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-vi
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ABC is equal ABCD Algebra angle ABC angle ACB angle BAC angles equal applied base base BC bisected centre circle ABC circumference common consequently contained Deduction demonstrated described diameter divided double draw equal equal angles equiangular equilateral equimultiples exceed extremes fall figure fore four fourth Geometry given circle given right line greater greater ratio half hence inscribed join less Let ABC magnitudes manner meet multiple opposite parallel parallelogram pass perpendicular PROBLEM produced proportional PROPOSITION Q. E. D. PROPOSITION ratio reason rectangle rectangle contained rectilineal figure remaining angle right angles right line AC segment shown sides similar square taken THEOREM third triangle ABC unequal whence Wherefore whole
Page 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.
Page 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Page 33 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Page 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.
Page 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.
Page 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.
Page 28 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Page 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.
Page 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.
Page 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.