## 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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Side 35

The complements of the parallelograms , which are about the

parallelogram , are equal to one another . Steps of the Demonstration . 1 . Prove

that A ABC = A ACD , that A AEK + AKGC = A AKH + A KCF , that complement Bk

...

The complements of the parallelograms , which are about the

**diameter**of anyparallelogram , are equal to one another . Steps of the Demonstration . 1 . Prove

that A ABC = A ACD , that A AEK + AKGC = A AKH + A KCF , that complement Bk

...

Side 53

If a point be taken in the

drawn from it to the circumference , the greatest is that in which the centre is , and

the other part of that

If a point be taken in the

**diameter**of a circle , of all the straight lines which can bedrawn from it to the circumference , the greatest is that in which the centre is , and

the other part of that

**diameter**is the least ; and of any others , that which is the ... Side 60

The

is nearer to the centre is always greater than the K more remote ; and conversely

, the greater is nearer to the centre than G the less . Part 1st . That the

The

**diameter**is the greatest straight line in a circle ; and I of all others , that whichis nearer to the centre is always greater than the K more remote ; and conversely

, the greater is nearer to the centre than G the less . Part 1st . That the

**diameter**... Side 61

The straight line drawn at right angles to the

extremity of it , falls without the circle ; and no straight line can be drawn between

that straight line and the circumference from the extremity , so as not to cut the

circle .

The straight line drawn at right angles to the

**diameter**of a circle from theextremity of it , falls without the circle ; and no straight line can be drawn between

that straight line and the circumference from the extremity , so as not to cut the

circle .

Side 78

In a given circle to place a straight line , not greater than the

. DProved by showing , from the construction , that AC = Ce ; and CE = D .

PROPOSITION II . Problem . In a given circle to inscribe a triangle equiangular to

a ...

In a given circle to place a straight line , not greater than the

**diameter**of the circle. DProved by showing , from the construction , that AC = Ce ; and CE = D .

PROPOSITION II . Problem . In a given circle to inscribe a triangle equiangular to

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.