It's interesting that although Euclid delayed any explicit use of the 5th postulate until Proposition 29, some of the earlier Propositions tacitly rely on it. For example, Proposition 16 saysIn any triangle, if one of the sides be [extended], the exterior angle is greater than either of the interior and opposite angles.A * * * *D B C So he's saying the angle ACD is greater than ABC and BAC. This is not necessarily true in non-Euclidean geometry (as with triangles drawn on the surface of a sphere). Heath's comment on this proposition seems slightly middled. He saysAs is well known, this proposition is not universally true, under the Riemann hypothesis of a space endless in extent but not infinite in size.This is presumably a roundabout way of referring to a finite but unbounded space, such as Riemann proposed as a space of constant positive curvature (e.g., the surface of a sphere), but it seems to me the proposition doesn't depend on the boundedness of the space, per se, but on the possibility of intrinsic curvature, at least if we allow the curvature to be variable. In other words, the proposition fails locally (not globally) under the assumption of sufficient curvature. For example, suppose we split up the earth's Northern Hemisphere into four quadrants, each of which is a triangle with three right angles. If one of the edges is extended, the exterior angle is also a right angle, so it is not greater than the opposite interior angles. This isn't because of the boundedness or finiteness of the earth's surface (as embedded in 3-d space), it's because of the intrinsic curvature. That's what allows triangles to have a sum of angles different than pi if the space is not flat. It's the same effect that allows the exterior angle to be not necessarily greater than the opposite interior angles. Still, Proposition 16 occurs long before Euclid explicitly invokes the parallel postulate (in Proposition 29), so some people assume it must be part of "absolute geometry", i.e., propositions that do not depend on the parallel postulate. I think it's more accurate to say that Euclid actually tacitly assumed the parallel postulate prior to invoking it explicitly in Proposition 29. This is just one of several example of logical problems in the Elements. Nevertheless, in spite of its imperfections, it remains a remarkable document. Considering how many bright men over the centuries were convinced that the 5th postulate was actually redundant, it was a great vindication for Euclid when non-Euclidean geometry was discovered (ironically). It's easy to see why Newton concluded that "the ancients" knew more than they were telling. Just for fun, here's a summary of the functional dependence between the propositions of Euclid's Book I, based on the Dover 2nd edition of T.E. Heath's translation: 23 Definitions: d1 through d23 5 Postulates: p1 through p5 5 Common Notions: cn1 through cn5 48 Propositions: PR1 through PR48 PR1 = f (p3, p1, d15, d15, cn1) PR2 = f (p1, PR1, p2, p3, cn3, cn1) PR3 = f (PR2, p3, d15, cn1) PR4 = f (cn4) PR5 = f (p2, PR3, p1, PR4) PR6 = f ( ) <------[see below] PR7 = f (PR5) PR8 = f (PR7) PR9 = f (PR3, PR8) PR10 = f (PR1, PR9, PR4) PR11 = f (PR3, PR1, PR8, d10) PR12 = f (p3, PR10, p1, PR8, d10) PR13 = f (d10, PR11, cn2, cn1) PR14 = f (PR13, p4, cn1, cn3) PR15 = f (PR13, p4, cn1, cn3) PR16 = f (PR10, PR3, p1, p2, PR15, PR4, cn5) PR17 = f (p2, PR13) PR18 = f (PR3, PR16) PR19 = f (PR5, PR18) PR20 = f (PR5, cn5, PR19) PR21 = f (PR20, PR16) PR22 = f (PR20, PR3) PR23 = f (PR22, PR8) PR24 = f (PR23, PR4, PR5, PR19) PR25 = f (PR4, PR24) PR26 = f (PR4, PR16) PR27 = f (PR16, d23) PR28 = f (PR15, PR27, PR13, PR27) PR29 = f (PR13, *p5*, PR15, cn1, cn2) <--- 1st appearance of PR30 = f (PR29, cn1) the 5th Postulate PR31 = f (PR23, PR27) PR32 = f (PR31, PR29, PR13) PR33 = f (PR29, PR4, PR27) PR34 = f (PR29, PR26, cn2, PR4) PR35 = f (PR34, cn1, cn2, PR29, PR4, cn3) PR36 = f (cn1, PR33, PR34, PR35) PR37 = f (PR31, PR35, PR34) PR38 = f (PR31, PR36, PR34) PR39 = f (PR31, PR37, cn1) PR40 = f (PR31, PR38, cn1) PR41 = f (PR37, PR34) PR42 = f (PR23, PR31, PR41) PR43 = f (PR34, cn2, cn3) PR44 = f (PR42, PR31, PR29, p5, PR31, PR43, cn1, PR15) PR45 = f (PR42, PR44, cn1, PR29, PR14, cn2, PR34, PR30, PR33) PR46 = f (PR11, PR31, PR34, PR29) PR47 = f (PR46, PR14, cn2, PR4, PR41) PR48 = f (PR47, PR8) Interestingly, Proposition 6 does not explicitly invoke any axioms, definitions, common notions, or prior postulates in the Dover edition of Heath's translation, nor is it cited by any of the subsequent propositions in Book I (although it is cited later, e.g., in Book II, Proposition 4). Here is the immaculately conceived Proposition 6:If in a triangle two angle be equal to one another, the sides which subtend the equal angles will also be equal to one another.Euclid's proof, as translated by Heath, is as follows: "Let ABC be a triangle having the angle ABC equal to the angle ACB; I say the side AB is also equal to the side AC. For, if AB is unequal to AC, one of them is greater. Let AB be greater; and from AB the greater, let DB be cut off equal to AC the less; let DC be joined [as shown below]. A /\ / \ / \ D/ \ / '. \ / '. \ / '. \ / '. \ / '. \ / '.\ /____________________\ B C Then, since DB is equal to AC, and BC is common, the two sides DB, BC are equal to the two sides AC, CB respectively, and the angle DBC is equal to the angle ACB. Therefore, the base DC is equal to the base AB, and the triangle DBC will be equal to the triangle ACB, the less to the greater, which is absurd. Therefore AB is not unequal to AC; it is therefore equal to it." This is the first appearance of "reductio ad absurdum" in The Elements. Given a triangle ABC with equal angles ABC and ACB, we make the supposition that AB and AC have unequal lengths, from which it follows that one is longer than the other [presumably by the definition of equality]. Without loss of generality we assume that AB is longer than AC, which implies that there is a point D on the line such that the length of BD equals the length of AC. Euclid also tacitly assumes that this point D lies BETWEEN the points A and B, and that this "betweenness" implies that the triangle DBC is smaller than the triangle ABC. It's been pointed out by Hilbert and others that this really needs to be a postulate, but there are no postulates of "betweenness" in The Elements. In any case, the argument continues by noting that we have two triangles, DBC and ACB, such that DB equals AC [by construction] and BC equals CB [by an interesting tacit assumption of what might be called "commutativity" of distances between points], and of course we have our basic fact that the angles DBC and ACB are equal [assuming that since D is on the interior segment AB the angle DBC equals the angle ABC]. Now we invoke Proposition 4 (although the Dover edition of Heath fails to include this reference), which saysIf 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 subtend.Thus, Euclid claims that the triangles DBC and ACB must be equal, and yet we know DBC is smaller than ACB, so we have a contradiction. QED This particular proposition has been the subject of considerable discussion over the years, since the rigor of the argument can be challenged in various ways. See, for example, the related articles Are All Triangles Isosceles? Do Equal Bisectors Imply Isosceles? Focusing just on the relations between the Propositions, we could summarize Book I as follows: 1 5 10 15 20 25 30 35 40 45 | | | | | | | | | | ************************************************ 1 2 * 3 * 4 5 ** 6 ** 7 * 8 * 9 * * * 10 * * * 11 * * * 12 * * 13 * 14 * 15 * 16 ** * * 17 * * 18 * * * 19 * * 20 * * * 21 * * 22 * * 23 * * 24 *** * * 25 * * 26 ** * 27 ** * * 28 * * * 29 * * * 30 * 31 * * 32 * * * 33 * * * 34 * * * 35 * * * 36 *** 37 * ** 38 * * * 39 * * 40 * * 41 * * 42 * * * * * 43 * * * 44 * * * ** 45 * ** ** * * 46 * * * * * 47 * * * * * 48 * * * * Of course Book I is just a small part of The Elements. Here's a brief Table of Contents: I. Basic Propositions on Lines, Triangles, and Squares II. Gnomon and Geometric Algebra III. Geometry of the Circle IV. Rectilinear Figures Inscribed or Circumscribed in Circles V. Eudoxus' Theory of Proportions VI. Application of the Theory of Proportions to Plane Geometry VII. Number Theory (Greatest Common Divisor, Euclidean Algorithm) VIII. Geometric Progressions IX. Number Theory (unique factorization, infinitude of primes, perfect numbers) X. The Theory of Incommensurable (Irrational) Magnitudes XI. Introduction to Three-Dimensional Geometry XII. Method of Exhaustion For Areas and Volumes XIII. Construction of the Platonic Solids Isaac Newton's assistant at Cambridge claimed that during five years he saw Newton laugh only once. Newton had loaned a copy of Euclid to an acquaintance, and the gentleman asked what use it was to study Euclid, "upon which Sir Isaac was very merry".

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