Modern geometry [ed.] with an appendix by W.B. Jack1876 |
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Page 175
... incommensurable magni- tudes , is interminable . If the process be interminable , the magnitudes are in- commensurable . By a precisely similar method may be found the greatest common measure of two angles , or of any two magnitudes of ...
... incommensurable magni- tudes , is interminable . If the process be interminable , the magnitudes are in- commensurable . By a precisely similar method may be found the greatest common measure of two angles , or of any two magnitudes of ...
Page 176
... incommensurable . THEOREM LIX . When three or more parallels fall across two straight lines and divide one into equal parts , they also divide the other into equal parts . Let AD , BE , CF be parallels cutting the line AC in A , B , and ...
... incommensurable . THEOREM LIX . When three or more parallels fall across two straight lines and divide one into equal parts , they also divide the other into equal parts . Let AD , BE , CF be parallels cutting the line AC in A , B , and ...
Page 180
... incommensurable . Extension of Preceding Illustration . -The properties of proportion illustrated by the particular numbers 56 , 7 , 48 and 6 , may be shown to belong to all numbers , and therefore to all magnitudes 180 MODERN GEOMETRY ...
... incommensurable . Extension of Preceding Illustration . -The properties of proportion illustrated by the particular numbers 56 , 7 , 48 and 6 , may be shown to belong to all numbers , and therefore to all magnitudes 180 MODERN GEOMETRY ...
Page 184
... incommensurable . Let A and B be the magnitudes ; let B be divided into A B m n equal parts and one of these parts measured off along A. Suppose m times this part to be less than A , and m + 1 times to be greater than A ; then the ratio ...
... incommensurable . Let A and B be the magnitudes ; let B be divided into A B m n equal parts and one of these parts measured off along A. Suppose m times this part to be less than A , and m + 1 times to be greater than A ; then the ratio ...
Page 187
... incommensurable . If possible let it not be true that B : A :: D : C ; but a let B : A ' :: D : C. Let ad A and a b = A ' . d C b However small db , the difference of A and A ' , may be , B may be divided into parts less than db ...
... incommensurable . If possible let it not be true that B : A :: D : C ; but a let B : A ' :: D : C. Let ad A and a b = A ' . d C b However small db , the difference of A and A ' , may be , B may be divided into parts less than db ...
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Common terms and phrases
ABCD adjacent angles alternate angles angles equal angular points base bisect bisector CBEF centre chord circumference circumscribing circle common measure Construct a triangle contra-positive converse describe a circle diagonals diameter distance divided draw a straight equal angles equidistant extremities figure Find the area given circle given point given straight line gonals greater half Hence hypothenuse inches incommensurable inscribed circle interior angles isosceles triangle less Let ABC line drawn locus of points magnitudes middle points number of sides number of units parallel parallelogram pentagon perimeter perpendicular point of intersection proportion propositions proved quadrilateral radii radius ratio rect rectangle contained regular polygon respectively equal rhombus right angles right-angled triangle secant segment semicircle side opposite similar similar triangles square tangent termed Theo theodolite THEOREMS ON CHAPTER three given touch triangles are equal vertex vertical angle دو دو دو
Popular passages
Page 242 - 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 240 - 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.
Page 240 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.
Page 242 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Page 242 - If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.
Page 241 - If the first has to the second the same ratio which the third has to the fourth...
Page 240 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 242 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Page 18 - 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 239 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.