Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Appendix by Thos. Kirkland. the first six books |
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Page 43
... shews in his Corollary to Prop . 11 , Book 1 . The following definition of straight lines has also been proposed , " Straight lines are those which , if they coincide in any two points , coin- cide as far as they are produced . " But ...
... shews in his Corollary to Prop . 11 , Book 1 . The following definition of straight lines has also been proposed , " Straight lines are those which , if they coincide in any two points , coin- cide as far as they are produced . " But ...
Page 45
... the given line . Euc . 1. 2 , shews how , from any given point , to draw a straight line equal to another straight line which is given in magnitude and position . ON THE AXIOMS . AXIOMS are usually defined to be NOTES TO BOOK I. 45.
... the given line . Euc . 1. 2 , shews how , from any given point , to draw a straight line equal to another straight line which is given in magnitude and position . ON THE AXIOMS . AXIOMS are usually defined to be NOTES TO BOOK I. 45.
Page 59
... shew that the intersection of two lines is a point . 5. Give Euclid's definition of a plane rectilineal angle . What are the limits of the angles considered in Geometry ? Does Euclid consider angles greater than two right angles ? 6 ...
... shew that the intersection of two lines is a point . 5. Give Euclid's definition of a plane rectilineal angle . What are the limits of the angles considered in Geometry ? Does Euclid consider angles greater than two right angles ? 6 ...
Page 60
... side of the equilateral tri- angle DAB be produced both ways and cut the circle whose center is B and radius BC in two points G and H ; shew that either of the dis tances DG , DH may be taken as the radius 60 EUCLID'S ELEMEN IS .
... side of the equilateral tri- angle DAB be produced both ways and cut the circle whose center is B and radius BC in two points G and H ; shew that either of the dis tances DG , DH may be taken as the radius 60 EUCLID'S ELEMEN IS .
Page 61
... Shew how a given straight line may be bisected by Euc , 1. 1 . 43. In what cases do the lines which bisect the interior angles of plane triangles , also bisect one , or more than one of the corresponding opposite sides of the triangles ...
... Shew how a given straight line may be bisected by Euc , 1. 1 . 43. In what cases do the lines which bisect the interior angles of plane triangles , also bisect one , or more than one of the corresponding opposite sides of the triangles ...
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Common terms and phrases
A₁ ABCD AC is equal Algebraically angle ABC angle ACB angle BAC angle equal Apply Euc base BC chord circle ABC constr describe a circle diagonals diameter divided draw equal angles equiangular equilateral triangle equimultiples Euclid exterior angle Geometrical given circle given line given point given straight line gnomon greater hypotenuse inscribed intersection isosceles triangle less Let ABC line BC lines be drawn multiple opposite angles parallelogram parallelopiped pentagon perpendicular plane polygon produced Prop proportionals proved Q.E.D. PROPOSITION quadrilateral quadrilateral figure radius ratio rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn similar similar triangles solid angle square on AC tangent THEOREM touch the circle triangle ABC twice the rectangle vertex vertical angle wherefore
Popular passages
Page 93 - If a straight line be bisected and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the square on the line made up of the half and the part produced. Let the straight line AB be bisected in C, and produced to D ; The squares on AD and DB shall be together double of the squares on AC and CD. CONSTRUCTION. — From the point C draw CE at right angles to AB, and make it equal...
Page 118 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.
Page 145 - If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.
Page 88 - 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 on the line between the points of section, is equal to the square on half the line.
Page 26 - ... upon the same side together equal to two right angles, the two straight lines shall be parallel to one another.
Page 36 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 144 - The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle.
Page 92 - If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts...
Page xv - In every triangle, the square of the side subtending either of the acute angles is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.
Page 67 - A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions.