## Geometry Without Axioms; Or the First Book of Euclid's Elements. With Alterations and Familiar Notes; and an Intercalary Book in which the Straight Line and Plane are Derived from Properties of the Sphere ...: To which is Added an Appendix ... |

### Inni boken

Resultat 1-5 av 36

Side 3

And because the assumption cannot be true without the consequence being true

also ; if the assumption is found to involve a false or

the assumption cannot be true . This is the rationale ( or reasonable principle ] of

...

And because the assumption cannot be true without the consequence being true

also ; if the assumption is found to involve a false or

**impossible**consequence ,the assumption cannot be true . This is the rationale ( or reasonable principle ] of

...

Side 5

Neither the converse , the negative , nor the contrary of any proposition , is to be

admitted to be true , till it has been demonstrated as a distinct proposition . For till

this be done , it is

Neither the converse , the negative , nor the contrary of any proposition , is to be

admitted to be true , till it has been demonstrated as a distinct proposition . For till

this be done , it is

**impossible**to know whether it is true or not . For example , it is ... Side 8

Which is

begin with . The assumption " , therefore , which involves this

consequence , cannot be true ; or one remainder cannot be greater than the

other .

Which is

**impossible**; for the things by the hypothesis were • I. Nom.26 . equal tobegin with . The assumption " , therefore , which involves this

**impossible**consequence , cannot be true ; or one remainder cannot be greater than the

other .

Side 9

11 ) be greater than the double of the other ; which is

26 . equal . The assumption " , therefore , cannot be true ; or the one magnitude ,

that was doubled , is not greater than the other ; and because one is not greater ...

11 ) be greater than the double of the other ; which is

**impossible**, for it is * I.Nom.26 . equal . The assumption " , therefore , cannot be true ; or the one magnitude ,

that was doubled , is not greater than the other ; and because one is not greater ...

Side 15

Nom.15 . be interior to the other ; that is to say , one sphere will be greatert than

the other ; which is

therefore , cannot be unequal ; that is , they are equal . PROPOSITION IV .

THEOREM .

Nom.15 . be interior to the other ; that is to say , one sphere will be greatert than

the other ; which is

**impossible**, for they are equal . The central distances ,therefore , cannot be unequal ; that is , they are equal . PROPOSITION IV .

THEOREM .

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Geometry Without Axioms; Or the First Book of Euclid's Elements. With ... Thomas Perronet Thompson Uten tilgangsbegrensning - 1833 |

Geometry Without Axioms; Or the First Book of Euclid's Elements. with ... Thomas Perronet Thompson Ingen forhåndsvisning tilgjengelig - 2015 |

### Vanlige uttrykk og setninger

ABCD added alternate angle ABC angle BAC applied aret assigned axis base bisected body Book called centre change of place circle coincide common consequently Constr continually demonstrated described distance double drawn equal Euclid exterior angle extremities fall figure follows formed four right angles Geometry given straight line greater half impossible instance INTERC interior join kind less magnitude manner meet moved Note opposite parallel parallelogram parity of reasoning pass perpendicular plane portion prolonged proof Prop PROPOSITION proved radius remaining angle respectively rest right angles Second self-rejoining line shown situation space sphere sphere whose centre square straight line succession surface taken terminated thing third side touch triangle triangle ABC true turned unequal universally Wherefore whole

### Populære avsnitt

Side 51 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.

Side 109 - PARALLELOGRAMS upon the same base, and between the same parallels, are equal to one another...

Side 111 - Parallelograms upon the same base and between the same parallels, are equal to one another.

Side 120 - If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angle contained by these two sides is a right angle.

Side 72 - Any two sides of a triangle are together greater than the third side.

Side 55 - From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, whereof AB is the greater.

Side 70 - Any two angles of a triangle are together less than two right angles.

Side 138 - ... the exterior angle equal to the interior and opposite on the same side of the line ; and likewise the two interior angles on the same side of the line together equal to two right angles.

Side 106 - THE straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel.