## 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, The Book of Euclid's Data, in Like Manner Corrected. viz. The first six books, together with the eleventh and twelfth |

### From inside the book

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**construction**being made , because EA , EB from the center are equal to one another , the angle EAF is equal to the angle EBF ; and the right angle AFE is equal to the right angle BFE . therefore in the two triangles EAF , EBF there are ... Page 72

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**construction**be ing made , it may , as before , be demon- strated that AB is double of AF and CD double of CG , and that the squares of EF , FA are equal to the squares of EG , GC ; of which the square of FE is equal to the square of EG ... Page 73

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**construction**being made , EH is less than EK . because BC is greater than FG , BH likewife is greater than FK . and the squares of BH , HE are equal to the squares of FK , KE , of which the square of BH is greater than the square of FK ... Page 125

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**construction**, EF is F A greater than D ; therefore C F A C the whole EG is greater than K and D together . but K together with D is equal to L ; therefore EG is greater than L ; but FG is not greater than L ; and EG , FG are ... Page 149

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**construction**be made as in the Proposition , the Demonftra- tion will be the fame . a PROP . II . THEOR . straight line be drawn parallel to one of the fides Sce N. of a triangle , it shall cut the other fides , or these produced ...### Other editions - View all

### Common terms and phrases

AC is equal alfo alſo angle ABC angle BAC bafe baſe BC is given becauſe the angle biſected Book XI cafe cauſe circle ABCD circumference cone conſtruction cylinder demonftrated deſcribed diameter drawn EFGH equal angles equiangular equimultiples Euclid exceſs faid fame multiple fame ratio fame reaſon fegment fides fimilar firſt folid angle fore given angle given in magnitude given in poſition given in ſpecies given magnitude given ratio given ſtraight line gnomon greater join leſs line BC oppoſite parallel parallelepipeds parallelogram paſs perpendicular proportionals Propoſition pyramid Q. E. D. PROP rectangle contained rectilineal figure right angles ſame ſecond ſhall ſhall be equal ſhewn ſide ſpace ſphere ſquare of AC ſtraight line AC THEOR theſe thoſe thro tiple triangle ABC vertex wherefore

### Popular passages

Page 156 - 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 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.

Page 323 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Page 92 - If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it ; if the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, be equal to the square- of the line which meets it, the line which meets shall touch the circle.

Page 80 - EA : and because AD is equal to DC, and DE common to the triangles ADE, CDE, the two sides AD, DE are equal to the two CD, DE, each to each ; and the angle ADE is equal to the angle CDE, for each of them is a right angle ; therefore the base AE is equal (4.

Page 52 - If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced.

Page 36 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 2 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

Page 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.

Page 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...