## 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 ... |

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Side vii

But there are others where the reasoning depends altogether on the supposition

of a continuity of motion; as, notably, in the very pinch and nip of the argument on

Parallel Lines in

But there are others where the reasoning depends altogether on the supposition

of a continuity of motion; as, notably, in the very pinch and nip of the argument on

Parallel Lines in

**Prop**. XXVIII D in the First Book, where the conclusion rests PR ... Side viii

... the conclusion rests entirely on the impossibility of a certain line ceasing to cut

a series of other lines during a continuous motion, in a way incapable of being

supplied by any succession of insulated positions; as likewise in part of

... the conclusion rests entirely on the impossibility of a certain line ceasing to cut

a series of other lines during a continuous motion, in a way incapable of being

supplied by any succession of insulated positions; as likewise in part of

**Prop**. Side 6

Let A and B be equal, and let B be equal to C. A shall also be equal to C. For A is

equal to B, and C is equal to B; therefore (by

CoR. 2. If of equals, one be greater, or less, than some thing else; the rest are ...

Let A and B be equal, and let B be equal to C. A shall also be equal to C. For A is

equal to B, and C is equal to B; therefore (by

**Prop**. I. above) A and C are equal.CoR. 2. If of equals, one be greater, or less, than some thing else; the rest are ...

Side 7

I.) A is equal to D. But C is also equal to D ; therefore (by

equal. CoR. 4. Of magnitudes, if to equals be added the same, the sums are

equal. Let A and B be equal magnitudes, to each of which, one after the other, is

added ...

I.) A is equal to D. But C is also equal to D ; therefore (by

**Prop**. I.) A and C areequal. CoR. 4. Of magnitudes, if to equals be added the same, the sums are

equal. Let A and B be equal magnitudes, to each of which, one after the other, is

added ...

Side 9

... A is equal to C, the sum of A and C is equal to the double of A; and for the like

reason, the sum of B and D is equal to the double of B. Therefore (by Cor. 5), the

double of A, added to the double of B, is equal to the sum of A, B,

... A is equal to C, the sum of A and C is equal to the double of A; and for the like

reason, the sum of B and D is equal to the double of B. Therefore (by Cor. 5), the

double of A, added to the double of B, is equal to the sum of A, B,

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Geometry Without Axioms Or the First Book Euclid's Elements: With ... 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 adjacent angles alternate angles angle ABC angle ACB angle BAC angular points aret equal assigned point axis base BC bisected called CEGDHF central distances change of place circle coincide throughout Constr demonstrated double equal straight lines equal to AC equal to EF equilateral triangle Euclid exterior angle extremities four right angles Geometry given straight line greater hard body inclose a space instance INTERC Intercalary Book ist equal join line AC magnitude manner meet opposite angles parallelogram parity of reasoning pass perpendicular prolonged Prop PROPOSITION proved quadrilateral radii radius rectilinear figure remain unmoved remaining angle remains at rest respectively Scholium self-rejoining line shown side BC side opposite situation sphere whose centre straight line BC tessera THEOREM.–If third side triangle ABC tryo turned unlimited length Wherefore willf

### 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.