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Unformatted text preview: 09/07/05 WED 00:02 FAX 512 471 9038 UT DEPT MATH 001 Appendix A Summary
from Eueﬁd’s Elements" THE TEN STATED ASSUMPTIONS The Postulates Let the following be postulated: _
1. To draw a straight line from any point to any point. 2. To produce a ﬁnite straight line continuously in a straight line.
3. To describe a circle with any centre and distance. 4. That all right angles are equal to one another.
5. 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
produced indeﬁnitely, meet on that side on which are the angles less than the two right angles. The Common Nations (or Axioms) 1. Things which. are equal to the same thing are also equal to one
another. ‘ ‘ 2. If equals be added to equals. the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. * Taken, with permission, from T. L. Heath, The Thirteen ﬂocks of Enclr'd’r Elements
‘ (Cambridge University Press, New York, 1926). . 257 r 09/07/05 WED 00:03 FAX 512 471 9038 UT DEPT MATH 002 ass _ APPENDIX THE DEFINITIONS OF BOOK I 1. A point is that which has no part.* 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. _ 7. A plane surface is a surface which lies evenly with the straight lines
on itself. 8. 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. 9. And when the lines containing the angle are straight, the angle is
called rectilineal. . 10. 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. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less thana right angle. 13. A boundary is that which is an extremity of anything. 14. A ﬁgure is that which is contained by any boundary or boundaries. 15. 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. 16. And the point is called the centre of the circle. 17 . 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. 18. A semicircle is the ﬁgure contained by the diameter and the circum
ference cut off by it. And the centre of the semicircle is the same as that of
the circle. “ F 19. 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. 20. 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. 21. Further, of trilateral ﬁgures, a right—angled triangle is that which has
a right angle, an obtuseangled triangle that which has an obtuse angle, and
and acuteangled triangle that which has its three angles acute. * This deﬁnition and several others are extremely vague. Euclid never uses them. 09/07/05 WED 00:03 FAX 512 471 9038 UT DEPT MATH 003 A SUMMARY FROM EUCLID’S ELEMENTS ‘ 259 22. 0f quadrilateral ﬁgures, a Square is that which is both equilateral
and rightangled; an oblong that which is rightangled but not equilateral; a
rhombus that which is equilateral but not rightangled; and a rhombaid that
which has its opposite sides and angles equal to one another but is neither
equilateral nor rightangled. And let quadrilaterals other than these be called
tropezia. . ‘ 23. 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. THE l’ROPDSlTIOHS 0F BOOK  1. On a given ﬁnite straight line to construct an equilateral triangle. 2. To place at a given point (as an extremity) a straight line equal to a ‘
given straight line. 3. Given two unequal straight lines, to cut oil" from the greater a straight
line equal to the less. 4. If two triangles have the two sides equal to two sides respectively,
and have the angles contained by the equal straight lines equal, they will
also have the base equal to the base, the triangle will be equal to the triangle,
and the remaining angles will be equal to the remaining angles respectively,
namely those which the equal sides suhtcnd. 5. In isosceles triangles the angles at the base are equal to one another,
and, if the equal straight lines he produced further, the angles under the base
will be equal to one another. 6. If in a triangle two angles be equal to one another, the sides which ‘ subtend the equal angles will also be equal to one another. 7. Given two straight lines constructed on a straight line (from its
extremities)” and meeting in a point, there cannot be constructed on the same
straight line (from its extremities), and on the same side of it, two other
straight lines meeting in. another point and equal to the former two respec~
tively, namely each to that which has the same extremity with it. 8. If two triangles have the two sides equal to two sides respectively,
and have also the base equal to the base, they will also have the angles equal
which are contained by the equal straight lines. 9.. To bisect a given rectilinea'l angle. 10. To bisect a given ﬁnite straight line. 11. To draw a straight line at right angles to a given straight line from
a giVen point on it. 12. To a given inﬁnite straight line, from a given point which is not on
it, to draw a perpendicular straight line. 09/07/05 WED 00:03 FAX 512 471 9038 UT DEPT MATH 004 250 _ APPENDIX 13. If a straight line set up on a straight line make angles, it will make
either two right angles or angles equal to two right angles. ' 14. If with any straight line, and at a point on it, two straight lines not
lying on the same side make the adjacent angles equal to .two right angles,
the two straight lines will be in a straight line with one another. 15. If two straight lines out one another, they make the vertical angles
equal to one another. _ . ‘ 16. In any. triangle, if one of the sides be produced, the exterior angle is
greater than either of the interior and opposite angles. ‘ 17. In any triangle two angles taken together in any manner are less than
two right angles. . ' 18. In any triangle the greater side subtendsthe greater angle. 19. In any triangle the greater angle is subtended by the greater side. 20. In any triangle two sides taken together in any manner are greater
than the remaining one. 21. If on one of the sides of a triangle, from its extremities, there be
constructed two straight lines meeting within the triangle, the straight lines
so constructed will be less than the remaining two sides of the triangle, but
will contain a greater angle. ‘ ‘ 22. Out of three straight lines, which are equal to three given straight
lines, to construct a triangle: thus it is necessary that two of the straight lines
taken together in any manner shouldbe greater than the' remaining one. 23. On a given straight line and at a point on it to construct a rectis
lineal angle equal to a given rectilineal angle. ' 24. If two triangles have the two sides equal to two sides respectively,
but have the one of the angles contained by the equal straight lines greater
than the other, they will also have the base greater than thebase. 25. If two triangles have the two sides equal to two sides IESPECtiveIY:
but have the base greater than the base, they will also have the one of the
angles contained by the equal straight lines greater than the other. 26. If two triangles have the two angles equal to two angles respectively,
and one side equal to one side, namely, either the side adjoining the equal
angles, or that subtending one of the equal angles,"they will also have the
remaining sides equal to the remaining sides and the remaining angle to the
remaining angle. 27. If a straight line falling on two straight lines make the alterw
nate angles equal to one another, the straight lines will be parallel to one
another. ‘ '28. If a straight line falling on two straight lines make the exterior angle
equal to the interior and opposite angle on the same side, or the interior angles
on the same side equal to two right angles, the straight lines will be parallel
to one another. Emma armed talisman—i. "— as name (rsih the SH“ ‘Pcis’halial‘ew tilt; rs {mg—t. Magi 
a2 FWD"? 0? $l‘oPeS'tfln‘m Z‘l . 09/07/05 WED 00:04 FAX 512 471 9038 UT DEPT MATH 005 ‘_ A SUMMARY FROM EUCLID’S ELEMENTS 261 29. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and
opposite angle, and the interior angles on the same side equal to two right
angles. 30. Straight lines parallel to the same straight line are also parallel to
one another. ‘ 31. Through a given point to draw a straight line parallel to a given
straight line.  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. . 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. 34. In parallelogramnlic areas the opposite sides and angles are equal
to one another, and the diameter bisects the areas. 35. Parallelograms which are on the same base and in the same parallels
are equal to one another. I 36 Parallelograrns which are on equal bases and in the same parallels
are equal to one another. ' 37. Triangles which are on the same base and in the same parallels are
equal to one another. _ ’ 38. Triangles which are on equal bases and in the same parallels are '
equal to one another. 39. Equal triangles which are on the same base and on the same side
are also equal in the same parallels. 40. Equal triangles which are on equal bases and on the same side are
also in the same parallels. ‘ 41. If a parallelogram have the same base with a triangle and be in the
same parallels, the parallelogram is double of the triangle. 42;. To construct, in a given rectilineal angle, a parallelogram equal to a
given triangle. 43, In any parallelogram the complements of the parallelogram about
the diameter are equal to one another. 44. To a given straight line to apply, in a given rectilineal angle, a
parallelogram equal to a given triangle. 45. To construct, in a given rectilineal angle, a parallelogram equal to a
given rectilineal ﬁgure. 46. On a given straight line to describe a square. 47. In rightangled triangles the square on the side subtending the right
angle is equal to the squares on the sides containing the right angle. 43. If in a triangle the square on one of the sides be equal to the squares
on the remaining two sides of thetriangle, the angle contained by the re"
inaining two sides of the triangle is right. ...
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