First principles of Euclid: an introduction to the study of the first book of Euclid's Elements1880 |
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Σελίδα 15
... equal to one another . 7. Things which are halves of the same thing are equal to one another . 8. Magnitudes which coincide with one another are equal to one another . 8a . ( Not given by Euclid , but assumed Axioms . 15 AXIOMS •
... equal to one another . 7. Things which are halves of the same thing are equal to one another . 8. Magnitudes which coincide with one another are equal to one another . 8a . ( Not given by Euclid , but assumed Axioms . 15 AXIOMS •
Σελίδα 16
... coincide with one another . 9. The whole is greater than its part . 10. Two straight lines cannot enclose a space . 11. All right angles are ... coincide their points coincide ( 16 First Principles of Euclid . A CHAIN OF SYLLOGISMS.
... coincide with one another . 9. The whole is greater than its part . 10. Two straight lines cannot enclose a space . 11. All right angles are ... coincide their points coincide ( 16 First Principles of Euclid . A CHAIN OF SYLLOGISMS.
Σελίδα 17
T S. Taylor. 2nd Syllogism . When lines coincide their points coincide ( as- sumed as an axiom ) . A B and CD coincide ( 1st conclusion ( a ) become a premiss ) . ( b ) . The points of A B and C D coincide ( 2nd conclusion ) . 3rd ...
T S. Taylor. 2nd Syllogism . When lines coincide their points coincide ( as- sumed as an axiom ) . A B and CD coincide ( 1st conclusion ( a ) become a premiss ) . ( b ) . The points of A B and C D coincide ( 2nd conclusion ) . 3rd ...
Σελίδα 28
... coincide with the line DF , the angles are equal . If the line AC fall within the angle EDF , the angle B A C is less than ED F. But if the line A C fall without the angle EDF , then the angle B A C is greater than ED F. Try this with ...
... coincide with the line DF , the angles are equal . If the line AC fall within the angle EDF , the angle B A C is less than ED F. But if the line A C fall without the angle EDF , then the angle B A C is greater than ED F. Try this with ...
Σελίδα 71
... coincide with the point E because AB is equal to DE ( Axiom 8a ) . ( If AB were less than DE , B would fall between D and E ; if AB were greater than DE , B would fall beyond E. ) Since AB falls on DE , ( b ) AC will fall on DF ...
... coincide with the point E because AB is equal to DE ( Axiom 8a ) . ( If AB were less than DE , B would fall between D and E ; if AB were greater than DE , B would fall beyond E. ) Since AB falls on DE , ( b ) AC will fall on DF ...
Συχνά εμφανιζόμενοι όροι και φράσεις
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
Δημοφιλή αποσπάσματα
Σελίδα 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...
Σελίδα 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.
Σελίδα 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...
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.