## Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes; a Series of Questions on Each Book; and a Selection of Geometrical Exercises from the Senate-house and College Examination Papers, with Hints, &c. Designed for the Use of the Junior Classes in Public and Private Schools. the first six books, and the portions of the eleventh and twelfth books read at Cambridge |

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Seite 204

Vl . Magnitudes which have the same ratio are called

Vl . Magnitudes which have the same ratio are called

**proportionals**. N.B. “ When four magnitudes are**proportionals**, it is usually expressed by saying , the first is to the second , as the third to the fourth . ' VII . Seite 205

X. When three magnitudes are

X. When three magnitudes are

**proportionals**, the first is said to have to the third , the duplicate ratio of that ... and so on , quadruplicate , & c . , increasing the denomination still by unity , in any number of**proportionals**. Seite 206

Componendo , by composition ; when there are four

Componendo , by composition ; when there are four

**proportionals**, and it is inferred that ihe first together with the second , is to the second , as the third together with the fourth , is to the fourth . Prop . 18 , Book v . XVI . Seite 212

If four magnitudes are

If four magnitudes are

**proportionals**, they are**proportionals**also when taken inversely . Let A be to B , as C is to D. Then also inversely , B shall be to A , as D to C. A D BE G H F Take of B and D any equimultiples whatever E and F ... Seite 221

Let A , B , C , D be four magnitudes of the same kind , which are

Let A , B , C , D be four magnitudes of the same kind , which are

**proportionals**, viz . as A to B , so C to D. They shall also be**proportionals**when taken alternately : that is , A shall be to C , as B to D. G A C B D F Take of A and B ...### Was andere dazu sagen - Rezension schreiben

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### Häufige Begriffe und Wortgruppen

ABCD Algebraically angle BAC Apply base bisected Book chord circle circumference common construction contained definition demonstrated described diagonals diameter difference distance divided double draw drawn equal equal angles equiangular equilateral triangle equimultiples Euclid extremities fall figure formed four fourth Geometrical given circle given line given point given straight line greater half Hence inscribed intersection join less Let ABC line drawn magnitudes manner mean meet multiple opposite sides parallel parallelogram pass perpendicular plane problem produced Prop proportionals PROPOSITION proved Q.E.D. PROPOSITION radius ratio reason rectangle rectangle contained regular remaining respectively right angles segment semicircle shew shewn sides similar solid square taken tangent THEOREM third touch triangle ABC twice units vertex wherefore whole

### Beliebte Passagen

Seite 6 - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...

Seite 118 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Seite 2 - 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.

Seite 317 - 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.

Seite 90 - 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.

Seite 88 - 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.

Seite 30 - ... twice as many right angles as the figure has sides ; therefore all the angles of the figure together with four right angles, are equal to twice as many right angles as the figure has sides.

Seite 9 - 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.

Seite 22 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other...

Seite 92 - 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...