Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes .... the first six books |
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Page 204
... equimultiples whatsoever of the first and third being taken , and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second , the multiple of the third is also less than that of ...
... equimultiples whatsoever of the first and third being taken , and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second , the multiple of the third is also less than that of ...
Page 206
... EQUIMULTIPLES of the same , or of equal magnitudes , are equal to one another . II . Those magnitudes , of which the same or equal magnitudes are equimultiples , are equal to one another . • Prop . 4. Lib . II . Archimedis de sphæra et ...
... EQUIMULTIPLES of the same , or of equal magnitudes , are equal to one another . II . Those magnitudes , of which the same or equal magnitudes are equimultiples , are equal to one another . • Prop . 4. Lib . II . Archimedis de sphæra et ...
Page 207
... equal to E and F together : therefore , whatsoever multiple AB is of E , the same multiple is AB and CD together , of E and F together . Therefore , if any magnitudes , how many soever , be equimultiples of as many , each of each ...
... equal to E and F together : therefore , whatsoever multiple AB is of E , the same multiple is AB and CD together , of E and F together . Therefore , if any magnitudes , how many soever , be equimultiples of as many , each of each ...
Page 208
... equimultiples ; these shall be equi multiples , the one of the second , and the other of the fourth . Let 4 the first be the same multiple of B the second , that С the third is of D the fourth : and of A , C let equimultiples EF , GH be ...
... equimultiples ; these shall be equi multiples , the one of the second , and the other of the fourth . Let 4 the first be the same multiple of B the second , that С the third is of D the fourth : and of A , C let equimultiples EF , GH be ...
Page 209
... equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth , viz , ' the equimultiple of the first shall have the same ratio to that of the second , which the equimultiple of ...
... equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth , viz , ' the equimultiple of the first shall have the same ratio to that of the second , which the equimultiple of ...
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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...
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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...