## 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 |

### Inni boken

Resultat 1-5 av 8

Side 6

Euclides. The learner should keep his Companion open before him while

studying a

proceeds ; and , in long

his Euclid .

Euclides. The learner should keep his Companion open before him while

studying a

**proposition**, and carefully observe the steps of the proof as heproceeds ; and , in long

**propositions**, he had better draw a line between them inhis Euclid .

Side 7

It will be seen that the lines which are most strongly marked form the immediate

object of the

traced ; whilst the dotted lines are those which are merely introduced for the sake

...

It will be seen that the lines which are most strongly marked form the immediate

object of the

**proposition**; lines of secondary importance to these are more faintlytraced ; whilst the dotted lines are those which are merely introduced for the sake

...

Side 14

The nature of the proof of this

ones . This is an instance of the argument ad absurdum , i . e . , it is proved that

the theorem cannot be supposed false , without leading us to a manifest

absurdity ...

The nature of the proof of this

**proposition**is different from that of the precedingones . This is an instance of the argument ad absurdum , i . e . , it is proved that

the theorem cannot be supposed false , without leading us to a manifest

absurdity ...

Side 16

... rectilineal angle ; that is , to divide it into two equal angles . D7 B /

IX . XI . and XII . depend wholly on

... rectilineal angle ; that is , to divide it into two equal angles . D7 B /

**PROPOSITION**X . Problem . To bisect a given finite. * The proofs of**Propositions**IX . XI . and XII . depend wholly on

**Proposition**VIII . and that of X . wholly on**Proposition**IV . Side 17

Euclides.

, to divide it into two equal parts . : B

straight line at right angles to a given straight line from a given point in the same .

Euclides.

**PROPOSITION**X . Problem . To bisect a given finite straight line ; that is, to divide it into two equal parts . : B

**PROPOSITION**XI . Problem . To draw astraight line at right angles to a given straight line from a given point in the same .

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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.