhedron in the**sphere**ABC has to thesolid polyhedron in the

**sphere**DEF , the triplicate ratio ( Cor . 17 . 12 . ) of thatwhich BC has to EF . But the

**sphere**ABC has to the**sphere**GHK , the triplicateratio of ...

Side 396

is a point in the superficies of the

the circumference of the circle are equal . II . A great circle of the

**SPHERICAL**TRIGONOMETRY . DEFINITIONS . The pole of a circle of the**sphere**is a point in the superficies of the

**sphere**, from which all straight lines drawn tothe circumference of the circle are equal . II . A great circle of the

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added altitude angle ABC angle BAC base BC is given centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise magnitude manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid sphere square square of BC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 36 - If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.

Side 145 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

Side 65 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Side 248 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Side 11 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Side 121 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Side 21 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Side 14 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.

Side 80 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

Side 133 - ... rectilineal figures are to one another in the duplicate ratio of their homologous sides.