## 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 292

B. I. There are two

but is as necessary as the other ... 5. which is neceffary to the Demonstration of

this

B. I. There are two

**cases**of this Proposition , one of which is not in the Greek text ,but is as necessary as the other ... 5. which is neceffary to the Demonstration of

this

**case**, can be of no use at all in the Elements , or any where else , but in this ... Page 302

There are two

angles are in a segment not greater than a ... fimple Demonstration is given than

that which is in Commandine , as being derived only from the first

the ...

There are two

**cases**of this Proposition , the second of which , viz . when theangles are in a segment not greater than a ... fimple Demonstration is given than

that which is in Commandine , as being derived only from the first

**case**, withoutthe ...

Page 311

There is also a third

in which AE in the first , or EB in the second of the two other

than D ; and in this any equimultiples , as the doubles , of AE , EB are to be taken

...

There is also a third

**case**, which is not mentioned in this demonstration , viz . thatin which AE in the first , or EB in the second of the two other

**cases**, is greaterthan D ; and in this any equimultiples , as the doubles , of AE , EB are to be taken

...

Page 318

A second

which the exterior angle of a triangle is bisected by a straight line . the

Demonstration of it is very like to that of the first

, probably ...

A second

**case**of this , as useful as the first , is given in Prop . A , tiz . the**case**inwhich the exterior angle of a triangle is bisected by a straight line . the

Demonstration of it is very like to that of the first

**case**, and upon this account may, probably ...

Page 440

But if the given angle ABC of the parallelogram AC be not a right angle , in this

sides AB , BC to the parallelogram p . 62. Dat . AC is given n ; and AC is given ...

But if the given angle ABC of the parallelogram AC be not a right angle , in this

**case**because ABC is a given angle , the ratio of the rectangle contained by thesides AB , BC to the parallelogram p . 62. Dat . AC is given n ; and AC is given ...

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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.