## Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids; to which are Added Elements of Plane and Spherical TrigonometryG. Long, 1819 - 333 pages |

### From inside the book

Results 1-5 of 83

Page vii

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**parallel**lines . In the third Book , the remarks con- cerning the angles made by a straight line , and the circumference of a circle , are left out , as tending to perplex one who has advanced no farther than the ele- ments of the ... Page 20

... figures besides these , are called Trapeziums . XXX .

... figures besides these , are called Trapeziums . XXX .

**Parallel**straight lines , are such as are in the same plane , and which , being produced ever so far both ways , do not meet . : POSTULATES . I. LET it be granted that a 20 ELEMENTS. Page 21

... to one another . XI . " Two straight lines which intersect one another , cannot be both < 6

... to one another . XI . " Two straight lines which intersect one another , cannot be both < 6

**parallel**to the same straight line . " PROPOSITION I. PROBLEM . To describe an equilateral triangle upon OF GEOMETRY . BOOK I. 21. Page 38

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**parallel**. Let the straight line EF , which falls upon the two straight lines AB , CD make the alternate angles AEF , EFD equal to one another ; AB is**parallel**to CD . For , if it be not**parallel**, AB and CD being produced shall meet ... Page 39

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**parallel**( 30. Def . ) to one another . AB therefore is**parallel**to CD . Where- fore , if a straight line , & c . Q. E. D. PROP . XXVIII . THEOR . If a straight line falling upon two other straight lines makes the exterior angle equal ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD altitude angle ABC angle ACB angle BAC angle EDF arch AC base BC bisected centre circle ABC circumference cosine cylinder demonstrated diameter draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given straight line greater hypotenuse inscribed join less Let ABC Let the straight line BC magnitudes meet opposite angle parallel parallelogram perpendicular polygon prism PROB produced proportionals proposition Q. E. D. COR Q. E. D. PROP radius ratio rectangle contained rectilineal figure remaining angle segment semicircle shewn side BC sine solid angle solid parallelepipeds spherical angle spherical triangle square straight line AC THEOR third touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 56 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 19 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

Page 33 - THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Page 62 - In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the...

Page 62 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle, is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. Let ABC be an obtuse-angled triangle, having the obtuse angle...

Page 130 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 76 - THE diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote ; and the greater is nearer to the centre than the less.* Let ABCD be a circle, of which...

Page 36 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.

Page 18 - 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 55 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.