This is the definitive edition of one of the very greatest classics of all time — the full Euclid, not an abridgement. Using the text established by Heiberg, Sir Thomas Heath encompasses almost 2, 500 years of mathematical and historical study upon Euclid. This unabridged republication of the original enlarged edition contains the complete English text of all 13 books of the Elements, plus a critical apparatus that analyzes each definition, postulate, and proposition in great detail. It covers textual and linguistic matters; mathematical analyses of Euclid's ideas; classical, medieval, Renaissance, modern commentators; refutations, supports, extrapolations, reinterpretations, and historical notes, all given with extensive quotes. "The textbook that shall really replace Euclid has not yet been written and probably never will be." — Encyclopaedia Britannica. Volume 1. 151-page Introduction: life and other works of Euclid; Greek and Islamic commentators; surviving mss., scholia, translations; bases of Euclid's thought. Books I and II of the Elements, straight lines, angles, intersection of lines, triangles, parallelograms, etc. Volume 2. Books III-IX: Circles, tangents, segments, figures described around and within circles, rations, proportions, magnitudes, polygons, prime numbers, products, plane and solid numbers, series of rations, etc. Volume 3. Books X to XIII: planes, solid angles, etc.; method of exhaustion in similar polygons within circles, pyramids, cones, cylinders, spheres, etc. Appendix: Books XIV, XV, sometimes ascribed to Euclid.
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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 than a 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 circumference cut off by it. And the centre of the semicircle is the same as that of the circle.
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 obtuse-angled triangle that which has an obtuse angle, and an acuteangled triangle that which has its three angles acute.
22.Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus 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.
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.
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.
COMMON NOTIONS.
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.
[7]4.Things which coincide with one another are equal to one another.
[8]5.The whole is greater than the part.
DEFINITION1.
A point is that which has no part.
An exactly parallel use of
in the singular is found in Aristotle, Metaph. 1035 b 32
, literally “There is a part even of the form”; Bonitz translates as if the plural were used, “Theile giebt es,” and the meaning is simply “even the form is divisible (into parts).” Accordingly it would be quite justifiable to translate in this case “A point is that which is indivisible into parts.”
Martianus Capella (5th C. A.D.) alone or almost alone translated differently, “Punctum est cuius pars nihil est,” “a point is that a part of which is nothing.” Notwithstanding that Max Simon (Euclid und die sechs planimetrischen Bücher, 1901) has adopted this translation (on grounds which I shall presently mention), I cannot think that it gives any sense. If a part of a point is nothing, Euclid might as well have said that a point is itself “nothing,” which of course he does not do.
Pre-Euclidean definitions.
It would appear that this was not the definition given in earlier textbooks; for Aristotle (TopicsVI. 4, 141 b 20), in speaking of “the definitions” of point, line, and surface, says that they all define the prior by means ...
