These are, however, which concur in rendering the application of this curve to the vibration of pendulums designed for the measures of time, the source of errors even greater than those which by its peculiar property it is intended to obviate, and it is now not used. Although the times of vibration of a pendulum in different arches be nearly equal, yet if the arches differ very consi derably, the vibrations will be performed in different times, and the difference, though very small, will become sensible in the course of one day or more. In clocks for astronomical purposes, the arc of vibration must be accurately ascertained, and if it be different from that described by the pendu lum when the clock keeps time, a correction must be applied to the time shown by the clock. This correction, expressed in seconds of time, will be equal to the half of three times the difference of the square of the given arc, and of that of the arc described by the pendulum when the clock keeps time, these arcs being expressed in degrees; and so much will the clock gain or lose according as the first of these arches is less or greater than the second. Thus, if a clock keeps true time when the pendulum vibrates in an arch of 3o, it will lose 10 seconds daily in an arch of 4o, and 24 seconds in an arc of 5o for 4'-3' × }=7 × 10 and generally B2— A2 × } gives the time lost or gained, See Simpson's Fluxions, vol. ii. prob. xxviii, In all that has been hitherto said, the power of gravity has been supposed constantly the same. But, if the said power varies, the lengths of pendulums must vary in the same proportion, in order that they may vibrate in equal times; for we have shewn, that the ratio of the times of vibration and descent through half the lengths is given, and consequently the times of vibration and descent through the whole length is given; but the times of vibration are supposed equal, therefore the times of descent through the lengths of the pendu lum are equal. But bodies descending through unequal spaces, in equal times, are impelled by powers that are as the spaces described, that is, the powers of gravity are as the lengths of the pendulums. Pendulums' length in latitude of London, to swing Rule. To find the length of a pendulum to make any number of vibrations, and vice versa. Call the pendulum making 60 vibrations the standard length; then say, as the square of the given number of vibrations is to the square of 60; so is the length of the standard to the length sought. If the length of the pendulum be given, and the number of vibrations it makes in a minute be required; say, as the given length is to the standard length, so is the square of 60, its vibrations in a minute, to the square of the number required. The square root of which will be the number of vibrations made in a minute. The greatest inconvenience attending this most useful instrument is, that it is constantly liable to an alteration of its length, from the effects of heat and cold, which very sensibly expand and contract all metalline bodies. See HEAT, PYROMETER, &c. To remedy this inconvenience, the com mon method by applying the bob of the pendulum with a screw; so that it may be at any time made longer or shorter, according as the bob is screwed downwards or upwards, and thereby the time of its vibrations kept always the same. Again, if a glass or metalline tube, uniform throughout, filled with quicksilver, and 58.8 inches long, were applied to a clock, it would vibrate seconds for 39.2 of 58.8, and such a pendulum admits of a twofold expansion and contraction, viz. one of the metal and the other of the mercury, and these will be at the same time contrary, and therefore will correct each other. For by what we have shewn, the metal will extend in length with heat, and so the pendulum will vibrate slower on that account. The mercury also will expand with heat, and since by this expansion it must extend the length of the columa upward, and consequently raise the centre of oscillation; so that by this means its distance from the point of suspension will be shortened, and therefore the pendulum on this account will vibrate quicker; wherefore, if the circumstances of the tube and mercury are skilfully adjusted, the time of the clock might, by this means, for a long course of time, continue the same, without any sensible gain or loss. This was the invention of Mr. Graham, in the year 1721, who made a clock of this sort, and compared it with one of the best of the common sort for three years together, and found the errors of the former but about one-eighth part of the latter. Mr. Graham also made a pendulum consisting of three bars, one of steel between two of brass, and the steel bar acted upon a lever, so as to raise the pendulum, when lengthened by heat, and to let it down, when shortened by cold; but he found this clock liable to sudden starts and jerks in its motion. The ingenious Mr. Ellicott, in the Transactions of the Royal Society, describes a pendulum of his invention, composed of brass and iron, with the method of applying it, so as to avoid the many jerks to which the machine might be liable. But besides the irregularities arising from heat and cold, pendulum clocks are liable to others from friction and foulness; to obviate which, Mr. Harrison has several excellent contrivances, whereby his clocks are almost entirely free from friction, and never need to be cleaned. See LONGITUde. The gridiron pendulum is a contrivance for the same purpose. Instead of one rod, this pendulum is composed of any convenient odd number of rods, as five, seven, or nine; being so connected, that the effect of one set of them counteracts that of the other set; and therefore, if they are properly adjusted to each other, the centres of suspension and oscillation will always be equidistant. Fig. 11 represents a gridiron pendulum composed of nine rods, steel and brass alternately. The two outer rods, A B, CD, which are of steel, are fastened to the cross pieces AC, BD by means of pins. The next two rods, EF, GH, are of brass, and are fastened to the lower bar BD, and to the second upper bar EG. The two following rods are of steel, and are fastened to the cross bars E G and I K. The two rods adjacent to the central rod being of brass, are fastened to the cross pieces I K and LM; and the central rod, to which the ball of the pendulum is attach ed, is suspended from the cross piece LM, and passes freely through a perforation in each of the cross bars IK, BD. From this disposition of the rods, it is evident that, by the expansion of the extreme rods, the cross piece B D, and the two rods attached to it, will descend: but since these rods are expanded by the same heat, the cross piece EG will consequently be raised, and therefore also the two next rods; but be cause these rods are also expanded, the cross bar IK will descend: and by the expansion of the two next rods, the piece LM will be raised a quantity sufficient to counteract the expansion of the central rod. Whence it is obvious, that the effect of the steel rods is to increase the length of the pendulum in hot weather, and to diminish it in cold weather, and that the brass rods have a contrary effect upon the pendulum. The effect of the brass rods must, however, be equivalent not only to that of the steel rods, but also to the part above the frame and spring, which connects it with the cock, and to that part between the lower part of the frame and the centre of the ball. Another excellent contrivance for the same purpose is described in a French au thor on clock-making. It was used in the north of England by an ingenious artist about fifty years ago. This invention is as follows: a bar of the same metal with the rod of the pendulum, and of the same dimensions, is placed against the back-part of the clock-case: from the top of this a part projects, to which the upper part of the pendulum is connected by two fine pliable chains or silken strings, which just below pass between two plates of brass, whose lower edges will always terminate the length of the pendulum at the upper end. These plates are supported on a pedestal fixed to the back of the case. The bar rests upon an immoveable base at the lower part of the case; and is inserted into a groove, by which means it is always retained in the same position. From this construction, it is evident that the extension or contraction of this bar, and of the rod of the pendulum, will be equal, and in contrary directions. For suppose the rod of the pendulum to be expanded any given quantity by heat; then, as the lower end of the bar rests upon a fixed point, the bar will be expanded upwards, and raise the upper end of the pendulum just as much as its length was increased; and hence its length below the plates will be the same as before. Of this pendulum, somewhat improved by Mr. Crosthwaite, watch and clockmaker, Dublin, we have the following description, “A and B (fig. 12), are two rods of steel forged out of the same bar, at the same time, of the same temper, and in every respect similar. On the top of B is formed a gibbet C; this rod is firmly supported by a steel bracket D, fixed on a large piece of marble E, firmly set into the wall F, and having liberty to move freely upwards between cross staples of brass, 1, 2, 3, 4, which touch only in a point in front and rear (the staples having been carefully formed for that purpose); to the other rod is firmly fixed by its centre the lens G; of twentyfour pounds weight, although it should in strictness be a little below it. This pendulum is suspended by a short steel spring on the gibbet at C: all which is entirely independent of the clock. To the back of the clock-plate, I, are firmly screwed two cheeks nearly cycloidal at K, exactly in a, line with the centre of the verge L. The maintaining power is applied by a cylindrical steel-stud, in the usual way of regulators at M. Now, it is very evident, that any expansion or contraction that takes place in either of these exactly similar rods, is instantly counteracted by the other; whereas in all compensation pendulums composed of different materials, however just the calculation may seem to be, that can never be the case, as not only different metals, but also different bars of the same metal, that are not manufactured at the same time, and exactly in the same manner, are found by a good pyrometer to differ materially in their degrees of expansion and contraction, a very small change affecting one and not the other." The expansion or contraction of straight-grained fir-wood lengthwise, by change of temperature, is so small, that it is found to make very good pendulum rods. The wood called sapadillo is said to be still better. There is good reason to believe, that the previous baking, varnishing, gilding, or soaking of these woods in any melted matter, only tends to impair the property that renders them valuable. They should be simply rubbed on the outside with wax and a cloth. In pendulums of this construction the error is greatly dimi nished, but not taken away. PENGUIN. See APTENODYTES. PENELOPE, in natural history, a genus of birds of the order Gallina. By Latham, they are mostly arranged under the genus Meleagris, or the Turkey. Their legs, how ever, are without spurs. They inhabit principally South America, and particularly Brasil and Guiana. The P. cristata, or guan, is two feet six inches in length. P. cumanensis, or the yacou, is of the size of a hen turkey, and is found in Cayenne and Guiana. The Marail is found in flocks in Guiana, feeds on fruits, and roosts on trees. See Aves, Plate XI. fig. 5. PENIS. See ANATOMY, PENNANTIA in botany, so named in honour of Thouias Pennant, a genus of the Polygamia Dioecia, class and order. Essential character: calyx, none; corolla five petalled; stamens five: pericarpium, three sided, two-celled, with solitary subtriquetrous seeds. There is but one species, viz. P. corymbosa, a native of New Zealand. PENNATULA, in natural history, seapen, a genus of the Vermes Zoophyta class and order; animal not affixed, of various shapes, supported by a bony part within, naked at the base, the upper part with generally lateral ramifications, furnished with rows of tubular denticles producing radiate polypes from each tube. There are about eighteen species, of which P. coccinea is described as: stem round, radiating, with papillous polype-bearing sides, and clavate at the top. It is found in the White Sea, is soft, red, an inch and a half high, and as thick as the little finger, wrinkled, with the papillæ disposed in rows. P. phosphorea has a fleshy stem, with a rough midrib, and imbricate ramification. It inhabits most seas, and emits a very strong phosphoric light in the dark; about four inches long, red, stem villous, with a lanceolate rough midrib, and nearly incumbent rays, the tubes pointing all one way. P. reniformis; stem round, vermicular, supporting a kidney-shaped leaf-like head, producing polypes on one surface. It inhabits South Carolina: body expanded, kidney-shaped, flat, rising from a short round stenr, and covered on the upper surface with numerous tubular orifices, through which the polypes are obtruded at pleasure; the upper surface is of a rich purple, the under side brilliant, and sometimes yellowish. PENNY, an ancient silver coin, which, though now little used, was, the only one current among our Saxon ancestors. It was then equal to th part of a pound. In Etheldred's time the penny was the 20th part of the Troy ounce, hence the denomination penny-weight. Till the time of Edward the first, the penny was struck with a cross |