## The Elements of Euclid, Viz: 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. the first six books, together with the eleventh and twelfth |

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Page 12

An obtuse angled triangle , is that which has an obtuse angle . 44 XXIX . An acute

angled triangle , is that which has three acute angles . XXX . Of four fided figures ,

a

An obtuse angled triangle , is that which has an obtuse angle . 44 XXIX . An acute

angled triangle , is that which has three acute angles . XXX . Of four fided figures ,

a

**square**is that which has all its fides equal , and all its angles right angles . Page 48

To describe a

Let AB be the given straight line ; it is required to describe a

From the point A draw a AC at right angles to AB ; and make b AD equal to AB ,

and ...

To describe a

**square**upon a given straight line . a 11.1 . b 3. I. C 31. I. d 34. I. DLet AB be the given straight line ; it is required to describe a

**square**upon AB .From the point A draw a AC at right angles to AB ; and make b AD equal to AB ,

and ...

Page 49

FBC : Now the parallelogram BL is double 5 of the triangle g 41.1 , ABD ,

because they are upon the same base BD , and between the same parallels , BD

, AL ; and the

upon ...

FBC : Now the parallelogram BL is double 5 of the triangle g 41.1 , ABD ,

because they are upon the same base BD , and between the same parallels , BD

, AL ; and the

**square**GB is double of the triangle FBC , because these also areupon ...

Page 50

If the

equal to the Squares upon the other sides BA , AC ; the angle BAC is a right

angle . Fron , the point A draw • AD at right angles to AC , and make AD equal to

BA , and ...

If the

**square**described upon BC , one of the sides of the tri . angle ABC , beequal to the Squares upon the other sides BA , AC ; the angle BAC is a right

angle . Fron , the point A draw • AD at right angles to AC , and make AD equal to

BA , and ...

Page 52

IF a straight line be divided into any two parts , the rectangles contained by the

whole and each of the parts , are together equal to the

CB Let the straight line AB be divided into A any two parts in the point C ; the ...

IF a straight line be divided into any two parts , the rectangles contained by the

whole and each of the parts , are together equal to the

**square**of the whole line .CB Let the straight line AB be divided into A any two parts in the point C ; the ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe becauſe Book Book XI caſe circle circle ABCD circumference common cone cylinder definition demonſtrated deſcribed diameter divided double draw drawn equal equal angles equiangular equimultiples exceſs fame fides firſt folid fore four fourth given angle given in poſition given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane produced prop proportionals propoſition pyramid radius rectangle rectangle contained remaining right angles ſame ſecond ſegment ſhall ſides ſimilar ſolid ſphere ſquare ſquare of AC Take taken theſe third triangle ABC wherefore whole

### Popular passages

Page 32 - 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...

Page 165 - D ; wherefore the remaining angle at C is equal to the remaining angle at F ; Therefore the triangle ABC is equiangular to the triangle DEF.

Page 170 - 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 10 - When several angles are at one point B, any ' one of them is expressed by three letters, of which ' the letter that is at the vertex of the angle, that is, at ' the point in which the straight lines that contain the ' angle meet one another, is put between the other two ' letters, and one of these two is...

Page 55 - 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.

Page 32 - ... then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz.

Page 45 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 211 - AB shall be at right angles to the plane CK. Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK ; take any point F in CE, from which draw FG in the plane DE at right D angles to CE ; and because AB is , perpendicular to the plane CK, therefore it is also perpendicular to every straight line in that plane meeting it (3.

Page 38 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore 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 304 - Thus, if B be the extremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length : For if it have any, this length must either be part of the length of the line AB, or of the line KB.