Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson with Explanatory Notes ...

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Printed at the University Press, 1847 - Euclid's Elements - 478 pages
 

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Page 73 - If four magnitudes are in proportion, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference.
Page 44 - Upon a given straight line to describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.
Page 52 - ... which joins the centers of the two circles be produced to meet the circumferences, and let the extremities of this line and any other line from the point of contact be joined. From the center of the larger circle draw perpendiculars on the sides of the right-angled triangle inscribed within it. 79. In general, the locus of a point in the circumference of a circle which rolls within the circumference of another, is a curve called the Hypocycloid ; but to this there is one exception, in which the...
Page 60 - If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 14 - BD, the rectangle AB, BC is equal to the square of BD : And because from the point B, without the circle ACD, two straight lines BCA, BD are drawn to the circumference, one of which cuts, and the other meets the circle, and...
Page 40 - AB describe a segment of a circle containing an angle equal to the given angle, (in.
Page 69 - D the centre of the given circle. Through D draw CD perpendicular to BC, meeting the circumference in E, F. Join AF, and take FG to the diameter FE, as FC is to FA. The circle described passing through the two points A, G and touching the line BC in B is the circle required. Let H be the...
Page 11 - I. 25. 5. If none of the consequences so deduced be known to be either true or false, proceed to deduce other consequences from all or any of these, as in (2). 6. Examine these results, and proceed as in (3) and (4) ; and if still without any conclusive indications of the truth or falsehood of the alleged theorem, proceed still further, until such are obtained.
Page 39 - CC' describe a semicircle ; with center C, and radius equal to the sum of the radii of the two circles describe another circle cutting the semicircle in D, join CD cutting the circumference in A, through C draw CB parallel to CA and join AB. 84. The possibility is obvious. The point of bisection of the segment intercepted between the convex circumferences will be the center of one of the circles : and the center of a second circle will be found to be the point...
Page 31 - ... angle ACB double of the angle ABC, and let the perpendicular AD be drawn to the base BC. Take DE equal to DC and join AE. Then AE may be proved to be equal to EB. If ACB be an obtuse angle, then AC is equal to the sum of the segments of the base, made by the perpendicular from the vertex A.

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