## A companion to Euclid: being a help to the understanding and remembering of the first four books. With a set of improved figures, and an original demonstration of the proposition called in Euclid the twelfth axiom, by a graduate |

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Resultat 1-5 av 5

Side 16

To

B / PROPOSITION X . Problem . To

Propositions IX . XI . and XII . depend wholly on Proposition VIII . and that of X .

wholly on ...

To

**bisect**a given rectilineal angle ; that is , to divide it into two equal angles . D7B / PROPOSITION X . Problem . To

**bisect**a given finite. * The proofs ofPropositions IX . XI . and XII . depend wholly on Proposition VIII . and that of X .

wholly on ...

Side 17

To

PROPOSITION XI . Problem . To draw a straight line at right angles to a given

straight line from a given point in the same . A D CE CoR . ( Argument ad

absurdum ) ...

To

**bisect**a given finite straight line ; that is , to divide it into two equal parts . : BPROPOSITION XI . Problem . To draw a straight line at right angles to a given

straight line from a given point in the same . A D CE CoR . ( Argument ad

absurdum ) ...

Side 50

PROPOSITION III . POSITION Theorem . If a straight line drawn through the

centre of a circle

, it shall cut it at right angles ; and if it cuts it at right angles , it shall

Part 1st ...

PROPOSITION III . POSITION Theorem . If a straight line drawn through the

centre of a circle

**bisect**a straight line in it which does not pass through the centre, it shall cut it at right angles ; and if it cuts it at right angles , it shall

**bisect**it . APart 1st ...

Side 51

If in a circle 10 straight lines cut one another , which do not both pass through the

centre , they do not

AC , BD do

If in a circle 10 straight lines cut one another , which do not both pass through the

centre , they do not

**bisect**each other . Steps of the Demonstration . Suppose thatAC , BD do

**bisect**each other in E . ( State that this is evidently impossible if one ... Side 87

Prove that Fa , FB , FE ,

FC = FD , that the five right lines , FA , FB , FC , FD , FE = each other ; and : . a

described from centre F and distance any one of them , will pass through the

points A ...

Prove that Fa , FB , FE ,

**bisect**S CBA , BAE , AED , that _ FCD = 2 CDF , that : .FC = FD , that the five right lines , FA , FB , FC , FD , FE = each other ; and : . a

described from centre F and distance any one of them , will pass through the

points A ...

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alternate angle contained angle equal applied Argument ad absurdum base bisect BOOK centre circumference coincides construction Demonstration described diameter directed divided draw drawn Edition Engravings equal equiangular equilateral Euclid extremities fall figure given circle given point given rectilineal given straight line greater HISTORY impossible inscribe interior joins learner least less meet Nature necessary opposite parallel parallelogram pass pentagon point of contact Problem produced proof PROPOSITION PROPOSITION VIII PROPOSITION XV Proved by showing READINGS rectangle contained right angles right line right Zs segment shows the supposition sides similarly square Steps straight line Suppose supposition is false Theorem touch triangle VOLUME whole whole line YOUNG

### Populære avsnitt

Side 24 - 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, namely, either t}le sides adjacent to the equal...

Side 45 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part.

Side 18 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

Side 61 - From this it is manifest that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle...

Side 37 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

Side 76 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

Side 77 - If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square on GEOMETRY.

Side 72 - If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.

Side 27 - If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.