Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes .... the first six books |
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Page 44
... construction of Prop . 14 , Book II ; the first case of Prop . 33 , Book III , and Prop . 13 , Book vi . The definition of the segment of a circle is not once alluded to in Book 1 , and is not required before the dis- cussion of the ...
... construction of Prop . 14 , Book II ; the first case of Prop . 33 , Book III , and Prop . 13 , Book vi . The definition of the segment of a circle is not once alluded to in Book 1 , and is not required before the dis- cussion of the ...
Page 45
... construction assumed in the Elements ; and are , in fact , problems , the possibility of which is admitted to be self - evident , and to require no proof . It must , however , be carefully remarked , that the third postulate only admits ...
... construction assumed in the Elements ; and are , in fact , problems , the possibility of which is admitted to be self - evident , and to require no proof . It must , however , be carefully remarked , that the third postulate only admits ...
Page 48
... construction is required to be effected : and it is a theorem when some Geo- metrical property is to be demonstrated . Every proposition is natu- rally divided into two parts ; a problem consists of the data , or things given ; and the ...
... construction is required to be effected : and it is a theorem when some Geo- metrical property is to be demonstrated . Every proposition is natu- rally divided into two parts ; a problem consists of the data , or things given ; and the ...
Page 49
... constructions which may be admitted as possible by direct appeal to our experience ; and the axioms assert general theoretic truths so simple and self - evident as to require no proof , but to be admitted as the assumed first principles ...
... constructions which may be admitted as possible by direct appeal to our experience ; and the axioms assert general theoretic truths so simple and self - evident as to require no proof , but to be admitted as the assumed first principles ...
Page 50
... in particular terms , as it is pointed out in the diagram , and directs attention to the demonstration , by pronouncing the thing sought . 4. The construction applies the postulates to prepare the diagram 50 EUCLID'S ELEMENTS .
... in particular terms , as it is pointed out in the diagram , and directs attention to the demonstration , by pronouncing the thing sought . 4. The construction applies the postulates to prepare the diagram 50 EUCLID'S ELEMENTS .
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Common terms and phrases
A₁ ABCD AC is equal Algebraically angle ABC angle ACB angle BAC Apply Euc axiom base BC chord circle ABC constr describe a circle diagonals diameter divided double draw equal angles equiangular equilateral triangle equimultiples Euclid Euclid's Elements exterior angle Geometrical given circle given line given point given straight line given triangle gnomon greater hypotenuse inscribed intersection isosceles triangle less Let ABC line AC lines be drawn meet the circumference multiple opposite angles parallelogram pentagon perpendicular porism problem produced Prop proportionals proved Q.E.D. PROPOSITION quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn similar similar triangles square on AC tangent THEOREM touch the circle trapezium triangle ABC twice the rectangle vertex vertical angle wherefore
Popular passages
Page 54 - If two triangles have two sides of the one equal to two sides of the...
Page 89 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square...
Page 38 - Let ABCD be the given rectilineal figure, and E the given rectilineal angle. It is required to describe a parallelogram equal to ABCD, and having an angle equal to E.
Page 144 - 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.
Page 18 - Any two angles of a triangle are together less than two right angles.
Page 266 - 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.
Page 152 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.
Page 7 - 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.
Page 3 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 96 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side...