The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected and Some of Euclid's Demonstrations are Restored. Also, to this Second Edition is Added the Book of Euclid's Data. In Like Manner Corrected. viz. the first six books, together with the eleventh and twelfthR. and A. Foulis, 1762 - 466 sider |
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Side 43
... rectangle contained by the two straight lines , is equal to the rectangles contained by the undivided line , and the feveral parts of the divided line . Let Book II . 2. II . I. b . 3. OF EUCLID . 43.
... rectangle contained by the two straight lines , is equal to the rectangles contained by the undivided line , and the feveral parts of the divided line . Let Book II . 2. II . I. b . 3. OF EUCLID . 43.
Side 44
... rectangle contained by the straight lines A , BC is equal to the rect- angle contained by A , BD ; and to that contained by A , DE ; and alfo to that contained by A ... rectangle rectangle contained by BA , AC ; for it is 44 THE ELEMENTS.
... rectangle contained by the straight lines A , BC is equal to the rect- angle contained by A , BD ; and to that contained by A , DE ; and alfo to that contained by A ... rectangle rectangle contained by BA , AC ; for it is 44 THE ELEMENTS.
Side 45
... rectangle contained by AB , AC toge- ther with the rectangle AB , BC , is equal to the fquare of AB . If therefore a straight line , & c . Q. E. D. IF PROP . III . THEOR . Fa ftraight line be divided into any two parts , the rectangle ...
... rectangle contained by AB , AC toge- ther with the rectangle AB , BC , is equal to the fquare of AB . If therefore a straight line , & c . Q. E. D. IF PROP . III . THEOR . Fa ftraight line be divided into any two parts , the rectangle ...
Side 46
... rectangle contained by AC , CB , for GC is equal to CB ; therefore GE is alfo equal to the rectangle AC , CB ; wherefore AG , GE are equal to twice the rectangle AC , CB . and HF , CK are the fquares of AC , CB ; wherefore the four ...
... rectangle contained by AC , CB , for GC is equal to CB ; therefore GE is alfo equal to the rectangle AC , CB ; wherefore AG , GE are equal to twice the rectangle AC , CB . and HF , CK are the fquares of AC , CB ; wherefore the four ...
Side 47
... rectangle AD , DB together with the fquare of CD , is equal to the fquare of CB . Book II . Upon CB defcribe the fquare CEFB , join BE , and thro ' D draw a . 46. r . DHG parallel to CE or BF ; and thro ' H draw KLM parallel to b . 31 ...
... rectangle AD , DB together with the fquare of CD , is equal to the fquare of CB . Book II . Upon CB defcribe the fquare CEFB , join BE , and thro ' D draw a . 46. r . DHG parallel to CE or BF ; and thro ' H draw KLM parallel to b . 31 ...
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alfo alſo angle ABC angle BAC bafe baſe BC is equal BC is given becauſe the angle becauſe the ratio bifected Book XI cafe circle ABCD circumference cone confequently conftruction cylinder defcribed demonftrated drawn equal angles equiangular equimultiples Euclid excefs faid fame multiple fame ratio fame reaſon fecond fegment fhall fhewn fides fimilar firft firſt folid angle fome fore fquare of BC given angle given in fpecies given in magnitude given in pofition given magnitude given ratio given ſtraight line gnomon greater join lefs leſs Let ABC likewife line BC muſt oppofite parallel parallelepipeds parallelogram perpendicular polygon prifms Propofition proportionals pyramid Q. E. D. PROP reafon rectangle rectangle contained rectilineal figure right angles ſhall ſphere ſquare ſtraight line AB thefe THEOR theſe thro tiple triangle ABC wherefore
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Side 5 - Let it be granted that a straight line may be drawn from any one point to any other point.
Side 163 - 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.
Side 48 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.
Side 73 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle...
Side 105 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Side 3 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Side 167 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Side 54 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.
Side 47 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Side 37 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.