Commentary on Proposition I.1¶
Back to Proposition I.1.
It is a commonplace that Euclid has no right to assume, without premising some postulate, that the two circles will meet in a point C. To supply what is wanted we must invoke the Principle of Continuity (see note thereon above, p. 235). It is sufficient for the purpose of this proposition and of I. 22, where there is a similar tacit assumption, to use the form of postulate suggested by Killing. “If a line [in this case e.g. the circumference ACE] belongs entirely to a figure [in this case a plane] which is divided into two parts [namely the part enclosed within the circumference of the circle BCD and the part outside that circle], and if the line has at least one point common with each part, it must also meet the boundary between the parts [i.e. the circumference ACE must meet the circumference BCD].”
Zeno’s remark that the problem is not solved unless it is taken for granted that two straight lines cannot have a common segment has already been mentioned (note on Post. 2, p. 196). Thus, if AC, BC meet at $F$ before reaching C, and have the part $FC$ common, the triangle obtained, namely $FAB$, will not be equilateral, but $FA$, $FB$ will each be less than AB. But Post. 2 has already laid it down that two straight lines cannot have a common segment.
Proclus devotes considerable space to this part of Zeno’s criticism, but satisfies himself with the bare mention of the other part, to the effect that it is also necessary to assume that two circumferences (with different centres) cannot have a common part. That is, for anything we know, there may be any number of points C common to the two circumferences ACE, BCD. It is not until III. 10 that it is proved that two circles cannot intersect in more points than two, so that we are not entitled to assume it here. The most we can say is that it is enough for the purpose of this proposition if one equilateral triangle can be found with the given base; that the construction only gives two such triangles has to be left over to be proved subsequently. And indeed we have not long to wait; for I. 7 clearly shows that on either side of the base AB only one equilateral triangle can be described. Thus I. 7 gives us the number of solutions of which the present problem is susceptible, and it supplies the same want in I. 22 where a triangle has to be described with three sides of given length; that is, I. 7 furnishes us, in both cases, with one of the essential parts of a complete διορισμός, which includes not only the determination of the conditions of possibility but also the number of solutions (ποσαχῶς ἐγχωρεῖ, Proclus, p. 202, 5). This view of I. 7 as supplying an equivalent for III. 10 absolutely needed in I. 1 and I. 22 should serve to correct the idea so common among writers of text-books that I. 7 is merely of use as a lemma to Euclid’s proof of I. 8, and therefore may be left out if an alternative proof of that proposition is adopted.
Agreeably to his notion that it is from I. 1 that we must satisfy ourselves that isosceles and scalene triangles actually exist, as well as equilateral triangles, Proclus shows how to draw, first a particular isosceles triangle, and then a scalene triangle, by means of the figure of the proposition. To make an isosceles triangle he produces AB in both directions to meet the respective circles in D, E, and then describes circles with A, B as centres and AE, BD as radii respectively. The result is an isosceles triangle with each of two sides double of the third side. To make an isosceles triangle in which the equal sides are not so related to the third side but have any given length would require the use of I. 3; and there is no object in treating the question at all in advance of I. 22. An easier way of satisfying ourselves of the existence of some isosceles triangles would surely be to conceive any two radii of a circle drawn and their extremities joined.
There is more point in Proclus’ construction of a scalene triangle. Suppose AC to be a radius of one of the two circles, and D a point on AC lying in that portion of the circle with centre A which is outside the circle with centre B. Then, joining BD, as in the figure, we have a triangle which obviously has all its sides unequal, that is, a scalene triangle.
The above two constructions appear in an-Nairīzī’s commentary under the name of Heron; Proclus does not mention his source.
In addition to the above construction for a scalene triangle (producing a triangle in which the “given” side is greater than one and less than the other of the two remaining sides), Heron has two others showing the other two possible cases, in which the “given” side is (1) less than, (2) greater than, either of the other two sides.