In general, the use of conics is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solution'. Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conics in every case where that was possible'. Heiberg3 suggests that the allusion of Aristotle to vevoeis perhaps confirms this supposition, as Aristotle nowhere shows the slightest acquaintance with conics. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by "plane" methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane vevoels. But one thing is certain, namely that Euclid deliberately excluded this class of problem, doubtless as not being essential in a book of Elements. Definitions not afterwards used.

Lastly, Euclid has definitions of some terms which he never afterwards uses, e.g. oblong (éтepóμnces), rhombus, rhomboid, trapezium. The "oblong" occurs in Aristotle; and it is certain that all these definitions are survivals from earlier books of Elements.

1 Cf. the chapter on veures in The Works of Archimedes, pp. c—cxxii.

2 Zeuthen, Die Lehre von den Kegelschnitten im Altertum, ch. 12, p. 262.
Heiberg, Mathematisches zu Aristoteles, p. 16.

4 Pappus VII. pp. 670-2.

I.

2.

3.

4.

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.

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.

II. 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 :

I. 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.

DEFINITION I.

Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.

A point is that which has no part.

An exactly parallel use of pépos (or) in the singular is found in Aristotle, Metaph. 1035 b 32 μépos μèv ovv ẻσTì Kaì TOû dovs, 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 (Topics VI. 4, 141 b 20), in speaking of "the definitions" of point, line, and surface, says that they all define the prior by means of the posterior, a point as an extremity of a line, a line of a surface, and a surface of a solid.

The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus, p. 95, 21), who defined it as a "monad having position" or "with position added" (μovas πpooλaßovoa Béow). It is frequently used by Aristotle, either in this exact form (cf. De anima 1. 4, 409 a 6) or its equivalent: e.g. in Metaph. 1016 b 24 he says that that which is indivisible. every way in respect of magr itude and qua magnitude but has not position is amonad, while that which is similarly indivisible and has position is a point. Plato appears to have objected to this definition. Aristotle says (Metaph.