The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and ExercisesMacmillan, 1867 - 400 páginas |
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Página 134
... Ratio is a mutual relation of two magnitudes of the same kind to one another in respect of quantity . 4. Magnitudes are said to have a ratio to one another , when the less can be multiplied so as to exceed the other . 5. The first of ...
... Ratio is a mutual relation of two magnitudes of the same kind to one another in respect of quantity . 4. Magnitudes are said to have a ratio to one another , when the less can be multiplied so as to exceed the other . 5. The first of ...
Página 135
... ratio than the third has to the fourth ; and the third is said to have to the fourth a less ratio than the first has to the second . 8. Analogy , or proportion , is the similitude of ratios . 9. Proportion consists in three terms at ...
... ratio than the third has to the fourth ; and the third is said to have to the fourth a less ratio than the first has to the second . 8. Analogy , or proportion , is the similitude of ratios . 9. Proportion consists in three terms at ...
Página 136
... ratio com- pounded of the ratios of E to F , G to H , and K to L. In like manner , the same things being supposed , if M has to N the same ratio that A has to D ; then , for the sake of shortness , M is said to have to N the ratio com ...
... ratio com- pounded of the ratios of E to F , G to H , and K to L. In like manner , the same things being supposed , if M has to N the same ratio that A has to D ; then , for the sake of shortness , M is said to have to N the ratio com ...
Página 140
... & c . Q.E.D. PROPOSITION 4. THEOREM . If the first have the same ratio to the second that the third has to the fourth , and if there be taken any equi- multiples whatever of the first and the third , and 140 EUCLID'S ELEMENTS .
... & c . Q.E.D. PROPOSITION 4. THEOREM . If the first have the same ratio to the second that the third has to the fourth , and if there be taken any equi- multiples whatever of the first and the third , and 140 EUCLID'S ELEMENTS .
Página 141
... ratio to the multiple of the second , that the multiple of the third has to the multiple of the fourth . Let A the first have to B the second , the same ratio that C the third has to D the fourth ; and of A and C let there be taken any ...
... ratio to the multiple of the second , that the multiple of the third has to the multiple of the fourth . Let A the first have to B the second , the same ratio that C the third has to D the fourth ; and of A and C let there be taken any ...
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Términos y frases comunes
ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral radius rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Pasajes populares
Página 286 - 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.
Página 30 - 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...
Página 34 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles...
Página 37 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Página 12 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Página 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Página 220 - 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.
Página 354 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.
Página 104 - 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.
Página 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.