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 8
... greater of two given straight lines to cut off a part equal to the less Let AB and C be the two given straight lines , of which AB is the greater : it is required to cut 8 EUCLID'S ELEMENTS .
... greater of two given straight lines to cut off a part equal to the less Let AB and C be the two given straight lines , of which AB is the greater : it is required to cut 8 EUCLID'S ELEMENTS .
Page 9
... greater : it is required to cut off from AB , the greater , a part equal to C the less . From the point A draw the straight line AD equal to C ; [ I. 2 . and from the centre A , at the distance AD , describe the circle DEF meeting AB at ...
... greater : it is required to cut off from AB , the greater , a part equal to C the less . From the point A draw the straight line AD equal to C ; [ I. 2 . and from the centre A , at the distance AD , describe the circle DEF meeting AB at ...
Page 12
... greater than the other . Let AB be the greater , and from it cut off DB equal to AC the less , and join DC . Then , because in the triangles DBC , ACB , DB is equal to AC , and BC is common to both , B [ I. 3 . [ Construction . the two ...
... greater than the other . Let AB be the greater , and from it cut off DB equal to AC the less , and join DC . Then , because in the triangles DBC , ACB , DB is equal to AC , and BC is common to both , B [ I. 3 . [ Construction . the two ...
Page 13
... greater than the angle BCD ; much more then is the angle BDC greater than the angle BCD . Again , because BC is equal to BD , the angle BDC is equal to the angle BCD . [ Hypothesis . [ I. 5 . But it has been shewn to be greater ; which ...
... greater than the angle BCD ; much more then is the angle BDC greater than the angle BCD . Again , because BC is equal to BD , the angle BDC is equal to the angle BCD . [ Hypothesis . [ I. 5 . But it has been shewn to be greater ; which ...
Page 20
... greater than either of the interior opposite angles . Let ABC be a triangle , and let one side BC be pro- duced to D : the exterior angle ACD shall be greater than either of the interior opposite angles CBA , BAC . Bisect AC at E , [ I ...
... greater than either of the interior opposite angles . Let ABC be a triangle , and let one side BC be pro- duced to D : the exterior angle ACD shall be greater than either of the interior opposite angles CBA , BAC . Bisect AC at E , [ I ...
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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.