## The Elements of Euclid |

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Side 4

Now , upon the 10th Definition of this Book depend the 25th and 28th

8th of the same ; and this 8th , and the Corollary of

Now , upon the 10th Definition of this Book depend the 25th and 28th

**Propositions**of it ; and upon the 25th and 26th ... 12th Book depends upon the8th of the same ; and this 8th , and the Corollary of

**Proposition**17th and**Proposition**18th of ... Side 238

... 16th and 31 st

editor : on which account , especially since they are quite useless , these

definitions are distinguished from the rest by inverted double commas 238 BOOK

I . NOTES .

... 16th and 31 st

**propositions**of book 3 , are the editions of some less skilfuleditor : on which account , especially since they are quite useless , these

definitions are distinguished from the rest by inverted double commas 238 BOOK

I . NOTES .

Side 240

13 , book 1 . This part must therefore have been added to prop . 5 , upon account

of some

it except the 7th

13 , book 1 . This part must therefore have been added to prop . 5 , upon account

of some

**proposition**betwixt the 5th and 13th , but none of these stand in need ofit except the 7th

**proposition**, on account of which it has been added : besides ... Side 241

... because , without D this restriction , there might be three different cases of the

the second edition of his Elements of Geometry , printed anno 1760 , observes in

...

... because , without D this restriction , there might be three different cases of the

**proposition**, as Campanus and others make . Mr . Thomas Simson , in p . 262 ofthe second edition of his Elements of Geometry , printed anno 1760 , observes in

...

Side 249

In this , as also in the 8th and 13th

indirectly , that the circle touches a straight ... he divides the

cases ; whereas , one and the same construction and demonstration serves for ...

In this , as also in the 8th and 13th

**propositions**of this book , it is demonstratedindirectly , that the circle touches a straight ... he divides the

**proposition**into threecases ; whereas , one and the same construction and demonstration serves for ...

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added altitude angle ABC angle BAC base BC is given 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 magnitude given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise magnitude manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid sphere square square of BC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 36 - 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 at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.

Side 145 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

Side 65 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Side 248 - 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 11 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Side 121 - 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 21 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Side 14 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.

Side 80 - 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...

Side 133 - ... rectilineal figures are to one another in the duplicate ratio of their homologous sides.