## 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 CambridgeLongman, Green, Longman, Roberts, and Green, 1868 - 410 pages |

### From inside the book

Results 1-5 of 98

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**similar**reason , EF is equal to BC ; wherefore AD is equal to EF ; ( ax . 1. ) and DE is common ; therefore the whole , or the remainder AE , is equal to the whole , or the remainder DF ; ( ax . 2 or 3. ) and AB is equal to DC ; ( 1. 24 ... Page 49

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**similar**way may be supplied the reserved premiss in every enthy- meme . The conclusion of two enthymemes may form the major and minor premiss of a third syllogism , and so on , and thus any process of reasoning is reduced to the ... Page 51

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**similar**construction the less of two given straight lines may be produced , so that the less together with the part produced may be equal to the greater . Prop . III . This problem admits of two solutions , and it is left unde- termined ... Page 55

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**similar**way , it may be shewn that BC cannot be otherwise than equal to EF . If ACB , DFE be both right angles : the case falls under Euc . 1. 26 . Prop . xxvII . Alternate angles are defined to be the two angles which two straight ... Page 61

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**similar**limitation necessary with respect to the three angles ? 52. Is it possible to form a triangle with three lines whose lengths are 1 , 2 , 3 units or one with three lines whose lengths are 1 , √2 , √3 ? 53. Is it possible to ...### Common terms and phrases

A₁ ABCD AC is equal Algebraically angle ABC angle ACB angle BAC Apply Euc base BC chord circle ABC constr demonstrated describe a circle diagonals diameter divided double draw equal angles equiangular equilateral triangle equimultiples Euclid exterior angle Geometrical given circle given line given point given straight line gnomon greater hypotenuse inscribed intersection isosceles triangle less Let ABC line BC lines be drawn multiple opposite angles parallelogram parallelopiped pentagon perpendicular plane polygon problem produced Prop proportionals proved Q.E.D. PROPOSITION quadrilateral figure radius ratio rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn similar similar triangles solid angle square on AC tangent THEOREM touch the circle trapezium triangle ABC twice the rectangle vertex vertical angle wherefore

### Popular passages

Page 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...

Page 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.

Page 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.

Page 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.

Page 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.

Page 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.

Page 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.

Page 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.

Page 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...

Page 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...