## 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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**perpendicular**to a given given point without it . Let AB be the given straight line , which may be produced to any length both ways , and let C be a point without it . It is required to draw a straight line**perpendicular**to AB from the ... Page 55

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**perpendicular**falls , and the straight line intercept- ed without the triangle between the**perpendicular**and the obtuse angle . Let ABC be an obtuse angled triangle , having the obtuse angle ACB , and from the point A let AD be drawn ... Page 56

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**perpendicular**let fall upon it from the opposite angle , and the acute angle . Let ABC be any triangle , and the angle at B one of its acute angles , and upon BC one of the fides containing it let fall the**perpendicular**AD from the ... Page 57

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**perpendicular**to BC ; then is BC the ftraight line between the**perpendicular**and the acute angle at B. and it is manifest that the squares of AB , BC are e- qual to the square of AC , and twice the square of BC . Therefore in every ... Page 74

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**perpendicular**to FA , and let it meet the cir- cumference in H. and because AGD is a right angle , and DAG less than a right angle , DA is great- erd than DG . but DA is equal to DH ; therefore DH is greater than DG , the less than the ...### 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...