## The Elements of Euclid |

### Inni boken

Side 95

Geometers make use of the following technical words to signify certain ways of

changing either the order or magnitude of

still to be

Geometers make use of the following technical words to signify certain ways of

changing either the order or magnitude of

**proportionals**, so as that they continuestill to be

**proportionals**. ' XIII . Permutando , or alternando , by permutation , or ... Side 136

If four straight lines be

equal to the rectangle contained by the ... Let the four straight lines AB , CD , E , F

, be

If four straight lines be

**proportionals**, the rectangle contained by the extremes isequal to the rectangle contained by the ... Let the four straight lines AB , CD , E , F

, be

**proportionals**, viz . as AB to CD , so is E to F ; the rectangle contained by ... Side 141

And if to AB , FG , two of the homologous sides , a third

AB has ( 10 . def . ... Therefore , universally , it is manifest , that if three straight

lines be

the ...

And if to AB , FG , two of the homologous sides , a third

**proportional**M be takenAB has ( 10 . def . ... Therefore , universally , it is manifest , that if three straight

lines be

**proportionals**, as the first is to the third , so is any rectilineal figure uponthe ...

Side 195

Let A B , C , be three

to KM ; that is , the sides about the equal angles are reciprocally

therefore the parallelogram LM is equal ( 14 . 6 . ) to EF : and because EDF , LKM

...

Let A B , C , be three

**proportionals**, viz . A to B , as B to C . The solid described ...to KM ; that is , the sides about the equal angles are reciprocally

**proportional**;therefore the parallelogram LM is equal ( 14 . 6 . ) to EF : and because EDF , LKM

...

Side 253

cited place , censures Commandine , and that very justly , for demonstrating this

proposition by help of the 16th of the fifth ; yet he himself gives no demonstration

of it , but thinks it plain from the nature of

...

cited place , censures Commandine , and that very justly , for demonstrating this

proposition by help of the 16th of the fifth ; yet he himself gives no demonstration

of it , but thinks it plain from the nature of

**proportionals**, as he writes in the end of...

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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.