The elements of geometry, in eight books; or, First step in applied logic1874 |
From inside the book
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Page 4
... difference , in the second , ratio . The study of differences and ratios belongs to the province of Arithmetic , and is not dealt with in Geometry save in connexion with form , for the completion of the theory of similarity and to ...
... difference , in the second , ratio . The study of differences and ratios belongs to the province of Arithmetic , and is not dealt with in Geometry save in connexion with form , for the completion of the theory of similarity and to ...
Page 15
... difference . Evidently the difference of two quantities contains as many times their common unit as the unit is contained more in one than in the other . It is also evident that if the difference of two quantities be added to the lesser ...
... difference . Evidently the difference of two quantities contains as many times their common unit as the unit is contained more in one than in the other . It is also evident that if the difference of two quantities be added to the lesser ...
Page 16
... difference , of two quantities is not altered when one of them is replaced by an equal quantity . THEOREM I. ( Eucl . Ax . 1. ) Two quantities , each of which is equal to a third , are equal to one another . Let there be three ...
... difference , of two quantities is not altered when one of them is replaced by an equal quantity . THEOREM I. ( Eucl . Ax . 1. ) Two quantities , each of which is equal to a third , are equal to one another . Let there be three ...
Page 17
... difference of A and C is equal to that of B and C ; therefore the sum , or difference , of A and C is equal to that of B and C , which W. T. B. D. Inversely If , after adding the same quantity to , or subtracting it from , two other ...
... difference of A and C is equal to that of B and C ; therefore the sum , or difference , of A and C is equal to that of B and C , which W. T. B. D. Inversely If , after adding the same quantity to , or subtracting it from , two other ...
Page 18
... difference , shall be greater than the second . THEOREM IV . ( Eucl . Ax . 2 and 3. ) If two quantities are equal to two others , each to each , the sum and difference of the first two quantities are respectively equal to the sum and ...
... difference , shall be greater than the second . THEOREM IV . ( Eucl . Ax . 2 and 3. ) If two quantities are equal to two others , each to each , the sum and difference of the first two quantities are respectively equal to the sum and ...
Common terms and phrases
A B and C D A B C and D E F adjacent angles adjacent sides altitude angle A B C angle ABC angles formed apothem bisect bisectrix centre angle centre line chord circular segment coincide Const Conversely COROLLARY II diagonals diameter divided equal angles equal circumferences equal to half equally distant equilateral equilateral polygon equivalent Eucl extremities given straight line greater homologous hypothenuse inscribed angle intercepts intersection isosceles trapezium isosceles triangle Let A B C magnitude middle line middle perpendicular middle point parallel parallelogram perimeter plane figure point H point of tangence points of section portions produced quadrangle quantities radii radius reasoning would prove rectangle regular polygon right-angled triangle Scholium segment sides A B square straight angle symmetric points tangent THEOREM transversal trapezium triangle A B C unequal vertex W. W. T. B. D. COROLLARY W. W. T. B. D. Inversely
Popular passages
Page 226 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
Page 202 - Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 230 - 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 143 - 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. 11. An obtuse angle is that which is greater than a right angle. 12. An acute angle is that which is less than a right angle. 13. A term or boundary is the extremity of any thing.
Page 218 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
Page 202 - The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
Page 268 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their areas are to each other as the squares of those sides (Prop.
Page 43 - The projection of a point on a plane is the foot of the perpendicular drawn from the point to the plane.
Page 335 - Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles...
Page 382 - FLC, there -are two angles -of the one equal to two angles of the other, each to each ; and the side FC which is adjacent to the equal...