Let a line AX (Fig. 3), proceeding to infinity in the direction of X, be divided into equal parts AB, BC, ... and let the lines AA', BB', ... each produced to infinity, make equal angles with AX. Then the infinite strips A'ABB', B'BC'C',... can all be superposed and have equal areas, but it requires infinitely many of these strips to make up the area A'AX, contained between the lines AA' and AX, each produced to infinity. Again, let the angle A'AX be divided into equal parts A'AP, PAQ, .... Then all these sectors can be superposed and have equal areas, but it requires only a finite number of them to make up the area A'AX. Hence, however small the angle A'AP may be, the area A'AP is greater than the area A'ABB', and cannot therefore be contained within it. AP must therefore cut BB' ; and this result is easily recognised as Euclid's axiom. The fallacy here consists in applying the principle of superposition to infinite areas, as if they were finite magnitudes. If we consider (Fig. 4) two infinite rectangular strips A'ABB' and A'PQB' with equal bases AB, PQ, and partially superposed, then the two strips are manifestly unequal, or else the principle of 1. 7] INFINITE AREAS 9 superposition is at fault. Again, suppose we have two rectangular strips A'ABB', C'CDD' (Fig. 5). Mark off equal lengths AA1, A12, ... along AA', each equal to CD, and equal lengths CC1, C1C 2, ..... along CC', each equal to AB, and divide the strips at these points into rectangles. Then all the rectangles are equal, and, if we number them consecutively, then to every rectangle in the one strip there corresponds the similarly numbered rectangle in the other strip. Hence, if the ordinary theorems of congruence and equality of areas are assumed, we must admit that the two strips are equal in area, and that therefore the area is independent of the magnitude of AB. Such deductions are just as valid as the deduction of Euclid's axiom from a consideration of infinite areas. 7. It suffice to give one other example of the attempts to base the theory of parallels on intuition. Suppose that, instead Euclid's definition of parallels as "straight lines, which. g in the same plane, and being produced indefinire in both directions, do not meet one another in either direction," we define them as "straight lines which are everywhere equidistant," then the whole Euclidean theory of parallels comes out with beautiful simplicity. In particular, the sum of the angles of any triangle ABC (Fig. 6) is proved equal to two right angles by drawing through the vertex A a parallel to the base BC. Then, if we draw perpendiculars from A, B, C on the opposite parallel, these perpendiculars are all equal. The angle EAB = /_B_and the angle CAF = LC. It is scarcely necessary to point out, however, that this definition contains the whole debatable assumption. We have no warrant for assuming that a pair of straight lines can exist with the property ascribed to them in the definition. To put it another way, if a perpendicular of constant length move with one extremity on a fixed line, is the locus of its free extremity another straight line? We shall find reason later on to doubt this. In fact, non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term parallel' Greek πapáλλŋλos = running alongside) originally connoted equidistance, but the term is used by Euclid rather in the sense "asymptotic " (Greek à-σúμπTwTos non-intersecting), and this term has come to be used in the limiting ᅲ 1. 8] SACCHERI 11 case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In noneuclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean. Other instances will present themselves. 8. First glimpses of Non-Euclidean geometry. Among the early postulate-demonstrators there stands a unique figure, that of a Jesuit, Gerolamo SACCHERI (16671733), contemporary and friend of Ceva. This man devised an entirely different mode of attacking the problem, in an attempt to institute a reductio ad absurdum.1 At that time the favourite starting-point was the conception of parallels as equidistant straight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. He starts with two equal perpendiculars AC and BD to a line AB. When the ends C, D are joined, it is easily proved that the angles at C and D are equal; but are they right angles? Saccheri keeps an open mind, and proposes three hypotheses: (1) The Hypothesis of the Right Angle. The object of his work is to demolish the last two hypotheses and leave the first, the Euclidean hypothesis, supreme; 1 1Euclides ab omni naevo vindicatus, Milan, 1733. English trans, by Halsted, Amer. Math. Monthly, vols. 1-5, 1894-98; German by Stäckel and Engel, Die Theorie der Parallellinien, Leipzig, 1895. (This book by Stäckel and Engel contains a valuable history of the theory of parallels.) but the task turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following: If one of the three hypotheses is true in any one case, the same hypothesis is true in every case. On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. On the hypothesis of the right angle two straight lines intersect, except in the one case in which a transversal cuts them at equal angles. On the hypothesis of the obtuse angle two straight lines always intersect. On the hypothesis of the acute angle there is a whole pencil of lines through a given point which do not intersect a given straight line, but have a common perpendicular with it, and these are separated from the pencil of lines which cut the given line by two lines which approach the given line more and more closely, and meet it at infinity. The locus of the extremity of a perpendicular of constant length which moves with its other end on a fixed line is a straight line on the first hypothesis, but on the other hypotheses it is curved; on the hypothesis of the obtuse angle it is convex to the fixed line, and on the hypothesis. of the acute angle it is concave. Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less than two right angles); but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that |