The Elements of EuclidDesilver, Thomas, 1838 - 416 σελίδες |
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Σελίδα 94
... proportion , is the similitude of ratios . ' IX . Proportion consists in three terms at least . X. When three magnitudes are proportionals , the first is said to have to the third the duplicate ratio of that which it has to the second ...
... proportion , is the similitude of ratios . ' IX . Proportion consists in three terms at least . X. When three magnitudes are proportionals , the first is said to have to the third the duplicate ratio of that which it has to the second ...
Σελίδα 96
... proportion . It is demon- strated in 22d Prop . book 5 . XX . Ex æquali , in proportione perturbata , seu inordinata ; from equality , in perturbata or disorderly proportion ; * this term is used when the first magnitude is to the ...
... proportion . It is demon- strated in 22d Prop . book 5 . XX . Ex æquali , in proportione perturbata , seu inordinata ; from equality , in perturbata or disorderly proportion ; * this term is used when the first magnitude is to the ...
Σελίδα 144
... proportion- als ; and they are therefore similar to one another ( 1. def . 6. ) : for the same reason , the parallelogram ABCD is similar to the parallelogram FHCK . Wherefore each of the parallelograms GE , KH is similar to DB ; but ...
... proportion- als ; and they are therefore similar to one another ( 1. def . 6. ) : for the same reason , the parallelogram ABCD is similar to the parallelogram FHCK . Wherefore each of the parallelograms GE , KH is similar to DB ; but ...
Σελίδα 250
... proportion of numbers to one another is defined , and treated of , yet without giving any definition of the ratio of num- bers ; though such a definition was as necessary and useful to be given in that book , as in this : but indeed ...
... proportion of numbers to one another is defined , and treated of , yet without giving any definition of the ratio of num- bers ; though such a definition was as necessary and useful to be given in that book , as in this : but indeed ...
Σελίδα 261
... proportion- als , by reason of again neglecting permutation . On these ac- counts , a demonstration is given in Euclid's manner , like to that he makes use of in the 20th prop . of this book : and it is extended to five - sided figures ...
... proportion- als , by reason of again neglecting permutation . On these ac- counts , a demonstration is given in Euclid's manner , like to that he makes use of in the 20th prop . of this book : and it is extended to five - sided figures ...
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altitude angle ABC angle BAC base BC BC is equal BC is given bisected centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles gnomon greater join less Let ABC multiple parallel parallelogram parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopiped square of AC square of BC straight line AB straight line BC tangent THEOR triangle ABC triplicate ratio vertex wherefore
Δημοφιλή αποσπάσματα
Σελίδα 34 - 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 at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.
Σελίδα 143 - 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.
Σελίδα 63 - 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.
Σελίδα 246 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.
Σελίδα 9 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.
Σελίδα 119 - 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.
Σελίδα 19 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Σελίδα 12 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.
Σελίδα 78 - 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. either the sides adjacent to the equal...
Σελίδα 131 - ... rectilineal figures are to one another in the duplicate ratio of their homologous sides.