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 Exercises |
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Page 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 ...
Page 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 ...
Page 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 ...
Page 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 .
Page 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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Common terms and phrases
ABCD AC is equal angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC 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
Popular passages
Page 262 - 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.
Page 71 - 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 262 - To draw a straight line at right angles to a given straight line, from a given point in the same. Let AB be...
Page 182 - 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 8 - 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.
Page 298 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Page 58 - 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 60 - If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. Let the straight line AB be divided into two equal parts in the point C, and into two unequal parts in the point D ; The squares on AD and DB shall be together double of AD»+DB
Page 242 - Let AB and C be two unequal magnitudes, of which AB is the greater. If from AB there be taken more than its half, and from the remainder more than its half, and so on ; there shall at length remain a magnitude less than C. For C may be multiplied, so at length to become greater than AB.
Page 4 - Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.