week1.Euclid (propositions 1 and 32).pdf

# week1.Euclid (propositions 1 and 32).pdf - Elements all...

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Unformatted text preview: “Elements all thirteen books complete in one volume The Thomas L. Heath Translation Dana Densmore, Editor Green Lion Press Santa Fe, New Mexico 5305499199319? 10. 11. 12. 13. 14. 15. 16. 17. Euclid’s Elements Book I Definitions A point is that which has no part. A line is breadthless length. The extremities of a line are points. A straight line is a line which lies evenly with the points on itself. A surface is that which has length and breadth only. The extremities of a surface are lines. A plane surface is a surface which lies evenly with the straight lines on itself. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. And when the lines containing the angle are straight, the angle is called rectilineal. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. An obtuse angle is an angle greater than a right angle. An acute angle is an angle less than a right angle. A boundary is that which is an extremity of anything. A ﬁgure is that which is contained by any boundary or boundaries. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; And the point is called the centre of the circle. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. Euclid’s definitions, postulates, and common notions—if Euclid is indeed their author— were not numbered, separated, or italicized until transiators began to introduce that practice. The Greek text, however, as far back as the 1533 first printed edition, presented the definitions in a running narrative, more as a preface discussing how the terms would be used than as an axiomatic foundation for the propositions to come. We foliow Heath’s formatting here. —Ed. Book One: Definitions, Postulates, Common Notions 18. 19. A semicircle is the ﬁgure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. Rectilineal ﬁgures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. . Of~ trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal. . Further, of trilateral figures, a right—angled triangle is that which has a right angle, an obtuseeangled triangle that which has an obtuse angle, and an acute" angled triangle that which has its three angles acute. . Of quadrilateral figures, a square is that which is both equilateral and rightrangled; an oblong that which is right-angled but not equilateral; a rhornbus that which is equilateral but not right—angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia. . Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. Postulates Let the following be postulated: 1. To draw a straight line from any point to any point. To produce a ﬁnite straight line continuously in a straight line. To describe a circle with any centre and distance. That all right angles are equal to one another. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if pro- duced indefinitely, meet on that side on which are the angles less than the 7. two right angles. ' Common Notions Things which are equal to the same thing are also equal to one another. If equals be added to equals, the wholes are equal. If equals be subtracted from equals, the remainders are equal. Things which coincide with one another are equal to one another. The whole is greater than the part. :ircumference it of the circle. ines, triiateral ained by four, res. ts three sides 3 equal, and a :h has a right and an acute- uilateral and equilateral; a rhomboid that but is neither han these be he plane and me another in iterior angles lines, if pro— less than the __ e another. :her. Proposition 1 On a given ﬁnite straight line to construct an equilateral triangle. Let AB be the given ﬁnite straight line. Thus it is required to construct an equilateral triangle on the straight line AB. With centre A and distance AB let the circle BCD be described; [Post 3] D again, With centre B and distance BA let the circle ACE be described; [Post 3] and from the point C; in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. ‘ [Post 1] Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. :5] Again, since the point B is the centre of the circle CAB; BC is equal to BA. [Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [on 1] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another. Therefore the triangle ABC is equilateral; and it has been constructed on the given ﬁnite siIaight line AB. ‘ Being What it was required to do. Proposition 2 To place at a given point [as an extremity]1 a straight line equal to a given straight line. Let A be the given point, and BC the given straight line. Thus it is required to place at the point A [as an extremity} a straight line equal to the given straight line BC. From the point A to the point B let the straight line AB be joined; [Post 1] and on it let the equilateral triangle DAB be constructed. [I- 1] 1. Square brackets indicate material which Heath identified as having been supplied by him, adding Clariﬁcation but not literally present in the Greek text. —Ed. Book One: Propositions 1—2 Book One: Propositions 31—32 Proposition 31 Through a given point to draw a straight line parallel to a given straight line. Let A be the given point, and BC the given straight line; thus it is required to draw through the point A a straight line parallel to the straight line BC. Let a point D be taken at random on BC, and let AD be joined; . on the straight line DA, and at the pointA on it, let the angle DAB be constructed equal to the angle ADC; [L 23] and let the straight line AF be produced in a straight line with EA. Then, since the straight line AD falling on the two straight lines BC, EF has made the alternate angles BAD, ADC equal to one another, therefore BAP is parallel to BC. [|. 27] Therefore through the given point A the straight line BAP has been drawn parallel to the given straight line BC. QEE Proposition 32 In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. ' B C D Let ABC be a triangle, and let one side of it BC be produced to D; I say that the exterior angle ACD is equal to the two interior and opposite angles CAB, AB C, and the three interior angles of the triangle ABC, BCA, CAB are equal to two right angles. For let CE be drawn through the point C parallel to the straight line AB. [L 31] Then, since AB is parallel to CE, and AC has fallen upon them, the alternate angles BAC, ACE are equal to one another. [I. 29] yen straight line. .ne parallel to the 1; AE be constructed [I. 23] .1 EA. 5 BC, EF has made [L 27] has been draWn Q.E.F. zgle is equal to the" of the triangle are 3 D; [Cl opposite angles 3A, CAB are equal ht line AB. [L31] 1! iother. [L 29] Book One: Propositions 32—33 Again, since AB is parallel to CE, and the straight line BD has fallen upon them, the exterior angle B CD is equal to the interior and opposite angle ABC. [L 29] But the angle ACE was also proved equal to the angle BAC; therefore the whole angle ACD is equal to the two interior and opposite angles BAC, ABC. Let the angle ACB be added to each; therefore the angles ACD, ACB are equal to the three angles ABC, B CA, CAB. But the angles ACD, ACB are equal to two right angles; [I. 13] therefore the angles ABC, B CA, CAB are also equal to two right angles. Therefore etc. QED. Proposition 33 The straight lines joining equal and parallel straight lines [at the extremities which are] in the same directions [respectively] are themselves also equal and parallel. B A Let AB, CD be equal and parallel, and let the straight lines AC, BD join them [at the extremities which are] in the same directions [respectively]; I say that AC, BD are also equal and parallel. Let BC be joined. Then, since AB is parallel to CD, and BC has fallen upon them, the alternate angles ABC, BCD are equal to one another. [I. 29] And, since AB is equal to CD, and BC is common, the two sides AB, BC are equal to the two sides DC, CB; and the angle ABC is equal to the angle B CD; therefore the base AC is equal to the base BD, and the triangle ABC is equal to the triangle DCB, and the remaining angles Will be equal to the remaining angles respectively, namely those which the equal sides subtend; [L4] therefore the angle ACE is equal to the angle CBD. And, since the straight line BC falling on the two straight lines AC, BD has made the alternate angles equal to one another, AC is parallel to BD. {L 27] And it was also proved equal to it. Therefore etc. QED. 25 ...
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