conclusive with direct demonstration. For this there is no foundation; though it must be admitted that direct demonstrations are more pleasing and more elegant. But it is obvious that, if everything which contradicts a proposition be false, the proposition itself must be true. The student of logic must distinguish between that which is only contradictory, and that which is contrary to a proposition. Thus, to the proposition that "all squares are equal," it is contradictory that some squares are not equal," and contrary, that "no squares are equal." The contrary is the most complete contradictory, and affirms that the proposition is true in no one instance. It is not correct to say that, if a proposition be false, its contrary is true; for example, it is false that all squares are equal, and equally false that no squares are equal. But of a proposition and its contradictory one must be true; thus either all squares are equal or some squares are not equal. Hence, whatever disproves a proposition proves something contradictory, and whatever disproves everything contradictory proves the proposition. The Reductio ad Absurdum is, therefore, as conclusive as direct demonstration.

The Reductio ad Absurdum, in Euclid, is wholly unnecessary to all who can see that contra-positive propositions are identically the same. The following forms are contra-positive: Every A is B

Every not-B is not-A.

Thus (Euclid I. 4) two sides equal to two sides understood, proves that equal angles give equal areas: that is, unequal areas give unequal angles. He then has to prove I. 6, which he does by Reductio ad Absurdum. His form is, equal base angles give equal opposite sides: its equivalent contra-positive is, unequal sides give unequal opposite angles. From the unequal sides it may immediately be shown, as in Euclid, that two triangles having two pairs of sides equal, each to each, have unequal areas, and therefore unequal angles. Thus it is shown that the angles opposite unequal sides are unequal: which is but saying that the sides opposite equal angles are equal. Had logic been cultivated concurrently with geometry, the Reductio ad Absurdum would long ago have disappeared, in nearly all the cases in which it is now used. ABUTMENT, in building, is that which receives the end of, and gives support to, anything having a tendency to thrust outwards in a horizontal direction. The piers against which an arch that is less than a semi-circle rests are abutments; while the supports of an arch of any other figure, which springs at right angles to the horizon, are imposts. The piers of the arches of Southwark and Vauxhall bridges are abutments or abutment-piers; whereas those of London, Blackfriars, and Waterloo bridges, and of the old Westminster bridge, are imposts or impost-piers. Nevertheless, the piers at the extremities of a bridge, of whatever form its arch or arches may be, are always termed its abutments; that is, abutments of the bridge itself.

ABUTMENT, in machinery, is a term applied to a fixed point from which resistance or re-action is obtained. In an ordinary steam-engine, for example, each end of the cylinder acts alternately as an abutment. The steam, being unable to expand itself in the direction of the fixed obstacle, that is, the end of the cylinder, expends the whole of its elastic force in the opposite direction, against the movable obstacle or piston. In like manner the breech of a gun forms an abutment for the expansive force of the ignited powder; although in this case, the abutment not being absolutely a fixed point, its recoil occasions some loss of power. Even a rotatory steam-engine, with a continuous circular action, must have an abutment to render the force of the steam effective. Springs, whether used to impel machinery, as in the case of a watch, or to measure or control force, as in the various contrivances noticed under SPRING-BALANCE, must have their abutments or points of resistance; as also must all mechanical combinations in which power is transmitted by means of screws, of which it is sufficient to cite as an example the nut in the fixed head of an ordinary screw-press. In all these cases an analogy may be traced with the use of the term abutment in architecture. With a similar meaning the name is applied in carpentry to a joint in which two pieces of timber meet so that the fibres of one piece run in a direction oblique or perpendicular to the joint, and those of the other parallel with it.

ABUTTALS, from the French abutter, to limit or bound, are the buttings and boundings of lands to the east, west, north, and south, showing by what other lands, highways, hedges, rivers, &c., such lands are in those several directions bounded.

The boundaries and abuttals of corporation and church lands, and of parishes, are usually preserved by an annual procession. ABYSSINIAN CHRISTIANS. The discovery of a body of Christians in so remote a country excited, in no small degree, the attention of Europe in the 15th century, which was again revived by Salt's last mission, in 1810. From the Tareek Negushti,' or Chronicle of the Abyssinian Kings,' combined with the evidence of the ecclesiastical writers, we learn that Christianity was introduced into Abyssinia in the time of Constantine, by Frumentius, or Fremonatos, as the chronicles call him. Frumentius, after residing some years in the country, was raised by Athanasius the patriarch of Alexandria, to the dignity of bishop. He arrived in Abyssinia, perhaps about the year A.D. 330, and probably in the reign of the King Aizanas, whose name still exists in the inscription of Axum. It is, however, not certain to which king of the Abyssinian chronicles we ought to apply the

names of Aizanas and his brother Saizanas, both of which occur in the inscription, and also in a letter of the Emperor Constantine, addressed to them A.D. 356. When the Greek merchant Cosmas visited Abyssinia, A.D. 525, it was completely a Christian country, and well provided both with ministers and churches. Of the Abyssinian churches, which probably belong to the earlier periods of their conversion, or at least are eight or nine hundred years old, there are still some remains. The most remarkable is Abuhasubha, hewn out of the solid rock, which at this place is soft and easily worked. The Portuguese, Alvariz, describes ten such churches as these, of which he has given a plan, and one of them is probably the same as that which Mr. Pearce visited at Jummada Mariam. (Salt, p. 302.) The great church at Axum is comparatively modern, though parts of it, such as the steps, clearly belong to a prior edifice. Mr. Salt describes the wellbuilt remains of a church or monastery near Yahee, which he assigns to the 6th century of the Christian era.

The monastic, and also the solitary life, spread into Abyssinia from the deserts of the Thebais, and when the Portuguese Jesuits entered the country they found it full of such devotees; many of them seemed, however, to be monks only as far as celibacy was concerned, for they cultivated the ground and lived in villages.

With the Christian religion, the Abyssinians received the Holy Scriptures, which they now possess in the ancient Ethiopic version, made, according to Ludolf, from the Greek Septuagint, though nothing is known of the date of this version. As to the New Testament (says Ludolf), no entire copy has been yet brought to Europe. Mr. Bruce brought with him from Abyssinia a complete copy of the Scriptures in the Ethiopic language, and also a set of the Abyssinian Chronicles. The Abyssinians divide the Scriptures, which they have entire, differently from what we do, making four principal parts of the Old Testament, and mixing what we call the Canonical with the Apocryphal books. The New Testament is also divided into four parts, to which they add the Book of Revelation as a supplement. The old written language is of the Semitic stock, and is written from left to right, but the language is not now spoken; there are two languages now in use, the Tigré and the Amharic. For other information respecting the Abyssinian liturgies, and the religious opinions of the Abyssinians, we refer to Ludolf, Book iii. chaps. 4, 5. Ludolf denies the existence of the Book of Enoch, because he had only seen a spurious copy. A knave who got possession of an Ethiopic book, wrote the name of Enoch upon it, and sold it to Peiresc for a considerable sum of money, and this was the book that Ludolf saw. Bruce brought home three copies of the book of Enoch; one of which he gave to the Bodleian Library at Oxford. This book was originally written in Greek, but the original is lost-all but one large fragment. In the epistle of Jude reference is made to the prophecies of Enoch; and Mr. Bruce says, "the quotation is word for word the same in the second chapter of the book." This, however, will not prove the genuineness of the prophecies of Enoch, as Mr. Bruce has very well argued. An English translation of the book of Enoch was published by Dr. Lawrence, Oxford, 8vo. 1822. The High Priest (or sole bishop) of Abyssinia is called Abuna, which signifies Our Father; and as Frumentius, the first bishop, received his appointment from the Patriarch of Alexandria, this dignitary has, probably, always been a foreigner. The king is the head of the Church. Polygamy, though not allowed by the ecclesiastical canon, is common enough in practice; and Mr. Salt mentions an instance of one gentleman who had five wives at once. The king, of course, marries as many as he pleases: the clergy, also, who are not monks, may marry, but only once. A second marriage renders thém unworthy of their sacred office, according to the ancient canons. Circumcision, according to Bruce, is practised in Abyssinia, and baptism of infants and agapæ or love-feasts have been in use ever since the introduction of Christianity. The creed of the Abyssinian Church is what is called the Monophyiste; i. e., admitting the divinity of our Saviour, but acknowledging in him only one nature.

It would appear, from what we know of the Abyssinian Church, that its priests, at present, are not well informed, nor are the people in general well acquainted with the principles of the Christian religion, though they may be Christians in name; yet some of their ceremonies are conducted with great decency, and very much resemble those of the Church of England. When Salt was at Chelicut, Lent was strictly observed for fifty-two days, and no flesh was eaten during this period. though fish and various dishes were always plentiful on the table: the people always fasted till sunset. A feast followed this severe and protracted fast, in which they all seemed anxious to make up for lost time, by over eating and drinking. The Sacrament is also administered in Abyssinia, in a very decorous manner; and red wine made of a grape which is common in some parts of the country, is used on the occasion. Formerly (says Mr. Salt) if a man married more than one wife, he was excluded from participating in this rite, but wealth and power have induced the Church to relax its severity in this respect. Marriage itself in Tigré, appears a mere civil institution: the woman keeps her name, and the parties can separate whenever they agree to do so. In this case the woman has her dowry back, which is not forfeited unless she is manifestly guilty of adultery. The higher classes are subject to no rule, but what may be considered as imposed by the relatives of the male and female. The priests are forbidden to marry after ordination. The Abyssinians bury their dead immediately after

'Diccionario de la Lengua Castellana,' six vols. fol., 1726-1739. The Royal Academy of Spanish History was commenced as a private association at Madrid in 1730, but was taken under the royal protection, and incorporated by Philip V. in 1738. It consists of twentyfour members. The first volume of its Transactions was published in 1796, under the title of 'Memorias de la Real Academia de la Historia.' It has also printed some ancient manuscripts, and given new editions of some historical works. There are also an Academy of History and Geography at Valladolid, and a Literary Academy at Seville, both founded in 1753. The principal Portuguese academy is the Academy of Science, Agriculture, Arts, Commerce, and general Economy, founded by Queen Maria in 1779. It has published several volumes of Transactions in different sets. There is also a Geographical Academy at Lisbon, established in 1799.

Of Austrian Academies, the most ancient is the 'Academia Naturæ Curiosorum,' established at Vienna in 1652. In 1687, during the reign of the Emperor Leopold I., it assumed the name of the Academia Cæsareo-Leopoldina. Its Transactions were at first published in separate treatises, but since 1684 they have appeared in volumes, under the title of Ephemerides et Acta Academiæ Cæsarea Naturæ Curiosorum. A history of this academy was published by Büchner, Halle, in 1756. The Academy of Arts and Sciences of Vienna was founded in 1705. In 1754 was established in the same city an Academy for the cultivation of the Oriental Languages.

The Royal Academy of Science and Belles Lettres of Berlin has long been one of the most eminent among the learned societies of Europe. It was established in 1700, by Frederick I., who appointed the celebrated Leibnitz its first president. The first volume of its Transactions appeared in 1710, under the title of Miscellanea Berolinensia,' and other volumes followed at intervals of three or four years, till the accession of Frederick the Great in 1740, who, in 1744, took it under his special protection, and proceeded to give it a new organisation, with the view of extending its usefulness, and raising it to a higher rank than it had hitherto enjoyed. A history of this academy was published in 1752. In 1754, was established by the Elector of Mainz, the Electoral Academy at Erfurt, for the promotion of the useful sciences. Its Transactions were originally published in Latin, under the title of 'Acta Academiæ Electoralis Moguntinæ Scientiarum Utilium;' but they have of late appeared in German. Of other German Academies the principal are the Academy of Sciences, otherwise called the Royal Society, of Göttingen, established in 1733; the Electoral Academy of Science and Bavarian History at Munich, first established in 1760, but greatly enlarged since the erection of Bavaria into a kingdom, and which has published its Transactions, since 1763, in German, under the title of 'Abhandlungen der Baierischen Akademie;' that of Mannheim, founded in 1755, by the Elector Charles Theodore, and now divided into three classeshistorical, physical, and meteorological; the Transactions of the two former of which have been published, under the title of 'Acta Academiæ Theodoro-Palatina-those of the last, under that of Ephemerides Societatis Meteorologica Palatinæ ;' and the Academy of Suabian History, established at Tübingen, in Würtemberg. The Royal Academy of Sciences, of Stockholm, was originally a private association, founded by Linnæus, and a few of his friends, in 1739, and was not incorporated by the Crown till two years afterwards. Its Transactions appear in quarterly parts, which form an octavo volume a year. The first forty volumes, from 1739 to 1779, are called the Old Transactions;' those which have appeared since, the 'New.' They are written in Swedish, but have also been translated into German. Stockholm also possesses an Academy of the Belles Lettres, founded in 1753; and an institution denominated the Literary Academy of Sweden, founded in 1786. The object of the latter is the cultivation and improvement of the national language. There is an Academy for the investigation of Northern Antiquities, at Upsal, which has published some valuable volumes of Memoirs. The Royal Academy of Sciences of Copenhagen was founded by the Count of Holstein in 1742, and incorporated the following year. Its Transactions appear in Danish; but they have been partly translated into Latin.

The Imperial Academy of St. Petersburg, like most of the valuable institutions of Russia, originated in the bold and contriving mind of Peter the Great. That monarch however did not live to carry into effect the scheme which he had arranged, and which is said to have been suggested to him by his inspection of the academies of France, when in that country in 1717, and to have been matured by consultations with Christian Wolff and Leibnitz. But immediately after his death, in 1725, his successor, Catherine I., proceeded to execute the intentions of her deceased husband; and the Academy was forthwith established, and held its first sitting in December of that year. Some of the most distinguished foreign mathematicians and philosophers of the day were wisely selected by the empress to grace the new foundation, and induced by liberal salaries to accept places in it under the title of professors. Among them were Wolff, Nicolas and Daniel Bernoulli, Bulfinger, &c. In its earlier days this institution underwent various fluctuations in reputation and efficiency, according as it happened to be patronised or neglected by the reigning sovereign; but since the accession, in 1741, of the Empress Elizabeth, who placed it upon a broader and more independent basis, it has generally maintained

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a high character. Its annual revenue is considerable; and one important service which it has thus been enabled to render, has been the exploration of various portions of the Russian empire, by means of the travellers Pallas, Stolberg, Klaproth, and others, whom it has sent out for that purpose. Its Transactions, down to the year 1747 inclusive, forming 14 volumes, are in Latin, and are entitled 'Commentarii Academiæ Scientia Imperialis Petropolitana.' Twenty volumes more, down to 1777, likewise in Latin, are entitled 'Novi Čommentarii.' Since 1777 they take the name of Acta,' and are partly in Latin and partly in French. Of the whole number of mathematical papers which appeared in these Transactions down to the year 1783, in which he died, the celebrated Euler is computed to have written fully one half; and he left behind him about a hundred additional memoirs, which have appeared in the volumes printed since that period. These papers of Euler's contributed, more than any other publications of the time, to the simplification and improvement of the modern analysis. The Imperial Academy possesses a library of some extent, which contains a considerable number of oriental manuscripts, as well as valuable collections of medals and of specimens of natural history. In 1783, an institution, on the model of the Académie Française, having for its object the improvement of the Russian language, was founded at St. Petersburg, and was soon after united with the Imperial Academy. Among the other European academies, may be mentioned the Medical Academy of Geneva, founded in 1715; the Académie des Sciences et des Belles Lettres of Brussels, which has published its Transactions, under the title of 'Mémoires,' since the year 1777; and the institution of the same name at Flushing, whose Transactions have also appeared. In the British dominions there are no associations for the cultivation of science or learning, which have this name, except the Royal Irish Academy, founded in 1782, and which has published its Transactions since 1787. In the United States of North America, as in England, such institutions are, for the most part, called Societies, but a few are styled Academies, such as those at Boston and Philadelphia, and have published their Transactions.

Academy is also the name usually given, both in this country and on the Continent, to an institution established for the cultivation and promotion of the fine arts, that is, of painting, sculpture, architecture, and music. Such institutions commonly partake both of the character of academies, in the sense already explained, and of schools or colleges, consisting, on the one hand, of an association of amateurs and distinguished proficients, professing to have in view the diffusion of a taste for the arts among the public generally, by publications, exhibitions, or any other means which may be made available for that end; and, on the other, of an establishment of teachers or professors, for the instruction of youth in the practice of some one or more of the branches in question. The latter object is effected by lectures, by prescribed tasks, and by the distribution of prizes and honours. Societies of painters, for the promotion and protection of their art, are of very ancient date. The Royal Academy of London originated in an association of painters, who obtained a charter, in 1765, under the title of the Incorporated Society of Artists of Great Britain. This society, however, was soon after broken up by disputes among its members; and in 1768, the Royal Academy of Arts was incorporated in its stead. It consists of forty artists bearing the title of academicians, of twenty associates, of two academician engravers, of five associate engravers, and of three or four individuals of distinction, under the name of honorary members, but who also hold certain nominal offices. From the academicians are selected the professors of painting, of sculpture, of architecture, and of perspective; and there is also a professor of anatomy, who is commonly a member of the medical profession. Nine of the academicians are likewise appointed annually to officiate in setting the models, and otherwise superintending the progress of the students. The sovereign is the patron of this institution; but its funds are, we believe, entirely derived from the money paid by the public for admission to the exhibition, which takes place every year, in the months of May, June, and July. A branch of the English Royal Academy was established some years ago at Rome. The Edinburgh Royal Academy of Painting was founded in 1754. A similar institution has also been established in Dublin, under the title of the Royal Hibernian Academy. An Academy of Ancient Music was established in London so early as the year 1710; but a disagreement among its members occasioned its dissolution after it had existed above twenty years. Some time after this the Royal Academy of Music was instituted, with Handel at its head, and for ten years, during which the operas of that great composer were performed under its superintendence in the Haymarket Theatre, enjoyed splendid success. But discord here also came at length, to divide and disperse the professors of harmony; and in the year 1729, the institution was broken up. A new Royal Academy of Music, which holds its meetings in Hanoversquare, was established in 1822. The French Opera, it may be added, is styled the Académie de Musique.

ACAʼNTHUS (in Architecture). The name by which the leaf used in the enrichment of the Corinthian capital is known. It is thus called because of its general resemblance to the leaves of a species of the acanthus plant; or rather because of a pretty traditional story which the Roman author Vitruvius tells of the fancied origin of the Corinthian capital, in which the leaves are said to be imitated from those of the acanthus. The same leaf, however, is commonly used in

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architectural and sculptural enrichments generally; in the enrichment of modillions, of mouldings, and of vases, as well as of foliated capitals; and we gather from Virgil, that the acanthus was by the ancients also employed as an ornament in embroidery. In the first book of the Eneid,' verse 649, and again at 711, a veil or vest is said to be interwoven or embroidered with the crocus-coloured or saffron acanthus. Pliny the elder, in his 'Natural History,' describes the acanthus in such a manner that it can only be recognised in the brank-ursine; and his nephew, in speaking of the successful cultivation of the same plant as an ornament to his garden, leaves little doubt that the brank-ursine is identical with the common architectural and sculptural acanthus. It is stated, however, that the brank-ursine (Acanthus mollis) does not grow in Greece, and it has been suggested that the plant from which the Greek architectural ornament was taken was the Acanthus spinosa, which grows there, and is still called the ǎkavea.

This ornament, in the ancient Greek and Roman models, is very characteristic of the styles of architectural enrichment of those nations; in the Roman it is full, and somewhat luxuriant, and in the Greek more restrained, but simple and graceful.

ACCELERATED MOTION, ACCELERATING FORCE, ACCELERATION. When the velocity of a moving body is continually increased, so that the lengths described in successive equal portions of time are greater and greater, the motion is said to be accelerated, which is the same thing as saying that the velocity continually increases. [VELOCITY.] We see instances of this in the fall of a stone to the earth, in the motion of a comet or planet as it approaches the sun, and also in the ebb of the tide. As it is certain that matter, if left to itself, would neither accelerate nor retard any motion impressed upon it, we must look for the cause of acceleration in something external to matter. This cause is called the accelerating force. [See INERTIA; FORCE; CAUSE to the remarks in the last of which articles we particularly refer the reader, both now and whenever the word cause is mentioned.] At present, the only accelerating force which we shall consider is the action of the earth, producing what is called weight, when not allowed to produce motion.

It is observed, that when a body falls to the ground from a height above it, the motion is uniformly accelerated; that is, whatever velocity it moves with at the end of the first second, it has half as much again at the end of a second and a half; twice as much at the end of two seconds; and so on. At least this is so nearly true, that any small departure from it may be attributed entirely to the resistance of the air, which we know from experience must produce some such effect. And this is the same with every body, whatever may be the substance of which it is composed, as is proved by the well-known experiment of the guinea and the feather, which fall to the bottom of an exhausted receiver in the same time. The velocity thus acquired in one second is called the measure of the accelerating force. On the earth it is about 32 feet 2 inches per second. If we could take the same body to the surface of another planet, and if we found that it there acquired 40 feet of velocity in the first second, we should say that the accelerating force of the earth was to that of the planet in the proportion of 32 to 40. By saying that the velocity is 324 feet at the end of the first second, we do not mean that the body falls through 32 feet in that second, but only that if the cause of acceleration were suddenly to cease at the end of one second, the body would continue moving at that rate. In truth, it falls through only half that length, or 16, in the first second. It may be proved mathematically, that if a body, setting out from a state of rest, have its velocity uniformly accelerated, it will, at the end of any time, have gone only half the length which it would have gone through had it moved, from the beginning of the time, with the velocity which it has acquired at the end of it. Thus, if a body have been falling from a state of rest during ten seconds (the resistance of the air having been removed), it will then have a velocity of 32 x 10 or 321 feet per second. Had it moved through the whole ten seconds with this velocity, it would have passed over 321 x 10 or 3216 feet. It really has described only the half, or 1608 feet. We may give an idea of the way in which this proposition is established, as follows:-The area of a rectangle [RECTANGLE]-that is, the number of square feet it contains, is found by multiplying together the numbers of linear feet in the sides. Thus, if A B be 4 feet, and A 0 5 feet, the number of square feet in the area is 4 x 5, or 20. Again, the number of feet described by a body moving with a uniform velocity, for a certain

number of seconds, is found by multiplying the number of seconds by the number of feet per second or the velocity. If, then, A B contain as many feet as there are seconds, and a c as many feet as the body moves through per second; so many feet as the body describes in its motion, so many square feet will there be in ABDC. That is, if we let A B represent the time of motion, and a c the velocity, the area A BDC will represent the length described in the time AB, with the velocity a C.

Not that ABDC is the length described, or A B the time of describing it; but A B contains a foot for every second of the time, and ABDC contains a square foot for every foot of length described. Similarly, if at the end of the time just considered, the body suddenly receive an accession of velocity D F, making its whole velocity BF per second; and if with this increased velocity it move for a time which contains as many seconds as BE contains feet, the length described in this second portion of time will contain as many feet as BEGF contains square feet; and the whole length described in both portions of time will be represented by the sum of the areas ABDO and BEGF. And similarly for another accession of velocity GI, and an additional time represented by EH. Now, let a body move for the time represented by AM; at the beginning of this time let it be at rest; and by the end let it have acquired the velocity MN: so that had it moved from the beginning with this velocity, it would have described the length represented by AMN P. Instead of supposing the velocity to be perpetually increasing, let us divide the time A M into a number of equal parts-say four, A B, BE, EH, HM-and let one-fourth of the velocity be communicated at

the beginning of each of these times, so that the body sets off from a, with the velocity a C, which continues through the time represented by A B, and causes it to describe the length represented by A B D C. We know from geometry that BD, EG, and H K, are respectively one-fourth, one-half, and three-quarters of M N, which is also evident to the eye, and may be further proved by drawing the figure correctly, which we recommend to such of our readers as do not understand geometry. Hence, GE or B F is the velocity with which the body starts at the end of the time AB; EI at the end of A E; and HQ at the end of A H. Consequently, the whole length described is a foot for every square foot contained in ABDC, EBFG, EIKH, and HQNM, put together. But this is not a uniformly accelerated velocity, for the body first moves through the time A B, with the velocity A C, and then suddenly receives the accession of velocity D F. But if, instead of dividing AM into four parts, we had divided it into four thousand parts, and supposed the body to receive one four-thousandth part of the velocity M N at the beginning of each of the parts of time, we should be so much nearer the idea of a uniformly accelerated velocity as this, that instead of moving through one-fourth of its time without acquiring more velocity, the body would only have moved one four-thousandth part of the time unaccelerated. And the figure is the same with the exception of there being more rectangles on A M, and of less width. Still nearer should we be to the idea of a perfectly uniform acceleration if we divided A м into four million of parts, and so on. Here we observe1, that the triangle A N M is the half of A P NM; 2, that the sum of the little rectangles ACDB, BFGE, &c., is always greater than the triangle AN M, by the sum of the little triangles A CD, DFG, &c.; 3, that the sum of the last-named little triangles is only the half of the last rectangle HQ N M, as is evident from the inspection of the dotted part of the figure. But by dividing A M into a sufficient number of parts, we can make the last rectangle HQ N M as small as we please, consequently we can make the sum of the little triangles as small as we please; that is, we can make the sum of the rectangles ACDB, &c., as near as we please to the triangle A N M. But the more parts we divide A M into, the more nearly is the motion of the body uniformly accelerated; that is, the more nearly the motion is uniformly accelerated, the more nearly is ANM the representation of the space described. Hence we must infer (and there are in mathematics accurate methods of demonstrating it), that if the acceleration were really uniform, ANM would really have a square foot for every foot of length described by the body; that is,

since A N M is half of APN M, and the latter contains a square foot for every foot of length which would have been described if M N had been the velocity from the beginning, we must infer that the length described by a uniformly accelerated motion from a state of rest, is half that which would have been described, if the body had had its last velocity from the beginning. If the body begin with some velocity, instead of being at rest, the space which it would have described from that velocity must be added to that which, by the last rule, it describes by the acceleration. Suppose that it sets out with a velocity of 10 feet per second, and moves for 3 seconds uniformly accelerated in such a manner as to gain 6 feet of velocity per second. Hence it will gain 18 feet of velocity, which, had it had at the beginning, would have moved it through 18 x 3 or 54 feet of length, and the half of this is 27 feet. This is what it would have described had it had no velocity at the beginning; but it has 10 feet of velocity per second, which, in 3 seconds, would move it through 30 feet. Hence 30 feet and 27 feet, or 57 feet, is the length really moved through in the 3 seconds.

Similarly we can calculate the effects of a uniform retardation of velocity. This we can imagine to take place in the following way. While the body moves uniformly from left to right of the paper, let the paper itself move with a uniformly accelerated velocity from right to left of the table. Let the body at the beginning of the motion be at the left edge of the paper, and let that edge of the paper be placed on the middle line of the table. Let the body begin to move on the

A

paper uniformly 10 inches per second, and let the paper, which at the beginning is at rest, be uniformly accelerated towards the left, so as to acquire 2 inches of velocity in every second. At the end of 3 seconds, the body will be at B, 30 inches from A, but the paper itself will then have acquired the velocity of 6 inches per second, and will have moved through the half of 18 inches or 9 inches; that is, a c will be 9 inches. Hence the distance of the body from the middle line will be c B, or 21 inches. Relatively to the paper, the velocity of the body is uniform, but relatively to the table, it has a uniformly retarded velocity. At the end of the fourth second, it will have advanced 40 inches on the paper, and the paper itself will have receded 16 inches, giving 24 inches for C B. At the end of the fifth second, a B will be 50 inches, ▲ c 25 inches, and C B 25 inches. At the end of the sixth second, A B will be 60 inches, a c 36 inches, and B c 24 inches, so that the body, with respect to the table, stops in the sixth second, and then begins to move back again. We can easily find when this takes place, for, since the velocity on the paper is 10 inches per second, and that of the paper gains 2 inches in every second, at the end of the fifth second the body will cease to move forward on the table. At the end of 10 seconds it will have returned to the middle line again, and afterwards will begin to move away from the middle line towards the left. At the end of the twelfth second, it will have advanced 120 inches on the paper, and the paper will have receded 144 inches, so that the body will be 24 inches on the left of the middle line.

The general algebraical formulæ which represent these results are as follow. Let a be the velocity with which the body begins to move, the number of seconds elapsed from the beginning of the motion, g the velocity acquired or lost during each second. Then the space described in a uniformly accelerated motion from rest is gt; when the initial velocity is a, the space described in an accelerated motion is at + gt2, and in a retarded motion the body will have moved through at-gt in the direction of its initial velocity if at be greater than gt2, or will have come back and passed its first position on the other side by gtat, if at be less than gt. In the last case it continues to move in the direction of its initial velocity for seconds and proceeds in that direction through the space

a2

For further explanation as to velocities which are accelerated or retarded, but not uniformly, see VELOCITY. ACCELERATION AND RETARDATION OF TIDES are certain deviations of the times of consecutive high-water at any place from those which would be observed if the tides occurred after the lapse of a mean interval. The interval between the culmination of the moon, or the occurrence of her principal phases, and the nearest time of highwater, is also called the retardation of the tide.

The tides are caused by the attractions exercised by both the sun and moon on the waters of the earth; but the effect produced by the moon exceeds that which is produced by the sun, and the difference is such, that the phenomena of the tides depend principally on the former. The mean interval between two consecutive returns of the moon, above and below the pole, to the meridian of any place, is 24h.

50m. 28-328.; and since, neglecting all causes of irregularity, two lunar high-tides occur in that time, the mean interval between two consecutive lunar tides should be 12h. 25m. 14-16s.; while the mean interval between two consecutive solar tides should be 12h. Hence, if at the time of a conjunction or opposition of the sun and moon, the high tides which are produced by the actions of the luminaries separately were coincident, the next lunar tide would be retarded with respect to the next solar tide, by 25m. 14:16s., that is, by the excess of half a lunar day above half a solar day. These retardations continuing daily, the lunar high-water would coincide, at the time of quadrature, with the solar low-water, and thus produce the neap or diminished tides; after which, the like retardation continuing, the solar and lunar high-waters would again become coincident at the times of syzygy, and so on. The observed daily retardation of the lunar high-tides varies however according to the position of the moon with respect to the sun, to the moon's declination, and to the distance of that luminary from the earth. At Brest, when the sun and moon are in conjunction or in opposition, at the summer or winter solstice, the retardation is equal to 40m. 51.69s., and at the time of the equinoxes 37m. 38 15s. Again, when the sun and moon are in quadrature at either solstice, the retardation is 1h. 7m. 27-49s., and at the time of the equinoxes 1h. 23m. 16:34s.

If the earth were a solid of revolution, and were covered by the sea, the high tides produced by the sun and moon separately would, at any place, occur at the instants when those celestial bodies are on the meridian of the place; but such is not the fact in the actual condition of the earth; and local circumstances produce, at different ports, great differences in the intervals between the culmination of the sun or moon at the time of high-water, even on the days when the luminaries are in conjunction or opposition. The interval between the instant that the sun passes the meridian of a place and the occurrence of the solar hightide, is found to be greater than the interval between the transit of the moon and the occurrence of the lunar high-tide; and this acceleration, as it is called, of the lunar tide, is with much probability ascribed by Dr. Young to a difference in the resistances experienced by the waters on account of the different velocities which are communicated to them by the separate actions of the sun and moon.

It should be observed however that at Ipswich the time of highwater is nearly coincident with the time at which the moon passes the meridian of that port; and both at Glasgow and Greenock, the hightide generally precedes the transit (Mr. Mackie's Report,' at the seventh meeting of the British Association); but such phenomena are of rare occurrence, and at almost every place the high-tide occurs some time after the moon has culminated.

From a series of observations continued during sixteen years, at Brest, La Place, taking the excesses of the height of the evening tide above that of the morning for the day of syzygy, for the day preceding it, and for four days following it, has ascertained that at the syzygies which occur about the vernal and autumnal equinox the highest tides at that port take place 1:48013 days after the instant of the conjunction or opposition; and at the syzygies which occur about the summer and winter solstices they take place 1.54684 days after conjunction or oppo-, sition. Again, taking the excesses of the height of the morning tide above that of the evening for six days, as above, he ascertained that at the quadratures which occur about the equinoxes the highest tides take place 1-50964 days after the instant of quadrature, and at the solsticial quadratures 1:51269 days after such instant.

Mr. Airy (Tides and Waves,' Encycl. Metrop.) observes that these retardations cannot be accounted for by delays in the transmission of the tide-waves, since no cause for such delay can be imagined to exist in the Southern Ocean, where the waves are formed; and it is known that the time of high-water at Brest is only fifteen hours later than at the Cape of Good Hope: he conceives, therefore, that the retardation must be ascribed to friction. By taking the means of the daily retardations of the morning and evening tides at Brest, La Place found that at the equinoctial syzygies such mean retardation was equal to 37m. 38s.; at the solsticial syzygies, 40m. 528; at the equinoctial quadratures, 83m. 16s.; and at the solsticial quadratures, 67m. 278.

From a series of observed heights of the tides, Sir John Lubbock has determined that the highest tides occur at London 2013 days after 1-68 days. (Phil. Trans.' 1831, 1835.) Also, from the observed the conjunction or opposition of the sun and moon; and at Liverpool, heights, Dr. Whewell has found that the highest tides occur at Bristol 1-667 days after the syzygics; and at Dundee 1639 days. (Phil. Trans.' 1838, 1839.) On the supposition that the mean retardation of the tide at London at the times of syzygy is 2:459 days, Mr. Airy has computed the moon's true hour-angle west of the meridian, at the time of high-water, for every half hour's difference in the time of her transit; and from the table it appears, that when the moon passes the meridian of London at noon (that is, at the time of conjunction), that angle, in time, is 1h. 57m. 17s.; when it passes at 3 P.M., the angle is 1h. 10m. 45s.; at 6 P.M., or at quadrature, Oh. 41m. 17s.; and at 9 P.M., 1h. 55m. 29s. The hour angle is the greatest at 104 P.M., when it is equal to 2h. 9m. 55s.; and at 111 P.M., or nearly at the time of opposition, it is 2h. 3m, 9s.: all these times are found to agree very nearly with the results of observation. From such results it is ascertained that, on the days following the times of syzygy and quadrature, the intervals between the time of the moon's