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, Books 1-6
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abc is equal adjacent angles Algebra angle abc angle bac angle bcd angles equal base bc bisect centre circle abc circumference diameter double draw equal angles equal circles equal right lines equal to f equi equimultiples Euclid EUCLID'S ELEMENTS exceed exterior angle fore four magnitudes fourth Geometry given circle given point given rectilineal angle given right line gnomon greater ratio hence inscribed join less Let abc multiple opposite angle parallel parallelogram perpendicular polygon proportional q. e. d. Deduction q. e. d. PROPOSITION rectangle contained remaining angle right line ac right line drawn segment side bc similar and similarly square of ac subtending Theorem third three right lines tiple touches the circle triangle abc triangle def whence wherefore the angle whole
Page 2 - 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.
Page 37 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Page 154 - 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 31 - 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.
Page 4 - 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.
Page 32 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Page 77 - 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.
Page 92 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.
Page 99 - 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.