## 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 twelfth |

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Page 44

Q. E. D. From this proposition it is manifest , that the

Q. E. D. From this proposition it is manifest , that the

**difference**of the squares of two unequal lines , AC , CD , is equal to the rectangle contained by ... Page 99

... contains three ; therefore BC , their

... contains three ; therefore BC , their

**difference**, contains two of the same parts : bisect * BC in E ; therefore BE , EC are each of them the fifteenth ... Page 268

... greater than the

... greater than the

**difference**of the other two : from this he shews the circles must meet one another , in one case ; and says , that it may be proved ... Page 283

... because it was fine , and might sharpen one's genius to invent others like it : whereas there is not the least

... because it was fine , and might sharpen one's genius to invent others like it : whereas there is not the least

**difference**between the two demonstrations ... Page 298

... that are equal to one another : so that there " is no

... that are equal to one another : so that there " is no

**difference**betwixt the compounding of ratios , " and the duplication or triplication of them which ...### What people are saying - Write a review

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added altitude angle ABC angle BAC base Book centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter difference divided double draw drawn equal equal angles equiangular equimultiples Euclid excess figure fore four fourth given angle given in position given in species given magnitude given ratio given straight line greater Greek half join less likewise logarithm magnitude manner meet multiple opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition proved pyramid radius reason rectangle rectilineal figure remaining right angles segment shewn sides similar sine solid sphere square square of AC taken THEOR third triangle ABC wherefore whole

### Popular passages

Page 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.

Page 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...

Page 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.

Page 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.

Page 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.

Page 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.

Page 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...

Page 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.

Page 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.

Page 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.