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(6) The Catenary.
This is the curve which a perfectly flexible chain will assume when suspended from two points in the same horizontal line; I must refer the reader to works on statics for an investigation of its equation, which is
y ( +).
Its most important geometrical properties are analogous to properties of the circle. Thus, the part of the normal intercepted between the curve and the axis of x is equal to the radius vector, but measured in the opposite direction ; and if we represent an ordinate corresponding to the abscissa x by f (x), and the corresponding area by cF (r), we shall readily find from the preceding equation that
cf (x + x') = f (x) f (x') + F(x) F ('),
cF(x – x') = F(v) f (x) - f (x) F (x'). It is obvious that the preceding formulæ are analogous to those connecting sines and cosines of circular arcs. For these and other properties of the catenary connected with the involute of the parabola, see a paper by Professor Wallace in the Edinburgh Transactions, Vol. xiv. p. 625.
(7) The Quadratrix of Dinostratus.
If the radius CQ of the circle ABD (fig. 18) revolve uniformly round from A to B, while the ordinate NM also moves uniformly parallel to itself from A to C, the locus of their intersection will be the quadratrix of Dinostratus. To find its equation, let AM = x, PM = Y, AC = a. Then from the uniformity of the motion of CQ and MN, we have
ACQ : ACB = AM : AC';
But PM = CM tan ACQ, therefore the equation to the
This curve was used by Dinostratus (a mathematician of the school of Plato) for the purpose of dividing an angle into any number of parts, and also of squaring the circle, from which it derives its name. The following is the property which enables the curve to be so employed.
When x = a, we have (by Chap. vi. Ex. 29) y =CE = -, so that CE is a third proportional to the quadrant and the radius, and thus if the point E could be determined by means of the straight line and circle, the circle could be squared.
Léotaud, in his treatise on this curve appended to his Cyclomathia, showed that it is not confined within the semicircle ABD, but that it has two infinite branches extending below the axis of X, and bounded by asymptotes parallel to the axis of y at distances - a and 3a from the origin. In addition to this, the curve has an infinite number of infinite branches, which are bounded by asymptotes parallel to the axis of y at distances 5a, 7a, &c., - 3a, - 5a, &c. from the origin, and which cut the axis of dat distances 4a, 6a, &c., - 2a, – 4a, &c. from the origin. The farther these points are removed from the origin the more nearly is the curve perpendicular to the axis of X, the value of
at the intersection being + (2n – 1) 5, 2na being the abscissa of the point where the curve cuts the axis of a.
Rolling Curves. (8) The Cycloid.
This curve is generated by a point P in the circumference of a circle 6 Pc (fig. 19), which rolls along a line AA. To find its equation put
It is easy to see both from geometrical and analytical considerations that the cycloid is not limited to the space between A and A', but that it consists of an infinite number of portions equal and similar to AC A and touching each other in cusps as in the figure.
After the Conic Sections there is no curve in geometry which has more exercised the ingenuity of mathematicians than the cycloid, and their labours have been rewarded by the discovery of a multitude of interesting properties, important both in geometry and in dynamics.
The invention of this curve is usually ascribed to Galileo, but Wallis in a letter to Leibnitz* says, that it is mentioned by Cardinal de Cusa in a work published in 1510, and that in the MSS. the date of which is about 1454, it is “pulchrè delineatam”, therein differing from the printed copies. Roberval proved that the whole area of the cycloid is three times that of the generating circle, and this discovery, which was the cause of many disputes between rival claimants to the honour of making it, drew the attention of mathematicians to the study of the properties of this new curve. Among others, Descartes occupied himself with the subject, and he
• Leibn. Opera, Vol. 111. p. 95.
showed how to draw tangents to the curve, and proved that the tangent at any point P (fig. 19) is perpendicular to the corresponding chord BQ of the generating circle, and consequently that it is parallel to CQ: from this also it readily follows that if QR be a tangent to the generating circle at Q, QR = PQ= arc BQ. Wren was the first who rectified the cycloid, and he showed that the length of an arc measured from the vertex is equal to twice the chord of the generating circle which is parallel to the tangent at the extremity, so that the whole length of the curve is equal to four times the diameter of the generating circle. Pascal discovered the means of finding the area and the centre of gravity of any segment of the curve as well as the content and surface of the solids formed by the revolution of the segment round the axis of the curve, and the base of the segment, and to the solution of these problems he challenged all mathematicians in a letter which he circulated under the name of Dettonville, offering at the same time a prize of forty pistoles to the first and one of twenty pistoles to the second person who should solve them. Wallis and Lalouère appeared as candidates for the prize, but none was awarded. To Huyghens is due the discovery that the evolute of the cycloid is an equal cycloid in an inverted position, and that the radius of curvature is double of the chord of the generating circle which is perpendicular to the tangent. He also discovered the important dynamical property of the tautochronism of a cycloidal pendulum ; that is to say, that a body under the action of gravity falling down an inverted cycloid with its base horizontal, will reach the lowest point in the same time froin whatever point it begins to fall. Two of the most remarkable properties of this curve were discovered by John Bernoulli: 1st, that it is the curve along which a body will, under the action of gravity, fall in the shortest time from one given point to another not in the same vertical : 2nd, that if any arc of a curve as AB (fig. 21), the tangents at the extremities of which are at right angles to each other, be evolved into a curve BA', beginning from B: and if the same operation be performed on AB, beginning from A', and so on in succession, the successive involutes will continually approximate to a common cycloid, the axis of which is parallel to AC* The preceding are only a few of the most important properties of this curve; for a detailed account of all which the industry of mathematicians has discovered, the reader must be referred to the treatises on the cycloid which have been written by various authors. Such are the Histoire de la Roulette of Pascal; the History of the Cycloid of Carlo Dati ; the Treatise de Cycloide of Wallis ; the Historia Cycloidis of Groningius in his Bibliotheca Universalis ; and the work of Lalouère called Geometria promota in VII de Cycloide libris.
(9) The Companion to the Cycloid.
If the ordinate QN (fig. 20) of a semicircle be produced till it be equal to the arc CQ, its extremity will lie in a curve which is called the companion to the cycloid. The co-ordinates of a point in this curve are, putting CO = a, CN = x, CN = Y, COQ = 0,
x = a (1 – cos 6), y = al. It has points of contrary flexure at the extremities D and d of an ordinate passing through the centre of the generating circle. The space COD is equal to the square of the radius; the whole arca ACa is equal to twice that of the generating circle, and if the line AC be drawn, the area AMD is equal to the area CLD.
(10) If instead of supposing the point P to be in the circumference of the generating circle we suppose it to be either within the arca of the circle or without it, the curve traced out is called a Trochoid. The equations to such a curve are
X = a (0 – n sin ),
y = a (1 – n cos 6), where n is the ratio of the distance of the tracing point from the centre of the generating circle to the radius of that circle.
John Bernoulli, Opera, Vol. iv. p. 98. Euler, Commen. Petrop. 1766. Legendre, Exercices du Calcul Integral, Tom. II. p. 491.