CATENARIA, in the higher geometry, the name of a curve line formed by a rope hanging freely from two points of sus.
pension, whether the points be horizon tal or not. The nature of this curve was Sought after in Galileo's time, but not discovered till the year 1690, when Mr. Bernoulli published it as a problem. Dr. Gregory, in 1697, published a method of investigation of the properties formerly discovered by Mr. Bernoulli and Mr. Leibnitz, together with some new pro perties of this curve. From him we take the following method of finding the general property of the catenaria.
1. Suppose a line heavy and flexible, the two extremes of which F and 1), Plate H. Miscellanies, fig. 8, are firmly fixed in those points; by its weight it is bent into a certain curve F A D, which is called the catenaria.
2. Let B D and b c be parallel to the horizon, A B perpendicular to B D, and D c parallel to A B, and the points B b infinitely near to each other. From the laws of mechanics, any three powers in equilibrio are to one another as the lines parallel to the lines of their direction, (or inclined in any given angle) and ter minated by their mutual concourses ; hence if D d express the absolute gravity of the particle D d, (as it will if we allow the chain to be every way uniform) then c will express that part of the gravity that acts perpendicularly upon D d ; and by the means of which this particle en deavours to reduce itself to a vertical position ; so that if this lineola d c be constant, the perpendicular action of gra vity upon the parts of the chain will be constant too, and may therefore be ex pressed by any given right line. Further,
the lineola. D c will express the force which acts against that conatus of the particle D d,, by which it endeavours to restore itself in a position perpendicular to the horizon, and hinders it from doing so. This force proceeds from the pon derous line D A drawing according to the direction D d; and is, ceteris paribus, proportional to the line D A which is the cause of it. Supposing the curve F A D, therefore, as before, whose vertex is A, axis A B, ordinate B D, fluxion of the ax is D C=B b, fluxion of the ordinate d c, the relation of these two fluxions is thus; viz. dc:Dd :: a : D A curve, which is the fundamental property of the curve, and may be thus expressed (putting AB=x and BD=y and AD=c