First principles of Euclid: an introduction to the study of the first book of Euclid's Elements1880 |
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Page 15
... unequal . 5. If equals be taken from unequals the remainders are unequal . 6. Things which are doubles of the same thing are equal to one another . 6a . ( Not given by Euclid , but assumed by him ) . Things which are doubles of equal ...
... unequal . 5. If equals be taken from unequals the remainders are unequal . 6. Things which are doubles of the same thing are equal to one another . 6a . ( Not given by Euclid , but assumed by him ) . Things which are doubles of equal ...
Page 35
... unequal . In fig . I they are equal . In fig . 2 they are unequal . .. In fig . I they are right angles . In fig . 2 they are not right angles . A Fig.2 D B In fig . I the straight line CD may be drawn either from D , a point in AB , or ...
... unequal . In fig . I they are equal . In fig . 2 they are unequal . .. In fig . I they are right angles . In fig . 2 they are not right angles . A Fig.2 D B In fig . I the straight line CD may be drawn either from D , a point in AB , or ...
Page 47
... but we prove that it is impossible they should be unequal . We prove that an assertion cannot be untrue , by a method which is called the reductio ad absurdum . absurdity . That is , by reducing to METHODS OF DEMONSTRATION.
... but we prove that it is impossible they should be unequal . We prove that an assertion cannot be untrue , by a method which is called the reductio ad absurdum . absurdity . That is , by reducing to METHODS OF DEMONSTRATION.
Page 48
... unequal . And then we find that , supposing the sides to be unequal , we are able to prove that a small triangle is equal to a larger triangle . As this is manifestly absurd , the supposition on which it is founded must also be absurd ...
... unequal . And then we find that , supposing the sides to be unequal , we are able to prove that a small triangle is equal to a larger triangle . As this is manifestly absurd , the supposition on which it is founded must also be absurd ...
Page 75
... unequal . Thus , in Fig . 2 , sides CA and DA are drawn equal . Then it can be seen that CB and DB are unequal . A Fig.2 A A Fig . 3 B Again , in Fig . 3 , sides CB and DB are drawn equal . And , as can be seen , CA and DA become unequal ...
... unequal . Thus , in Fig . 2 , sides CA and DA are drawn equal . Then it can be seen that CB and DB are unequal . A Fig.2 A A Fig . 3 B Again , in Fig . 3 , sides CB and DB are drawn equal . And , as can be seen , CA and DA become unequal ...
Common terms and phrases
1st conclusion 2nd Syllogism A B equal ABC is equal adjacent angles alternate angle angle A CD angle ABC angle B A C angle BAC angle contained angle DFE angle EDF angle GHD angles BGH angles equal Axiom 2a Axiom 9 base B C bisected CD is greater coincide Construction definition diameter enunciations of Euc equal angles equal to A B equal to angle equal to CD equal to side equilateral triangle EXERCISES.-I exterior angle figure given line given point given straight line greater than angle included angle interior opposite angle isosceles triangle Join Let us suppose line A B line CD major premiss parallel to CD parallelogram Particular Enunciation PROBLEM Euclid produced proposition prove that angle remaining angle Required right angles side A C sides equal square THEOREM Euclid triangle ABC
Popular passages
Page 83 - 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...
Page 18 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
Page 66 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let...
Page 34 - 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 94 - Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of' the base, equal to one another, and likewise those which are terminated in the other extremity.
Page 88 - 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 104 - If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Page 140 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Page 51 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.
Page 132 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.