## 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 |

### From inside the book

Results 1-5 of 100

Page 18

In BD take any point F , and from AE , the greater , cut off AG equal to AF , the less

, and

AC are equal to the two GA , AB , each to each ; and they contain the angle ...

In BD take any point F , and from AE , the greater , cut off AG equal to AF , the less

, and

**join**FC , GB . Because AF is equal to AG , and AB to AC ; the two sides FA ,AC are equal to the two GA , AB , each to each ; and they contain the angle ...

Page 19

For , if AB bę not equal to AC , one of them is greater than ibe other : Let AB be

the greater , and from it cut off DB e- a 3. ti qual to AC , the less , and

thereA fore , because in the triangles DBC , ACB , DB is equal to AC , and EC ...

For , if AB bę not equal to AC , one of them is greater than ibe other : Let AB be

the greater , and from it cut off DB e- a 3. ti qual to AC , the less , and

**join**DC ;thereA fore , because in the triangles DBC , ACB , DB is equal to AC , and EC ...

Page 21

Take any point D in AB , and from AC cut off AE 2-8 3. . qual to ĄD ;

upon iç describe an equilateral triangle DEF ; А bi . I then

line AF bifects the angle BAC . Because AĎ is equal to AE , and AF is common to

...

Take any point D in AB , and from AC cut off AE 2-8 3. . qual to ĄD ;

**join**DE , andupon iç describe an equilateral triangle DEF ; А bi . I then

**join**AF ; the straightline AF bifects the angle BAC . Because AĎ is equal to AE , and AF is common to

...

Page 26

Bisect a AC in E ,

also FC , and produce AC to G. Because AE is equal to EC , and BE to EF ; AE ,

EB E are equal to CE , EF , each to each ; and the angle AEB is equal to the

angle ...

Bisect a AC in E ,

**join**BE A and produce it to F , and make EF equal to BE ;**join**also FC , and produce AC to G. Because AE is equal to EC , and BE to EF ; AE ,

EB E are equal to CE , EF , each to each ; and the angle AEB is equal to the

angle ...

Page 27

Let ABC be a triangle , of which the Gde AC is greater than the side AB ; the

angle A. ABC is also greater than the angle BCA . Because AC is greater than D

AB , make AD equal to AB , and

of ...

Let ABC be a triangle , of which the Gde AC is greater than the side AB ; the

angle A. ABC is also greater than the angle BCA . Because AC is greater than D

AB , make AD equal to AB , and

**join**BD ; and because A DB is the exterior angleof ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Other editions - View all

### 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.