## 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 14

III . , to show him how to cut off this line , which , in fact , is not possible to be done

, it would be much more intelligible to say ,

and

III . , to show him how to cut off this line , which , in fact , is not possible to be done

, it would be much more intelligible to say ,

**Suppose**AB to be greater than Ac , —and

**suppose**that a part of AB as DB = AC , & c . , We shall therefore , in giving ... Side 19

line with it . Then prove , on that supposition , 1 . that ZS ABE + ABC = 2 rt . Zs , 2 .

that ZS ABE + ABC = Zs ABC + ABD , 3 . that Z ABE = ABD , i . e . less = greater ...

**Suppose**that bd is not in the same st . line with bc , but that BE is in the same st .line with it . Then prove , on that supposition , 1 . that ZS ABE + ABC = 2 rt . Zs , 2 .

that ZS ABE + ABC = Zs ABC + ABD , 3 . that Z ABE = ABD , i . e . less = greater ...

Side 52

and Eg each = EC , and : : = each other ; i . e . less = greater , which shows the

supposition to be false . PROPOSITION VI . ( Argument ad absurdum ) . Theorem

.

**Suppose**that E is the centre of both Os ; and prove , on that supposition , that EFand Eg each = EC , and : : = each other ; i . e . less = greater , which shows the

supposition to be false . PROPOSITION VI . ( Argument ad absurdum ) . Theorem

.

Side 56

Then

= DK ; and show , on that supposition , 2 . that DB = Dx ; i . e . a line near to the

least = one more remote , which , by the preceding part of the demonstration is ...

Then

**suppose**that from point d to the o another line on besides DB can be drawn= DK ; and show , on that supposition , 2 . that DB = Dx ; i . e . a line near to the

least = one more remote , which , by the preceding part of the demonstration is ...

Side 57

Theorem . One circumference of a circle cannot cut another in all more than two

points .

and prove , on that supposition , that k would be the centre of both Os , which ( by

v .

Theorem . One circumference of a circle cannot cut another in all more than two

points .

**Suppose**the FAB can cut O DEF in more than two points , as B , G , F ,and prove , on that supposition , that k would be the centre of both Os , which ( by

v .

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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.