## 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, to this Second Edition is Added the Book of Euclid's Data. In Like Manner Corrected. viz. the first six books, together with the eleventh and twelfth |

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Page 20

Book 1. than the angle ACB ; but it is not ; therefore the side AC is not less than

AB , and it has been shewn that it is not equal to AB . therefore AC is

AB . wherefore the

Book 1. than the angle ACB ; but it is not ; therefore the side AC is not less than

AB , and it has been shewn that it is not equal to AB . therefore AC is

**greater**thanAB . wherefore the

**greater**angle , & c , Q. E. D. b 18. 1 . PROP . XX . THEOR . Page 23

AB equal to DE , Book I. and AC to DF ; but the angle BAC

EDF . the base BC is also

DE be the side which is not

AB equal to DE , Book I. and AC to DF ; but the angle BAC

**greater**than the angleEDF . the base BC is also

**greater**than the base EF . Of the two sides DE , DF letDE be the side which is not

**greater**than the other , and at the point D in the ... Page 126

Let A , B have each of them the same ratio to C ; A is equal to B. for if they are not

equal , one of them is

was sewn in the preceeding Proposition , there are some equimultiples of A and

...

Let A , B have each of them the same ratio to C ; A is equal to B. for if they are not

equal , one of them is

**greater**than the other ; let A be the**greater**; then , by whatwas sewn in the preceeding Proposition , there are some equimultiples of A and

...

Page 130

Book V. B and D , N and K are equimultiples ; if M be

than K ; and if equal , equal ; and if less , less b ; but G is b . s . Def.s.

K , therefore M is

Book V. B and D , N and K are equimultiples ; if M be

**greater**than N , G is**greater**than K ; and if equal , equal ; and if less , less b ; but G is b . s . Def.s.

**greater**thanK , therefore M is

**greater**than N. but H is not**greater**than L ; and M , H are ... Page 336

Rook Xf . quired that B , C , D together be

mtaking away B , C , the remaining one D must be

above B and C. take therefore any magnitude D which is less than A , B , C

together ...

Rook Xf . quired that B , C , D together be

**greater**than A , from each of thesemtaking away B , C , the remaining one D must be

**greater**than the excess of Aabove B and C. take therefore any magnitude D which is less than A , B , C

together ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe BC is given 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 in ſpecies given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane priſm produced PROP proportionals Propoſition pyramid reaſon rectangle remaining right angles ſame ſecond ſegment ſhall ſhewn ſides ſimilar ſolid ſquare ſquare of AC taken THEOR theſe third thro triangle ABC wherefore whole

### Popular passages

Page 5 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 163 - 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 48 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 73 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle...

Page 105 - 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 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 167 - Similar triangles are to one another in the duplicate ratio of their homologous sides.

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 47 - 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 37 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.