The Elements of Euclid: The Errors by which Theon, Or Others, Have Long Vitiated These Books, are Corrected, and Some of Euclid's Demonstrations are Restored. Also the Book of Euclid's Data, in Like Manner Corrected. viz. the first six books, together with the eleventh and twelfthJ. Collingwood, 1827 - 513 Seiten |
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Seite 37
... square upon a given straight line . Let AB be the given straight line ; it is required to describe a square upon AB . C * 30. 1 . + Constr . 1 . Constr . +2 Ax . * 44. 1 . * 11.1 . * 31. 1 . 1 . * 34. 1 . From the point A draw * AC at ...
... square upon a given straight line . Let AB be the given straight line ; it is required to describe a square upon AB . C * 30. 1 . + Constr . 1 . Constr . +2 Ax . * 44. 1 . * 11.1 . * 31. 1 . 1 . * 34. 1 . From the point A draw * AC at ...
Seite 38
... square , and it is described upon the given straight line AB . Which was to be done . COR . Hence every ... AC . B D H E L K On BC describe * the square BDEC , and on BA , AC the squares GB , HC ; and through A draw * AL parallel to BD ...
... square , and it is described upon the given straight line AB . Which was to be done . COR . Hence every ... AC . B D H E L K On BC describe * the square BDEC , and on BA , AC the squares GB , HC ; and through A draw * AL parallel to BD ...
Seite 39
... squares GB , HC upon BA , AC ; therefore the square upon the side BC is equal to the squares upon the sides BA , AC . Therefore , in any right - angled trian- gle , & c . Q. E. D. PROP . XLVIII . THEOR . If the square described upon one ...
... squares GB , HC upon BA , AC ; therefore the square upon the side BC is equal to the squares upon the sides BA , AC . Therefore , in any right - angled trian- gle , & c . Q. E. D. PROP . XLVIII . THEOR . If the square described upon one ...
Seite 41
... AC shall be equal to the square of AB . * 34. 1 . 46. 1 . A св 31. 1 . Upon AB describe * the square ADEB , and through C draw * CF , parallel to AD or BE . Then AE is equal to the rectan- gles AF , CE : but AE is the square of AB ; and ...
... AC shall be equal to the square of AB . * 34. 1 . 46. 1 . A св 31. 1 . Upon AB describe * the square ADEB , and through C draw * CF , parallel to AD or BE . Then AE is equal to the rectan- gles AF , CE : but AE is the square of AB ; and ...
Seite 42
... square of the aforesaid part . Let the straight line AB be divided into any two parts in the point C : the rectangle AB , BC shall be equal to the rectangle AC , CB , together with the square of BC . A C F D B Upon BC describe the square ...
... square of the aforesaid part . Let the straight line AB be divided into any two parts in the point C : the rectangle AB , BC shall be equal to the rectangle AC , CB , together with the square of BC . A C F D B Upon BC describe the square ...
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ABC is given AC is equal altitude angle ABC angle BAC base BC bisected centre circle ABCD circumference common logarithm cone Constr cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gnomon greater join less Let ABC logarithm multiple parallel parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle contained rectilineal figure right angles ROBERT SIMSON segment shewn sides BA similar sine solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR triangle ABC vertex wherefore
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Seite 141 - 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.
Seite 40 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. Let...
Seite 26 - If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite upon the same side, and also the two interior angles upon the same side together equal to two right angles.
Seite 46 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.
Seite 28 - Cor. angles; that is * together with four right angles. There1s, 1. fore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Seite 21 - If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.
Seite 12 - IF two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle. which is contained by the two sides...
Seite 169 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise 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.
Seite 5 - LET it be granted that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any centre, at any distance from that centre.
Seite 97 - 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.