## 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 ... |

### From inside the book

Page ix

... the disputed principle of

... the disputed principle of

**Parallel**Lines has been reduced in bulk to one half , and in substance to the following . ... which two of the**opposite sides**are equal to one another and make equal interior angles with a side**between**them ... Page 82

It cannot be , therefore , that AB and CD being prolonged , will meet on the

It cannot be , therefore , that AB and CD being prolonged , will meet on the

**side**of B and D. And in the**same**manner ... which are in the**same**plane , and which being prolonged ever so far both ways do not meet , are called**parallel**. Page 83

makes the two interior angles on the

makes the two interior angles on the

**same side**, together equal to two right angles ; those two straight lines shall be**parallel**. For First ; where the exterior angle EGA E is equal to the interior and**opposite**angle A G B GHC . Page 84

Because the straight line AD meets the two straight lines BC and EF which are both in the

Because the straight line AD meets the two straight lines BC and EF which are both in the

**same**plane , and the alternate Constr . angles EAD and ADC aret equal to**one**another , EF and BC are # 1.27.Cor.1 ,**parallel**. Page 86

The

The

**side**of the quadrilateral figure which is**opposite**to the base , is bisected at right angles by the perpendicular drawn from the middle of the base ; and is**parallel**to the base . For it has been shown that EF bisects DC at right ...### What people are saying - Write a review

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Geometry Without Axioms; Or the First Book of Euclid's Elements. with ... Thomas Perronet Thompson No preview available - 2015 |

### Common terms and phrases

ABCD added alternate angle ABC angle BAC applied aret assigned assumed axis base bisected body Book called centre change of place circle coincide common consequently Constr continually demonstrated described double drawn equal establish 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 surface taken terminated thing third side touch triangle triangle ABC true turned unequal universally Wherefore whole

### Popular passages

Page 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.

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

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

Page 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.

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

Page 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.

Page 103 - ... twice as many right angles as the figure has sides.

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

Page 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.

Page 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.