1 PENANCE is a punishment, either voluntary or imposed by authority, for the faults a person has committed. Penance is one of the seven sacraments of the Romish church. Besides fasting, alms, abstinence, and the like, which are the general conditions of penance, there are others of a more particular kind; as the repeating a certain number of ave-mary's, paternosters, and credos, wearing a hair shirt, and giving one's self a certain number of stripes. In Italy and Spain it is usual to see Roman Catholics almost naked, loaded with chains and a cross, and lashing themselves at every step. PENATES, in Roman antiquity, a kind of tutelar deities, either of countries or particular houses; in which last sense they differed in nothing from the lares. See LARES. They were properly the tutelar gods of the Trojans, and were adopted by the Romans, who gave them the title of penates. PENCARROW, a cape of Cornwall, on the south coast of the English Channel; two miles east of the mouth of the Fowey. PENCIL, n. s. Lat. penicillum. A small brush of hair which painters dip in their colors: an instrument for writing with black lead. Painting is almost the natural man ; Shakspeare. But shows some touch, in freckle, streak, or stain, Of his unrivalled pencil. Cowper. PENCILS are of various kinds, and made of boars' bristles, the thick ends of which are bound various materials; the largest sorts are made of to a stick, bigger or less according to the uses they are designed for: these, when large, are called brushes. The finer sorts of pencils are made of camels, badgers, and squirrels' hair, and of the down of swans; these are tied at the upper end with a piece of strong thread, and enclosed in the barrel of a quill. All good pencils, on being drawn between the lips, come to a fine point. PENCILS, for drawing, are made of long pieces of black-lead or red chalk, placed in a groove cut in a slip of cedar; on which other pieces of cedar being glued, the whole is planed round, and, one of the ends being cut to a point, it is fit for use. PENCKUM, a town of Germany, in Anterior Pomerania; thirteen miles south-west of Old Stettin, and forty-four N. N. W. of Custrin. Long. 31° 59′ E. Ferro, lat. 53° 15′ N. PENDA, the first king of Mercia, founded that kingdom, A. D. 626. He was killed by Oswy, king of Northumberland, A. D. 655. See MERCIA. PENDA. See PEMBA. A Fr. pendant of Lat. pendens, pendeo. jewel or any thing suspended; a pendulum ; (a small flag: pendence, pendency, or pendulosity and pendulousness all mcan suspension; the state of being pendent or hanging: pending is depending; hence undecided: pendulous, synonymous with pendent. Quaint in green she shall be loose enrobed With ribbons pendent, flaring about her head. Shakspeare. A pendent rock, A forked mountain, or blue promontory With trees upon't, that nod unto the world, And mock her eyes with air. Id. All the plagues, that in the pendulous air Hang fated o'er men's faults, light on thy daughters. Id. The Italians give the cover a graceful pendence or slopeness, dividing the whole breadth into nine parts, whereof two shall serve for the elevation of the highest top or ridge from the lowest. Wotton. To make the same pendant go twice as fast as it did, or make every undulation of it in half the time it did, make the line, at which it hangs, double in geometrical proportion to the line at which it hanged before. Digby on the Soul. They brought by wond'rous art Milton's Paradise Lost. His slender legs he increased by riding, that is, the humours descended upon their pendulosity, having no support or suppedaneous stability. Browne's Vulgar Errours. A person pending suit with the diocesan, shall be defended in the possession. Ayliffe. The judge shall pronounce in the principal cause, nor can the appellant alledge pendency of suit. Id. The spirits Some thrid the mazy ringlets of her hair, Some hang upon the pendents of her ear. Pope. PENDANTS are often composed of diamonds, pearls, and other jewels. PENDANTS, in heraldry, parts hanging down - from the label, to the number of three, four, five, or six, at most, resembling the drops in the Doric freeze. When they are more than three, they must be specified in blazoning. PENDANTS OF A SHIP are those streamers, or long colors, which are split and divided into two parts, ending in points, and hung at the head of masts, or at the yard-arm ends. PENDENNIS, a peninsula of Cornwall, at the mouth of Falmouth haven, a mile and a half in compass. On this Henry VIII. erected a castle, opposite to that of St. Maw's, which he likewise built. It was fortified by queen Elizabeth, and served them for the governor's house. It is one of the largest castles in Britain, and is built on a high rock. It is stronger by land than St. Maw's, being regularly fortified, and having good outworks. PENDULUM, n. s. Fr. pendule; Lat pendulus. Any weight hung so that it may easily swing backwards and forwards. See below. Upon the bench I will so handle 'em, Hudibras. A PENDULUM is a vibrating body suspended from a fixed point. For the history of this invention, see CLOCK. The theory of the pendulum depends on that of the inclined plane. Hence, to understand the nature of the pendulum, it will be necessary to premise some of the properties of this plane. I. Let AC, fig. 1, Plate PENDULUM, be an inclined plane, A B its perpendicular height, and D any heavy body: then the force which impels the body D to descend along the inclined plane A C is to the absolute force of gravity as the height of the plane A B is to its length AC; and the motion of the body will be uniformly accelerated. II. The velocity acquired in any given time by a body descending on an inclined plane, A C, is to the velocity acquired in the same time, by a body falling freely and perpendicularly, as the height of the plane A B to its length AC. The final velocities will be the same; the spaces described will be in the same ratio; and the times of description are as the spaces described. III. If a body descend along several contiguous planes, AB, BC, CD (fig. 2), the final velocity, namely, that at the point D, will be equal to the final velocity in descending through the perpendicular A E, the perpendicular heights being equal. Hence, if these planes be supposed indefinitely short and numerous, they may be conceived to form a curve; and therefore the final velocity acquired by a body in descending through any curve AF, will be equal to the final velocity acquired in descending through the planes A B, BC, CD, or to that in descending through A E, the perpendicular heights being equal. IV. If, from the upper or lower extremity of the vertical diameter of a circle, a cord be drawn, the time of descent along this cord will be equal to the time of descent through the vertical diameter; and therefore the times of descent through all cords in the same circle, drawn from the extremity of the vertical diameter, will be equal. V. The times of descent of two bodies through two planes equally elevated will be in the subduplicate ratio of the lengths of the planes. If, instead of one plane, each be composed of several contiguous planes similarly placed, the times of descent along these planes will be in the same ratio. Hence, also, the times of describing similar arches of circles similarly placed will be in the subduplicate ratio of the lengths of the arches. VI. The same things hold good with regard to bodies projected upward, whether they ascend upon inclined planes or along the arches of circles. The point or axis of suspension of a pendulum is that point about which it performs its vibrations, or from which it is suspended. The centre of oscillation is a point in which, if all the matter in a pendulum were collected, any force applied at this centre would generate the same angular velocity in a given time as the same force when applied at the centre of gravity. The length of a pendulum is equal to the distance between the axis of suspension and centre of oscillation. Let PN (fig. 3) represent a pendulum suspended from the point P; if the lower part N of the pendulum be raised to A, and let fall, it will by its own gravity descend through the circular arch AN, and will have acquired the same velocity at the point N that a body would acquire in falling perpendicularly from C to N, and will endeavour to go off with that velocity in the tangent ND; but, being prevented by the rod or cord, will move through the arch Ñ B to B, where, losing all its velocity, it will by its gravity descend through the arch BN, and, having acquired the same velocity as before, will ascend to A. In this manner it will continue its motion forward and backward along the arch AN B, which is called an oscillatory or vibratory motion; and each swing is called a vibration. Prop. I. If a pendulum vibrates in very small circular arches, the times of vibration may be considered as equal, whatever be the proportion of the arches. Let PN (fig. 4) be a pendulum; the time of describing the arch AB will be equal to the time of describing CD; these arches being supposed very small. Join AN, CN; then since the times of descents along all cords in the same circles, drawn from one extremity of the vertical diameter, are equal; therefore, the cords AN, CN, and consequently their doubles, will be described in the same time; but the arches AN, CN, being supposed very small, will therefore be nearly equal to their cords: hence the times of vibrations in these arches will be nearly equal. PROP. II.-Pendulums which are of the same length vibrate in the same time, whatever be the proportion of their weights. This follows from the property of gravity, which is always proportional to the quantity of matter, or to its inertia. When the vibrations of pendulums are compared, it is always understood that they describe either similar finite arcs, or arcs of evanescent magnitude, unless the contrary is mentioned. PROP. III.—If a pendulum vibrates in the small arc of a circle, the time of one vibration is to the time of a body's falling perpendicularly through half the length of the pendulum as the I circumference of a circle is to its diameter. Let PE (fig. 5) be the pendulum which describes the arch ANC in the time of one vibration; let PN be perpendicular to the horizon, and draw the cords AC, AN; take the arc Ee infinitely small, and draw EFG, efg, perpendicular to PN, or parallel to A C; describe the semicircle BG N, and draw er, gs, perpendicular to EG; now lett time of descending through the diameter 2 P N, or through the cord A N; then the velocities gained by falling through 2 PN, and by the pendulum's descending through the arch A E, will be as√2PN and BF; and the space described in the time t, after the fall through 2 PN, is 4 PN. But the times are as the spaces divided by the velocities. EF PN × Ee; And KG = KD: FG::Gg: Hence Ee = PN X FG × Gg. And, by substituting this va : in its mean quantity for all the arches Gg, is 2 BN × √ 2 P N–NK × BGN; and the 2 BN X2 PN-NK = × 2 BGN = tx BN. 2 BN X2 PN Now, if t be the time of descent through 2 PN; then, since the spaces described are as the squares of the times,t will be the time of descent through PN: therefore the diameter B N is to the circumference, 2 BGN, as the time of falling through half the length of the pendulum is to the time of one vibration. PROP. IV. The length of a pendulum vibrating seconds is to twice the space through which a body falls in one second as the square of the diameter of a circle is to the square of its circumference. Let d diameter of a circle 1, c circumference 3.14159, &c., t to the time of one vibration, and p the length of the corresponding pendulum; then by last proposition cd: 1′′ : time of falling through half the length of the pendulum. Let s = space described by a body falling perpendicularly in the first second: then, since the spaces described are in the subduplicate ratio of the times of ded d c s Gs; therefore scription, therefore 1" :::/:/p. Hence tx PN x BFX FNX Gg 2 KD × √ PN + PF × FN × √✅ BF × 2 PN= tx √PNXGg tx√2PN x Gg 2 KDX/PN+PFX √2 4KD× √PN+PF tx √2PN x Gg. 2 BN X 2PN-NF But NF, |