## 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, 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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Side 84

And it is manifest , that when the centre of the circle falls within the triangle , each

of its angles is less than a right angle ... It is also rectangular ; for the straight line

BD , being the diameter of the

And it is manifest , that when the centre of the circle falls within the triangle , each

of its angles is less than a right angle ... It is also rectangular ; for the straight line

BD , being the diameter of the

**circle ABCD**, BAD is a semicircle ; wherefore the ... Side 202

CIRCLES are to one another as the squares of their diameters . * Let ABCD ,

EFGH be two circles , and BD , FH their diameters : as the square of BD to the

square of FH , so is the

square ...

CIRCLES are to one another as the squares of their diameters . * Let ABCD ,

EFGH be two circles , and BD , FH their diameters : as the square of BD to the

square of FH , so is the

**circle ABCD**, to the circle EFGH . For , if it be not so , thesquare ...

Side 203

excess of the circle EFGH above the space S : because , by the preceding lemma

, if from the greater of two unequal ... the polygon AXBOCPDR to the polygon

EKFLGMHN : but the

excess of the circle EFGH above the space S : because , by the preceding lemma

, if from the greater of two unequal ... the polygon AXBOCPDR to the polygon

EKFLGMHN : but the

**circle ABCD**is greater than the polygon contained in it ... Side 204

than the

circle EFGH . Therefore as the square of FH is to the square of BD , so is the

circle EFGH to a space less than the

to be ...

than the

**circle ABCD**, because the space T is greater by hypothesis , than thecircle EFGH . Therefore as the square of FH is to the square of BD , so is the

circle EFGH to a space less than the

**circle ABCD**, which has been demonstratedto be ...

Side 218

than the pyramid in the cone EN ; but it is less , as was shown , which is absurd :

therefore the

which is less than the cone EN . In the same manner it may be demonstrated that

...

than the pyramid in the cone EN ; but it is less , as was shown , which is absurd :

therefore the

**circle ABCD**is not to the circle EFGH , as the cone AL to any solidwhich is less than the cone EN . In the same manner it may be demonstrated that

...

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The Elements of Euclid, Viz; The First Six Books: Together With the Eleventh ... Euclid Euclid Ingen forhåndsvisning tilgjengelig - 2018 |

The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh ... Euclid Euclid Ingen forhåndsvisning tilgjengelig - 2017 |

### Vanlige uttrykk og setninger

added altitude angle ABC angle BAC base centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP radius reason rectangle rectangle contained rectilineal figure remaining right angles segment shown sides similar sine solid solid angle sphere square square of AC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 45 - Ir a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Side 41 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Side 54 - Ir any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference ; the straight line drawn from A to B shall fall within the circle.

Side 18 - ABD, the less to the greater, which is impossible ; therefore BE is not in the same straight line with BC.

Side 10 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line: it is required to draw from the point A a straight line equal to BC.

Side 8 - Let it be granted that a straight line may be drawn from any one point to any other point.

Side 256 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Side 129 - 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.

Side 23 - At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...

Side 20 - ANY two angles of a triangle are together less than two right angles.