## 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 16

Wherefore from the given point A a straight line AL has been drawn equal to the

given straight line BC . Which was to be done , PRO P. III . PROB . " ROM the

greater of two given straight lines to cut off a part equal to the

be ...

Wherefore from the given point A a straight line AL has been drawn equal to the

given straight line BC . Which was to be done , PRO P. III . PROB . " ROM the

greater of two given straight lines to cut off a part equal to the

**less**. Let AB and Cbe ...

Page 26

THE O R. A . NY two angles of a triangle are together

А Let ABC be any triangle ; any two of its angles together are

angles . Produce BC to D ; and because ACD is the exterior angle of the triangle

...

THE O R. A . NY two angles of a triangle are together

**less**than two right angles .А Let ABC be any triangle ; any two of its angles together are

**less**than two rightangles . Produce BC to D ; and because ACD is the exterior angle of the triangle

...

Page 27

I. BCA are

BAC , ACB , as also CAB , ABC are

angles , & c . Q. E. D. PRO P. XVIII . THEOR . THE greater side of every triangle is

...

I. BCA are

**less**than two right angles . In like manner , it may demonstrated thatBAC , ACB , as also CAB , ABC are

**less**than two right angles . Therefore any twoangles , & c . Q. E. D. PRO P. XVIII . THEOR . THE greater side of every triangle is

...

Page 28

Book I. ABC would be

fide AC is not

AB ; therefore AC is greater than AB . Wherefore the greater angle , & c . Q. E. D.

...

Book I. ABC would be

**less**than the angle ACB ; but it is not ; there , m fore thefide AC is not

**less**than AB ; and it has been shewn € 18 , 1 . that it is not equal toAB ; therefore AC is greater than AB . Wherefore the greater angle , & c . Q. E. D.

...

Page 32

... is it lefs ; becaufe then the base BC would be

B E F is not ; therefore the angle BAC is not

fhewn that it is not equal to it ; therefore the angle BAC is greater than the angle

EDF .

... is it lefs ; becaufe then the base BC would be

**less**than the base EF ; but itB E F is not ; therefore the angle BAC is not

**less**than the angle EDF ; and it wasfhewn that it is not equal to it ; therefore the angle BAC is greater than the angle

EDF .

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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.