Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Appendix by Thos. Kirkland. the first six books |
From inside the book
Results 1-5 of 58
Page 8
... angles contained by those sides equal to each other ; they shall likewise have their bases or third sides equal , and the two triangles ... BAC equal to the included angle F Then shall the base BC be equal to the base 8 EUCLID'S ELEMENTS .
... angles contained by those sides equal to each other ; they shall likewise have their bases or third sides equal , and the two triangles ... BAC equal to the included angle F Then shall the base BC be equal to the base 8 EUCLID'S ELEMENTS .
Page 9
... angle BAC is equal to the angle EDF ; therefore also the point C shall coincide with the point F , because AC is equal to DF ; but the point B was shewn to coincide with the point E ; wherefore the base BC shall coincide with the base ...
... angle BAC is equal to the angle EDF ; therefore also the point C shall coincide with the point F , because AC is equal to DF ; but the point B was shewn to coincide with the point E ; wherefore the base BC shall coincide with the base ...
Page 12
... angle ECD is greater than the angle BCD ; ( ax . 9. ) therefore also the angle FDC is greater than the angle BCD ... BAC shall be equal to the angle EDF . For , if the triangle ABC be applied to DEF , so that the point B be on E , and ...
... angle ECD is greater than the angle BCD ; ( ax . 9. ) therefore also the angle FDC is greater than the angle BCD ... BAC shall be equal to the angle EDF . For , if the triangle ABC be applied to DEF , so that the point B be on E , and ...
Page 13
... angle , that is , to divide it into two equal angles . Let BAC be the given rectilineal angle . It is required to bisect it . A A B In AB take any point D ; from AC cut off AE equal to AD , ( 1. 3. ) and join DE ; on the side of DE ...
... angle , that is , to divide it into two equal angles . Let BAC be the given rectilineal angle . It is required to bisect it . A A B In AB take any point D ; from AC cut off AE equal to AD , ( 1. 3. ) and join DE ; on the side of DE ...
Page 17
... angles ' shall be equal . Let the two straight lines AB , CD cut one another in the point E. Then the angle AEC ... BAC . A F B E G Bisect AC in E , ( 1. 10. ) and join BE ; produce BE to F , making EF equal to BE , ( 1. 3. ) and join FC ...
... angles ' shall be equal . Let the two straight lines AB , CD cut one another in the point E. Then the angle AEC ... BAC . A F B E G Bisect AC in E , ( 1. 10. ) and join BE ; produce BE to F , making EF equal to BE , ( 1. 3. ) and join FC ...
Other editions - View all
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.