Elements of Geometry: Containing the Principal Propositions in the First Six, and the Eleventh and Twelfth Books of Euclid. With Notes, Critical and Explanatory

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Johnson, 1803 - Euclid's Elements - 279 pages
 

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Page 63 - AB is the greater. If from AB there be taken more than its half, and from the remainder more than its half, and so on ; there shall at length remain a magnitude less than C. For C may be multiplied, so as at length to become greater than AB.
Page 31 - THE Angle formed by a Tangent to a Circle, and a Chord drawn from the Point of Contact, is Equal to the Angle in the Alternate Segment.
Page 23 - To find the centre of a given circle. Let ABC be the given circle ; it is required to find its centre. Draw within it any straight line AB, and bisect (I.
Page 63 - Lemma, if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there shall at length remain a magnitude less than the least of the proposed magnitudes.
Page 24 - 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* the straight line GAH touching the circle in the a 17. 3. point A, and at the point A, in the straight line AH, makeb b 23.
Page 21 - The radius of a circle is a right line drawn from the centre to the circumference.
Page 30 - To bisect a given arc, that is, to divide it into two equal parts. Let ADB be the given arc : it is required to bisect it.
Page 7 - Beciprocally, when these properties exist for 'two right lines and a common secant, the two lines are parallel.* — Through a given point, to draw a right line parallel to a given right line, or cutting it at a given angle, — Equality of angles having their sides parallel and their openings placed in the same direction.

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