## 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

Because by

together equal to four right angles , and A D the two angles BAD , ADC are

together equal to the two angles BCD , ABC . wherefore BAD , ADC are the half

of all the ...

Because by

**Prop**. 32. B. 1. all the angles of the quadrilateral figure ABCD aretogether equal to four right angles , and A D the two angles BAD , ADC are

together equal to the two angles BCD , ABC . wherefore BAD , ADC are the half

of all the ...

Page 457

14. and for a like reason

will make the thing clear ; suppose it were to be demonstrated , That if a

inagnitude A together with a given magnitude has a given ratio to another

magnitude B ...

14. and for a like reason

**Prop**. 15. has been added to the Data . one examplewill make the thing clear ; suppose it were to be demonstrated , That if a

inagnitude A together with a given magnitude has a given ratio to another

magnitude B ...

Page 458

mediately placed , as being of the same kind . it is now put into its proper place .

but

, which is here

...

mediately placed , as being of the same kind . it is now put into its proper place .

but

**Prop**. 21. in the Greek is left out , as being the fame with**Prop**. 14. in that text, which is here

**Prop**. 18 . PRO P. XXIV . This , which is**Prop**. 13. in the Greek , is...

Page 459

which is

its Demonstration is contained in that of the first part of

**Prop**. 63. in the Greek text is omitted , being only a case of**Prop**. 49. in tiiat text ,which is

**Prop**. 53. in this Edition .**PROP**. LVIII . This is not in the Greek text , butits Demonstration is contained in that of the first part of

**Prop**. 54. in that text ... Page 460

PRO P. LXX . This is

particular case of it , and is therefore omitted . Dr. Gregory in the Demonstration of

PRO P. LXX . This is

**Prop**. 62 in the Greek text ;**Prop**. 7 8. in that text is only aparticular case of it , and is therefore omitted . Dr. Gregory in the Demonstration of

**Prop**. 62. cites the 49 .**Prop**. Dat . to prove that the ratio of the figure AEB to the ...### What people are saying - Write a review

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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.