Euclid's Elements of geometry, transl. To which are added, algebraic demonstrations to the second and fifth books; also deductions in the first six, eleventh and twelfth books, with notes, by G. Phillips. Part 1, containing book i-vi1826 |
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Página 5
... wherefore the three , CA , AB , BC , are equal to one another ; and , consequently , the triangle ABC is equilateral , and it is described upon the given finite right line AB . 2. E. F. PROPOSITION II . PROBLEM . From a given point to ...
... wherefore the three , CA , AB , BC , are equal to one another ; and , consequently , the triangle ABC is equilateral , and it is described upon the given finite right line AB . 2. E. F. PROPOSITION II . PROBLEM . From a given point to ...
Página 6
... Wherefore each of them , AL , BC , is equal to CG . And things which are equal to the same thing are equal to one another . Whence AL is equal to BC . Therefore from a given point , AL has been drawn , & c . g . E. F. * PROPOSITION III ...
... Wherefore each of them , AL , BC , is equal to CG . And things which are equal to the same thing are equal to one another . Whence AL is equal to BC . Therefore from a given point , AL has been drawn , & c . g . E. F. * PROPOSITION III ...
Página 6
... Wherefore c will also coincide with F : for the right line ac is equal to the right line DF ; but the point в coincides with the point E. Therefore the base BC will also coincide with the base EF . Because if the point в coinciding with ...
... Wherefore c will also coincide with F : for the right line ac is equal to the right line DF ; but the point в coincides with the point E. Therefore the base BC will also coincide with the base EF . Because if the point в coinciding with ...
Página 6
... Wherefore , because the whole angle ABG has been proved to be equal to the FC This theorem was discovered by Thales , for he is first said to have perceived and proved , that the angles at the base of every isosceles triangle are equal ...
... Wherefore , because the whole angle ABG has been proved to be equal to the FC This theorem was discovered by Thales , for he is first said to have perceived and proved , that the angles at the base of every isosceles triangle are equal ...
Página 7
... Wherefore if two angles of a triangle be equal to one another , & c . Q. E. D. COROLLARY . B Hence every equiangular triangle is also equilateral . PROPOSITION VII . THEOREM . On the same right line cannot be constituted two right lines ...
... Wherefore if two angles of a triangle be equal to one another , & c . Q. E. D. COROLLARY . B Hence every equiangular triangle is also equilateral . PROPOSITION VII . THEOREM . On the same right line cannot be constituted two right lines ...
Términos y frases comunes
ABC is equal adjacent angles Algebra angle ABC angle ACB angle BAC angles equal base BC bisected centre circle ABC circumference diameter double draw equal angles equal circles equal right lines equal to F equi equiangular equimultiples Euclid EUCLID'S ELEMENTS exceed exterior angle fore four magnitudes fourth given circle given point given right line gnomon greater ratio hence inscribed isosceles triangle join less Let ABC multiple parallel parallelogram perpendicular polygon proportional Q. E. D. Deductions Q. E. D. PROPOSITION rectangle contained remaining angle right line AB right line AC right line drawn segment side AC similar and similarly square of AC subtending THEOREM three right lines tiple touches the circle triangle ABC triangle DEF whence whole
Pasajes populares
Página xxx - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.
Página 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Página 33 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Página 146 - 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.
Página 27 - And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, therefore the whole angle ABD is equal to the whole angle ACD • (ax.
Página 2 - Things which are double of the same are equal to one another. 7. Things which are halves of the same are equal to one another.
Página 28 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Página 73 - DH ; (i. def. 15.) therefore DH is greater than DG, the less than the greater, which is impossible : therefore no straight line can be drawn from the point A, between AE and the circumference, which does not cut the circle : or, which amounts to the same thing, however great an acute angle a straight line makes with the diameter at the point A, or however small an angle it makes with AE, the circumference must pass between that straight line and the perpendicular AE.
Página 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.
Página 95 - IN a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF. Draw (1 7. 3.) the straight line G AH touching the circle in the point A, and. at the point A, in the straight line AH, make (23.