## The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and Exercises |

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Side 140

Let A the first be the same

fourth ; and of A and C let the equimultiples EF and GH be taken : EF shall be the

same

...

Let A the first be the same

**multiple**of B the second , that C the third is of D thefourth ; and of A and C let the equimultiples EF and GH be taken : EF shall be the

same

**multiple**of B that GH is of D . For , because EF is the same**multiple**of A that...

Side 141

of the second and the fourth , then the

ratio to the

of ...

**multiples**whatever of the first and the third , and also any equimultiples whateverof the second and the fourth , then the

**multiple**of the first shall have the sameratio to the

**multiple**of the second , that the**multiple**of the third has to the**multiple**of ...

Side 142

THEOREM . If one magnitude be the same

taken from the first is of a magnitude taken from the other , the remainder shall be

the same

THEOREM . If one magnitude be the same

**multiple**of another that a magnitudetaken from the first is of a magnitude taken from the other , the remainder shall be

the same

**multiple**of the remainder that the whole is of the whole . Let AB be the ... Side 143

Then , because AE is the same

and that AG is equal to EB ; therefore AE is the same

FD . But AE is the same

EB ...

Then , because AE is the same

**multiple**of CF that AG is of FD , [ Construction .and that AG is equal to EB ; therefore AE is the same

**multiple**of CF that EB is ofFD . But AE is the same

**multiple**of CF that AB is of CD ; [ Hypothesis . thereforeEB ...

Side 144

F , Next let GB be a

the same

E that CH is of [ Hypothesis . and GB is the same

F , Next let GB be a

**multiple**of E : HD shall be the same**multiple**of F . Make CKthe same

**multiple**of F that GB is of E . Then , because AG is the same**multiple**ofE that CH is of [ Hypothesis . and GB is the same

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The Elements of Euclid for the Use of Schools and Colleges Isaac Todhunter Uten tilgangsbegrensning - 1872 |

The Elements of Euclid for the Use of Schools and Colleges: With Notes, an ... Isaac Todhunter Uten tilgangsbegrensning - 1880 |

The Elements of Euclid for the Use of Schools and Colleges: Comprising the ... Euclid,Isaac Todhunter Uten tilgangsbegrensning - 1867 |

### Vanlige uttrykk og setninger

ABCD AC is equal angle ABC angle BAC Axiom base bisected Book centre chord circle ABC circumference common Construction Corollary Definition demonstration describe a circle described diameter difference divided double draw drawn equal equal angles equiangular equilateral equimultiples Euclid extremities fall figure fixed formed four fourth given circle given point given straight line greater half Hypothesis inscribed intersect join less Let ABC magnitudes manner meet middle point multiple namely opposite sides parallel parallelogram pass perpendicular plane polygon PROBLEM produced proportionals Q.E.D. PROPOSITION quadrilateral radius ratio reason rectangle contained rectilineal figure remaining respectively right angles segment shew shewn sides similar square straight line drawn suppose taken tangent THEOREM third triangle ABC twice Wherefore whole

### Populære avsnitt

Side 264 - 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.

Side 73 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a Right Angle; and the straight line which stands on the other is called a Perpendicular to it.

Side 264 - To draw a straight line at right angles to a given straight line, from a given point in the same. Let AB be...

Side 184 - 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 10 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Side 300 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.

Side 60 - If 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 62 - If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D ; The squares on AD and DB shall be together double of AD»+DB

Side 244 - Let AB and C be two unequal magnitudes, of which AB is the greater. If from AB there be taken more than its half, and from the remainder more than its half, and so on ; there shall at length remain a magnitude less than C. For C may be multiplied, so at length to become greater than AB.

Side 6 - Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.