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PROCESSIONAL-PROCLUS.

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understood in the Latin Church to mean that, as the Son proceeds from the Father, the Holy Ghost proceeds from both Father and Son; and in the course of the 7th and 8th centuries, the words 'and from the Son,' for greater distinctness, came to be added to the creed in several churches-as the West. In the controversy with the Latins, Photius (q. v.) took exception to this addition, as unauthorised, and made the addition one of the grounds for his charge of heresy against them, which was resumed on the consummation of the schism under Michael Cerularius. In the union of the Greek and Latin churches at Florence (1437), an article of agreement on this head was adopted, and the words Filioque were sung twice over both in Latin and in Greek, in the solemn mass which celebrated the union. But this union had no root in the popular mind, and the dispute still continues

as of old to divide the churches.

PROCE'SSIONAL (Lat. processionale), the service-book which contains the prayers, hymns, and general ceremonial of the different processions. Many ancient books of this class have been preserved. The processional approved for common use is that of Rome, of which many editions have been published.

a horrible state-prison, recently rendered famous by Carlo Poerio, who was confined there in chains.

PROCLAMA'TION, a public notice given by the sovereign to his subjects. The power of issuing proclamations is part of the prerogative of royalty as the fountain of justice. They sometimes consist state, or act of the executive government affecting the duties and obligations of subjects. The demise of the crown, and accession of a new sovereign, a declaration of war, and the issue of new coin, are all occasions on which a royal proclamation is issued. A proclamation may also be issued to declare the intention of the crown to exercise some prerogative or enforce some law which has for a long time been by a proclamation may lay an embargo on shipping, dormant or suspended. In time of war, the crown and order the ports to be shut. But the most usual class of proclamations are admonitory notices for the prevention of offences, consisting of formal declarations of existing laws and penalties, and of the intention to enforce them; such as the procla mation against vice and immorality, appointed to be read at the opening of all courts of Quarter Sessions in England.

of an authoritative announcement of some matter of

Proclamations are only binding when they do not contradict existing laws, or tend to establish new ones, but only enforce the execution of those which are already in being, in such manner as the sovereign judges necessary. A proclamation must be under the Great Seal. Statute 31 Henry VIII. c. 8 declared that the king's proclamations should be as binding as acts of parliament; an enactment which, while it subsisted, made an entire revolution in the government; but was repealed by 1 Edw. VI. c. 12. In later times, it was attempted to be maintained by the crown lawyers that the king might suspend or dispense with an existing law by proclamation; a power, however, which act 1 Will. and Mary c. 2 declared not to exist.

PROCESSIONS, as solemn and religious rites, are of very great antiquity. With the Greeks and Romans, they took place chiefly on the festivals of Diana, Bacchus, Ceres, and other deities; also before the beginning of the games in the Circus; and in spring, when the fields were sprinkled with holy water to increase their fertility. The priests used to head them, carrying images of the gods and goddesses to be propitiated, and either started from certain temples or from the Capitol. Among the Jews, certain processions around the altar were (and still are to a certain extent) usual on the Feast of Tabernacles; and from them the Mohammedans have adopted their mode of encompassing the sanctuary seven times at Mecca (q. v.). Processions form a prominent part of the Buddhist worship. The practice was early adopted in the Christiani. Church. The Reformation abolished it; and even in the Roman Catholic Church, especially in mixed countries, processions are less frequent or popular now than in former years. They are there either supplicatory processions or cross processions, and are either directed to a certain distant place, to some miraculous image or object, or they are confined to the streets of the cities and the churches. Banners, crosses, and images are generally carried in front; the clergy follow; and the people make up the rear, singing hymns or reciting prayers. In some Protestant states, they are still permitted, under certain restrictions. There is no doubt that, whatever their general intrinsic value, they offer in many instances one of the most strikingly picturesque features of the Roman faith; and that they answer a certain instinctive want in the multitude. For extensive pilgrimages, as such, their history and rites, we refer to PILGRIM, MECCA, FESTIVALS, &C.

PROCHEIN AMI, the old Norman-French for next friend, still often used in English law, means the person in whose name an infant sues in a court of law, or a married woman in a court of equity. The chief object is to have a person responsible for

costs. See NEXT FRIEND.

PRO'CIDA, an islet of Italy, between the island of Ischia and the shores of Naples, and separated from both of these by sea-ways about a mile in width, is three miles long and one mile broad. Pop. 13,S10. On its shores is the city of the same name, with a commodious harbour, a fine regal palace, and

PRO'CLUS, called the SUCCESSOR (Diadochos)., of Syrianus, as the head of the Athenian school-a celebrated Neo-Platonist, was born in Constantinople in 412. He was of Lycian origin, and received his first instruction at Xanthus, in Lycia. He then studied at Alexandria under Arion, Leonaras, Hero, and especially under Heliodorus, with whom he applied himself chiefly to Aristotelian and Platonic philosophy. From thence he went to Athens, where a certain Plutarch, a philosopher, and his daughter, Asclepigieneia, became his instructors-the latter a priestess of Eleusis, chiefly in theurgic mysteries. The vivid imagination and enthusiastic temperament which in his childhood already had led him to believe in apparitions of Minerva and Apollo, naturally convinced him, when all the influences of the were brought to bear upon Mysteries (q. v.) him, still more of his immediate and direct intercommunication with the gods; and he diatinctly believed himself to be one of the few chosen links of the Hermaic chain through which divine revelation reaches mankind. His soul had, he thought, once lived in Nicomachus the Pytha gorean, and, like him, he had the power to com mand the elements to a certain extent, to produce rain, to temper the sun's heat, &c. The Orphic Poems (q. v.), the writings of Hermes, and all that strangely mystical literature with which the age abounded, were to him the only source of true philosophy, and he considered them all more or less in the light of divine revelations. That same cosmopolitan spirit in religious matters which pervaded Rome towards her end, had spread throughout all the civilised 'pagan' world of those

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days, and P. distinctly laid it down as an axiom, of his own ideas in a systematic form. that a true philosopher must also be a hierophant have a work-again preserved in Latin only-On of the whole world. Acquainted with all the creeds Providence and Fate, On the Ten Doubts about Proand rites of the ancient Pantheons of the different vidence, &c., On Platonic Theology, and other minor nations, he not only philosophised upon them in an works, extant in a more.or less fragmentary form, allegorising and symbolising spirit, as many of his and repeatedly edited, with translations and modern contemporaries did, but practised all the ceremonies, commentaries. The most important of his works, however hard and painful. More especially was however, is the Philosophical and Theological Insti the practice of fasting in honour of Egyptian deities, tution, in which P. geometrically, as it were, evolves while on the one hand, it fitted him more and more his doctrines by heading each of its 211 chapters by for his hallucinations and dreams of divine inter- a kind of proposition, which he proceeds to demoncourse, on the other hand more than once endangered strate, appending corollaries in some instances. Ile his life. Of an impulsive piety, and eager to win chiefly treats in it of unity and multiplicity, on disciples from Christianity itself, he made himself productive causes and effects, on the highest good, obnoxious to the Christian authorities at Athens, on that which suffices in itself, on immobility, perwho, in accordance with the spirit of religious fection, eternity, divinity, and intelligence; on the intolerance and fanaticism which then began to soul, &c. Next in importance stand his commenanimate the new and successful religion against taries on Plato's Timæus, which, however, now only which P. waged constant war, banished him from embraces a third of this dialogue, a similar comthis city. Allowed to return, he acted with some-mentary on Plato's Parmenides, in seven books, what more prudence and circumspection, and only allowed his most approved disciples to take part in the nightly assemblies in which he propounded his doctrines. He died in 485, in his full vigour, and in the entire possession of all his mental powers, for which he was no less remarkable than for his personal beauty and strength.

Respecting his system, some modern philosophers have exalted it to an extent which his own works would hardly seem to warrant. Victor Cousin holds that he has concentrated in it all the philosophical rays which emanated from the heads of the greatest thinkers of Greece, such as Pythagoras, Plato, Aristotle, &c. P. recognises a certain kind of unity of the Creator, or rather of the divine mind, of which he took the human to be a fragment; and he speaks of the One' and 'The First. The human soul he considered wrapped up in various more or less dense veils, according to the degree of perfection attained; and he further assumed a certain sort of solidarity between the souls of those who naturally, or by certain immutable circumstances, were linked together, such as children and parents, rulers and subjects; and he carried this doctrine so far as to assert, that the children must naturally participate in their parents' faults. Faith alone, he further held, was essential to the attainment of Theurgy, which, comprising mantic and supernatural inspiration, is preferable to all human wisdom; and in this he chiefly differs from Plotinus (q. v.), with whose system he agrees in most other respects. He further tries to recognise and to fathom the original mysterious One by combination of figures, strongly reminding us of Gnosticism and the later Kabbala. His way of developing the finite beings out of the infinite Unit is also peculiar. A whole series of triads, at the head of each of which again stands a unit, goes in various gradations through the creation, the lower powers emanating from the higher, which are the thinking and creative ideas, &c. See PLOTINUS, GNOSTICS.

Of his manifold works, there have survived several hymns, which, by the true poetical and religious spirit which pervades them, stand out most favourably among the generally inane Orphic hymns. Of his astronomical and mathematical writings, there have survived a short summary of the chief theories of Hipparchus, Aristarchus, Claudius Ptolemæus, and others, a work On the Heavenly Spheres, a Commentary on Euclid, and a work-only known in a Latin translation-On the Effects of the Eclipses of the Sun and Moon. His grammatical works consist of some commentaries on Homer, Hesiod, &c. The greater part of his writings is devoted to philosophy. These are partly commentaries and paraphrases of Platonic dialogues, and partly the embodiments

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on Cratylus, the First Alcibiades, and fragments on other Platonic writings. Some other works attributed to P. have by modern investigators been pronounced to be spurious.

the consulship, who was invested with powers PROCO'NSUL, a Roman magistrate not holding extending over the city and its vicinity. The pronearly approaching those of a consul, not, however, consul was, at first, one who had held the office of him to bring an unfinished campaign to a close. consul, whose imperium was prolonged to enable The duration of the office was a year. During the latter period of the republic, when the consuls were expected to spend the year of their consulate at Rome, they were generally appointed at its close to undertake, as proconsuls, either the conduct of a Occasionally, the office of proconsul, with the war in some province, or its peaceful administration. government of a province, was conferred on a person who had never held the consulship. Under Constantine, parts of certain dioceses came to be governed by proconsuls.

PROCOP, ANDREW, the Hussite leader, known as P. the Elder, or the Holy, or the Shaven, in allusion to his having received the tonsure in early life, was born towards the close of the 14th c., and belonged to a noble family of Prague. After having travelled with an uncle for some years through France and Spain, he returned to Bohemia at the outbreak of the religious wars, in which Ziska (q. v.) took so prominent a part, and at once entered the ranks of the insurgent Hussites. His military genius soon raised him to the rank of an influential commander; and on the death of Ziska in 1424, P. was elected by the Taborites, who formed an important section of the Hussites, as their leader, and from this period till 1427, his history presents an almost unbroken series of daring attacks upon the Austrians. In the meantime, another body of Taborites, who called themselves Orphans, had overrun Lausitz, and burned Lauban, under the leadership of a man, subsequently known as Procop the Lesser, or Younger, who now, in concert with the more distinguished P., attacked Silesia, and took part in those internal feuds of the Hussite factions by which Bohemia was almost wholly ruined. The threatened approach of three German armies, which had been levied by the neighbouring states to carry on an exterminating crusade against the heretics, was alone able to restore unanimity to the divided Hussites, who, under the leadership of the two Procops, offered a desperate and successful resistance to the larger numbers of the Germans, subsequently pursuing their enemies with fire and sword through Silesia, Moravia, and

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PROCOPIUS-PROCTORS.

which P. writes with the clearness, weight, and fulness of knowledge that might be expected of a man who had been an eye-witness of much of what he narrates, and who had occupied a position that fitted him to thoroughly understand what he had seen. He is the principal authority for the reign of Justinian. His style is pure, vigorous, and flexible. The best edition of his complete works is that by Dindorf (3 vols., Bonn, 1833-1838). PRO'CRUSTÉS (Gr. 'the Stretcher'), the surname of a celebrated robber of Attica, named Damastes, or Polypemon. According to the ancient legend, he was wont to place all persons who fell into his hands upon a bed which was made either too long or too short for them, and where he racked their limbs till they died. This he continued to do until Theseus overpowered him, and made him suffer the tortures he had inflicted on others. The story has given rise to a figurative expression. When an author is subjected by a critic to a cruel or unfair mode of criticism, he is said to be stretched on 'the bed of Procrustes.'

Hungary, as far as Presburg. In 1429, P. made of these productions is undoubtedly the first, in inroads into the German states as far as Magdeburg, and returned to Bohemia laden with spoil, and followed by a numerous band of captive nobles and knights; and in the following year, at the head of 50,000 men-at-arms, and half as many horsemen, he again broke into Misnia, Franconia, and Bavaria, and after having burned 100 castles and towns, and destroyed 1400 villages and hamlets, and carried off a vast amount of treasure, turned his arms against Moravia and Silesia. The Emperor Sigismund at this crisis offered to treat with him, but the imperial demand, that the Hussites should submit to the decision of a council, afforded P. a pretext for breaking off all negotiations with the imperial court. A second German crusading army now advanced in 1431, but was thoroughly defeated at Riesenburg. These successes, which were followed by others of nearly equal importance in Silesia, Hungary, and Saxony, where the princes had to purchase peace at the hands of the two Procops, on humiliating terms, induced the council of Basel to propose a meeting between the Hussite leaders and ten learned Catholic doctors. The meeting lasted fifty days, but was productive of no good result, better known as BARRY CORNWALL, was born in PROCTER, BRYAN WALLER, an English poet, and P. returned to Bohemia, where, combining his He studied law, forces with those of Procop the Lesser, he laid siege 1787, and educated at Harrow. to Pilsen. The council, on this, passed an act, was called to the bar in 1831, and for many years was one of the Commissioners of Lunacy, but known as the Basel Compact, by which the Hussites were allowed the use of the cup in the Lord's resigned in 1860, and was succeeded by his friend Supper, and the Bohemians were designated by the Mr Forster, the historical essayist. His Dramatic title of the First Sous of the Catholic Church.' Scenes and other Poems were published in 1819, and he subsequently produced several volumes both of The Taborites and Orphans, under the leadership of the two Procops, refused, however, to have anya Tragedy. As a poet, P. belongs to the school of verse and prose, the most important being Mirandola, thing to do with the pope, and hence dissensions arose between them and the more moderate of the Hussites. After many lesser encounters between these factions, a decisive battle was fought near Lipau in 1434, in which P. was induced, by a feint of the enemy, to leave his intrenchments. His followers at first fought desperately against the troops of the Bohemian nobles, who were commanded by Meinhard of Neuhaus; but at length, under the influence of a sudden panic, they gave way, and took to flight. P., after vainly striving to re-form their broken lines, threw himself into the midst of the enemy, and was killed. Procop the Lesser, following in his steps, was also slain, and with these two brave Hussite leaders the cause of the Taborites perished.

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PROCO'PIUS, an eminent Byzantine historian, was born at Cæsarea, in Palestine, about the beginning of the 6th c., went to Constantinople when still a young man, and acquired there so high a reputation as a professor of rhetoric, that Belisarius, in 527, appointed him his private secretary. P. accompanied the great warrior in all his important campaigns in Asia, Africa, and Italy, and appears to have displayed remarkable practical as well as literary talent, for we find him placed at the head both of the commissariat department and of the Byzantine navy. He returned to Constantinople shortly before 542, was highly honoured by Justinian, and appointed prefect of the metropolis in 562. His death occurred, it is thought, about three years later. P.'s principal works are his Historic, in 8 books (two on the Persian war, from 408 to 553; two on the war with the Vandals, from 395 to 545; four on the Gothic war, going down to 553); Ktismata, or six books on the buildings executed or restored by Justinian; and Anekdota, or Historia Arcana (of doubtful genuineness), a sort of chronique scandaleuse of the court of Justinian, in which the emperor, his wife Theodora, Belisarius, his wife Antonina, and other distinguished persons, are depicted in the darkest colours. The most valuable

Keats and Hunt, and through all his works the influence of the old English dramatists may be traced like Covent Garden Theatre, where it had considerable a vein through an agate. Mirandola was produced at success. It is not, however, on his Dramatic Scenes or his tragedies, but on his songs, that P.'s reputation rests. In 1866 appeared his Charles Lamb, a ter ADELAIDE ANNE, the poetess, died in February Memoir. Procter died October 4, 1874.-His daugh

1864.

PRO'CTOR (formed by contraction from Lat. procurator, one who cares for another) is the name given to the practitioners in Courts of Admiralty, and in the Ecclesiastical and Prerogative Courts. It corresponds to attorney or solicitor in the other

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By a recent statute, which abolished the exclusive jurisdiction of the Admiralty and Prerog ative Courts, now the Probate Court, all proctors were put on the same footing as attorneys and solicitors, and the power to practise in the new courts indifferently was given to each; and at the same time compensation was given for the loss of their monopoly. The mode by which one becomes a proctor is therefore the same as that by which one becomes an Attorney or Solicitor (q. v.).

PROCTORS, officers in the universities of Oxford and Cambridge (two in number in each), whose duties are to preserve the peace of the university, to repress disorders among the students, and inflict summary academical punishment. They have the command of the academical constabulary force, and have also an extensive police jurisdiction in the town. The proctors must be Masters of Arts, and are chosen by the colleges according to a certain rotation. They nominate two pro-proctors The summary to be their deputies and assistants. authority of the proctors extends both to undergraduates and Bachelors of Arts. They have also a legislative authority as assistants to the heads of houses, and vote in the election of some of the professors and other officers.

PROCURATOR-FISCAL-PROGRESSION.

PRO'FILE, the outline of a section through a cornice or other series of mouldings.-The outline of a capital when drawn geometrically; the outline of the human face in a section through the median line; &c.

PROGNO'SIS (from the corresponding Greek word) is the term employed in medicine to indicate the opinion or decision of the physician regarding the probable course and issue of a disease. The physician is guided in arriving at his decision by his knowledge of the course which the disease usually follows; and as some diseases almost always end in recovery, and others almost invariably terminate fatally, the final result may often be predicted with great confidence. In forming a prognosis, the physician must, however, not only take into his consideration the natural history of the individual disease, but numerous modifying influences, such as age, sex, mode of life, previous state of health, &c.

PRO'CURATOR-FI'SCAL, a legal practitioner in Scotland of some consequence, owing to his being the public prosecutor for a local district. He is generally a local procurator, or law-agent, and is appointed by the sheriff of the county, or in cities and towns by the magistrates. His business is to take the initiative in the prosecution of crimes. There being no coroner's inquisition in Scotland, he does the work which that functionary does in England. Whenever he has reason to believe a crime has been committed, his duty is to apply for a warrant to arrest the alleged criminal, or to summon him before the sheriff, when the witnesses are cited, and are precognosced-that is, they give what evidence they are in possession of. All the inquests and examinations of the procurator-fiscal are conducted privately; neither the press nor the public being allowed to be present. This arrangement, as tending to huddle up that which should be fully known-as, for example, the cause of catastrophes attended with loss of life-has latterly been the subject of earnest remonstrance. If the procurator-fiscal is informed of a crime which he thinks was either not committed, or of which there is no evidence satisfactory, he gives his concurrence merely to the private party who suggests it, but does not himself initiate the proceeding. When the procurator-fiscal takes the precognitions of the witnesses, he sends a copy of them to the crown counsel, of whom the Lord Advocate is the chief; and if these counsel think the evidence is strong enough, and warrants more than suspicion, the prosecution is proceeded with to trial. The procura-gression may be of various kinds, but the three tors-fiscal are now paid by salaries according to the population of the district.

PRODIGY. See OMEN.

PRODUCTION OF DOCUMENTS is often required in legal proceedings, or in the course of a suit, in Scotland, as well as in other countries; but it depends on the nature of the suit when and under what conditions the documents must be produced. As a general rule, whenever a right is founded on a document, that document must be produced or shewn to the court which has to determine the nature of the right.

PRODUCTIVE, and UNPRODUCTIVE, LABOUR. See LABOUR.

PROFESSOR, an officer in a university whose duty it is to instruct students, or read lectures on particular branches of learning. In the early times of universities, the degrees conferred on students were licenses to act as public teachers; and the terms Master, Doctor, and Professor were nearly identical in signification. As, however, the body of graduates ceased in the course of time to have any concern in public teaching, a separate class of recognised teachers sprang up, paid sometimes with salaries, in other instances by fees. These were called professors; and in the German and Scottish universities became the governing body, and sole recognised functionaries for the purpose of education. In the universities in which collegiate foundations prevailed, as Oxford and Cambridge, they became, on the other hand, only secondaries or auxiliaries, attendance on their lectures not being generally deemed indispensable, and the necessary business of instruction being carried on by the functionaries of the several colleges.

The word professor is occasionally used in a loose way to denote generally the teacher of any science or branch of learning, without any reference to a university. It has been assumed as a designation not only by instructors in music and dancing, but by conjurors.

PRO'GRESS OF TITLES, in Scotch Law, means the series or chain of conveyances by which a proprietor of lands establishes his right to property. As these titles are the sole evidences of property, the progress of titles-i. e., a short statement of the is first given to a purchaser, to shew that the nature of each conveyance, in their historical order-vendor is able to sell. See SALE OF LAND.

PROGRESSION, in Arithmetic, is the succession, according to some fixed law, of one number after another. A series of numbers so succeeding one another is said to be in progression.'

Pro

forms of most frequent occurrence are Arithmetical
Progression (q. v.), Geometrical Progression (q. v.), and
Harmonical Progression. The conditions of the
harmonical progression of a series are frequently
stated as follow: three numbers are in harmonical
progression, when the first has to the third the same
ratio that the excess of the first over the second has to
the excess of the second over the third, i. e., a, b, c are
in harmonical progression when a :c::a-bb-c;
but a much simpler conception of it is obtained by
means of one of its properties, viz., that if the terms
of a harmonical series be inverted, they form a series
in arithmetical progression; thus, 1, 2, 3, 4, 5, 6, &c.
&c. is a harmonical progression; 1, 4, 0,
is an arithmetical progression; and 1,,,, †, †,
1,
&c. is an arithmetical progression; and 1, 2, ∞o
(infinity), -2, -1, &c. is a harmonical progression.
This series is principally important in connection
with the theory of music, in determining the length
of the strings of instruments. See MUSIC.

PROGRESSION, MUSICAL, the regular succession of chords or movement of the parts of a musical composition in harmony, where the key continues unchanged, is called Progression; where a new key is introduced, it is not progression, but Modulation (q. v.). Musical compositions move from note to note either by degrees, when the interval does not exceed, or by skips where it does exceed a whole tone. Motion in music of two parts is of three kinds: oblique, when one part repeats or holds on the same note, while the other moves up or down; direct, when both parts move in the same direction; and contrary, when one moves up, and the other down. In progressing from chord to chord, it is in general desirable to retain every note common to both chords in the same part in which it appeared in the first chord, and to assign every new note to that part in the second chord which is nearest to it. There are certain chords which require to be followed by certain others in order to resolve them (see RESOLUTION); and there are certain progressions which must in ordinary cases be avoided, more particularly consecutive fifths and consecutive octaves,

PROHIBITION-PROJECTILES.

the latter being, however, admissible when employed the same time as it would, without being projected, to strengthen a part. have taken to fall to 4. A greater velocity of projection would make it take a wider flight; but at the end of four seconds, it must still be at some point in the same horizontal line-at g, for example.

PROHIBITION. Prohibitive duty refers to a practice, obsolete in this country, of prohibiting the importation of goods, with the view of encouraging native industry. See FREE TRADE.

PROHIBITION is a writ in England proceeding out of a superior court of law to prohibit or prevent an inferior court from proceeding to hear or dispose of a suit or matter over which it has no jurisdiction. -In Scotch law, the same word means a technical clause in a deed of entail by which the heir of entail is prohibited from selling the estate, or contracting debt, or altering the order of succession under pain of forfeiture, which forfeiture is declared by another Supplemental clause called a resolutive clause.

PROJECTILES, THEORY OF, is the investigation of the path or trajectory, as it is called, of a body which is projected into space in a direction inclined to that of gravitation. A body thus projected is acted upon by two forces, the force of projection, which, if acting alone, would carry the body onwards for ever in the same direction and at the same rate; and the force of gravity, which tends to draw the body downwards towards the earth. The force of projection acts only at the commencement of the body's motion; the force of gravity, on the contrary, continues to act effectively during the whole time of the body's motion, drawing it further and further from its original direction, and causing it to describe a curved path, which, if the body moved in a vacuum, would be accurately a parabola. This is readily seen by considering fig. 1, in which A represents the point from which the body is projected (suppose the embrasure of a fort); AB the direction of projection (horizontal in this instance); Al the distance which would be passed over by the projectile in unit of time if gravity did not act; 1-2, the distance which would

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similarly be described in second unit of time; 2-3, 3-4, &c. the distances corresponding to the third, fourth, &c. units of time-all these distances being necessarily equal, from the impulsive nature of the force of projection; Al', again, represents the distance which the projectile would fall under the action of gravity alone in the first unit of time; 1'--2' the distance due to gravity in the second unit of time; 2-3' the distance due to the third unit, &c., the distances A1, A2, A3', &c., being in the proportion of 1, 4, 9, &c. (see FALLING BODIES); hence, by the well-known principle of the Composition of Forces and Velocities (q. v.), we find at once, by completing the series of parallelograms, that at the end of the first unit of time the body is at c, at the end of the second at b, at the end of the third at e, &c. Now, as the lines l'c, 2'b, 3'e, &c. increase as the numbers 1, 2, 3, &c., and the lines A1, A2, A3', &c. as the numbers 12, 22, 32, it follows that the curve Acbe is a Parabola (q. v.). As, by the second law of motion, each force produces its full effect undisturbed by the other, it follows that the projectile reaches fin

In order to determine exactly the motion of a projectile, and to find its range, greatest altitude, its nature more technically for which some slight and time of flight, it will be necessary to examine knowledge of algebra and trigonometry is requisite. Let the body in this instance be projected obliquely to the direction of gravity, from the point A (fig. 2) in the direction ÄT, and let the velocity of projection v be sufficient, if gravity were not to act, to carry it to T in t units of time, and let the force of gravity, if allowed to act upon it at rest, carry

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it to G in the same time; then, as before, the body, under the action of both forces, will be found at P (which is found by completing a parallelogram of which AT and AG are the sides) at the end of t units of time, having fallen through a distance equal to TP (not at once, but in a constant succession of minute deflections, as indicated in fig. 1) in that time. Let t represent the time of flight, v the velocity due to projection, g the accelerating force of gravity, and let A be the angle of elevation TAB; then AT: vt, TP gt2, TM = vt sin. A; and conse quently PM (or y) = vt sin. A-gt2 (I.), and AM (or x)= vt cos. A. Now, if we find from the last two equations the values of t, and equate these values, we obtain, by an easy algebraic progx2 cess, the equation y=x tan. A 2v2 cos. A and if the height through which the body must fall to acquire a velocity equal to the velocity of projection be called h, then v3=2g.h, h=

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substituting which in the

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(II.),

equation, we obtain y = x tan. A as the equation to the path of a projectile, where x is the horizontal distance, and y the corresponding height above the level of the point of projection. Suppose, now, that we wish to find the time of flight on the horizontal plane, it is evident that at the end of its flight the projectile will be at B, and y will be equal to zero; hence, putting y = 0 in equation I., we obtain t = 2v sin. A The range or distance AB is similarly found by putting y = 0 in equation II., when x is found to be equal to 4h sin. A cos. A, or 2 sin. 2A. The greatest altitude

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