Some of the best known of the transcend ental curves are the spiral of Archimedes, r= aa the hyperbolic spiral, r19 ""a; the logarithmic spiral, r at% which cuts all its radii vectores at a constant angle; the logarith mic curve, y= as, or x = m log y, which is note worthy on account of the curious discontinuity of the negative branch (Salmon, H. P. C., chap. 7); the catenary, y= 2 c e e
the form as slimed by a chain hanging from two points of sup c+V
port; the tractrix, log V
which cuts all the tangents to the catenary orthogonally; this curve is of special interest on account of the use made of it by Beltrami ('Saggio d'interpretazione della geometria non euclidea,' 1868). (For detailed references con sult Lona,
All the curves hitherto mentioned have tan gents, which vary continuously from point to point; but there are curves, graphical repre sentations of certain functions, which differ in this respect. A curve may have a tangent at every point, which yet may not vary contin uously; e.g., 'polygonal y = f(x)dx, where f(x) is a certain arithmetic function; such a curve is composed of a number of seg ments of straight lines. (Grave, ( Compte rendus,' v. 127, pp. 1005-1007, 1898).
There are curves which are continuous, and yet have no definite tangents; the classic ex ample, due to Wcierstrass, is y = lbn cos an xir, (Wiener, v. 90, pp. 221-252, 1881).
The explanation of the impossibility of assign ing the tangent at any point is that in any finite interval the curve makes an infinity of oscilla tions.
Another possible deviation from the natural idea of a curve was discovered by Peano Ann.,' v. 36, pp. 157-160, 1890). He shows that it is possible to construct functions .(t), (t) of a single variable such that the points x=411(t), y---#(1) occupy all positions inside a given square; thus a curve can cover a plane area (For detailed discussion of such curves, consult E. H. Moore, Certain Crinkly Curves,' trans., Am. Math. Soc, v. 1, pp. 72 90, 1900).
These examples show clearly that the most general idea of a curve is far removed from the comparatively simple idea first presented in analytical geometry. The definition at present accepted of a plane curve without multiple points, due to Jordan and Hurwitz, has been thus expressed in English (by Osgood) : set of points which can be referred in a one-to-one manner and continuously to the points of a segment of a right line, inclusive of the ex tremities of the segment, if the curve is not dosed, and to the points of the circumference of a circle if the curve is closed." (Hurwitz, des ersten internationalen Mathematiker-Kongresses in Zurich,' 1897, pp, 102, 103).