Elements of Geometry, Containing the First Six Books of EuclidBaldwin, Cradock, and Joy, 1826 - 180 pages |
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Page 37
... rectangle , whose base is the diagonal of the trapezium , and perpendicular the aggregate of the perpendiculars drawn from the vertical angles unto the diagonal . 2. To describe a parallelogram , the area and perimeter of which shall be ...
... rectangle , whose base is the diagonal of the trapezium , and perpendicular the aggregate of the perpendiculars drawn from the vertical angles unto the diagonal . 2. To describe a parallelogram , the area and perimeter of which shall be ...
Page 40
... sides and angles of parallelograms are equal to one с * In like manner a rectangle may be described which is contained under two right lines . another . Therefore each of the opposite angles ABE , 40 [ Book I. EUCLID'S ELEMENTS .
... sides and angles of parallelograms are equal to one с * In like manner a rectangle may be described which is contained under two right lines . another . Therefore each of the opposite angles ABE , 40 [ Book I. EUCLID'S ELEMENTS .
Page 41
... rectangle . But it was shown to be equilateral , there- fore it is a square , and has been described on the given right line AB . Q. E. F. COROLLARY . Hence every parallelogram having one right angle is a rectangle . PROPOSITION XLVII ...
... rectangle . But it was shown to be equilateral , there- fore it is a square , and has been described on the given right line AB . Q. E. F. COROLLARY . Hence every parallelogram having one right angle is a rectangle . PROPOSITION XLVII ...
Page 45
... rectangle comprehended under the whole and divided line , is equal to the rectangles con- tained under the whole line , and the several segments of the divided line . Let A and BC be two right lines , and let BC be any how divided in ...
... rectangle comprehended under the whole and divided line , is equal to the rectangles con- tained under the whole line , and the several segments of the divided line . Let A and BC be two right lines , and let BC be any how divided in ...
Page 46
... rectangle DL is contained under A and ᎠᎬ ; because DK , that is d BG , is equal to a ; so likewise the rectangle EH is that contained under a and EC . Therefore the rectangle under A and BC is equal to the rectangles under A and BD ...
... rectangle DL is contained under A and ᎠᎬ ; because DK , that is d BG , is equal to a ; so likewise the rectangle EH is that contained under a and EC . Therefore the rectangle under A and BC is equal to the rectangles under A and BD ...
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Common terms and phrases
ABC is equal adjacent angles Algebra angle ABC angle ACB angle BAC angles equal base BC bisected centre circle ABC circum circumference BC 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 Geometry given circle given point given right line gnomon greater ratio hence inscribed join less Let ABC multiple parallel parallelogram perpendicular polygon proportional Q. E. D. Deduction Q. E. D. PROPOSITION rectangle contained remaining angle right angles right line AB right line AC sector HEF segment side BC similar and similarly square of AC subtending THEOREM tiple touches the circle triangle ABC triangle DEF whence whole
Popular passages
Page xxvi - 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 148 - 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 8 - To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line : it is required to divide it intotwo equal parts.
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 99 - To describe a square about a given circle. Let ABCD be the given circle ; it is required to describe a square about it. . Draw two diameters AC, BD of the circle ABCD, at right angles to one another, and through the points A, B, • 17.3. C, D, draw...
Page 7 - ... equal to them, of the other. Let ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz. AB equal to DE, and AC to DF ; but the base CB greater than the base EF ; the angle BAC is likewise greater than the angle EDF.
Page 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.