LOG-BOARD; two boards shutting together like a book, and divided into several columns, containing the hours of the day and night, the direction of the winds, and the course of the ship, with all the material occurrences that happen during the 24 hours, or from noon to noon, together with the latitude by observation. From this table, the officers work the ship's way, and compile their journals. The whole, being written with chalk, is rubbed out every day at noon.

LOG-BOOK; a book into which the contents of the log-board is daily transcribed at noon, together with every circumstance, deserving notice, that may happen to the ship, or within her cognizance, either at sea, or in a harbor, &c. The intermediate divisions or watches of a log-book, containing four hours each, are usually signed by the commanding officer thereof, in ships of war or East Indiamen. LOG-LINE; the line which is fastened to the log (q. v.).

LOGAN, James; born at Lurgan, in Ireland, Oct. 20, 1674, of Scottish parents. At the age of 13 years, having learned Latin, Greek, and some Hebrew, he was put apprentice to a linen-draper in Dublin; but, the country being involved in much confusion by the war of the revolution (1688), he returned to his parents, at Bristol, in England, where he devoted all the time which he could command to the improvement of his mind. In his 16th year, having happily met with a small book on mathematics, he made himself master of it without any manner of instruction. Having, also, further improved himself in the Greek and Hebrew, he acquired the French, Italian and Spanish languages. He was engaged in a trade between Dublin and Bristol, when Willian Penn made proposals to him to accompany him to Pennsylvania, as his secretary, which he accepted, and landed, with the proprietor, in Philadelphia, in the beginning of December, 1699. In less than two years, William Penn returned to England, and left his secretary invested with many important offices, which he discharged with fidelity and judgment. He filled the offices of provincial secretary, commissioner of property, chief justice, and, upon the demise of governor Gordon, governed the province for two years as president of the council. He had, for a long time, earnestly solicited from the proprietary family a release from the fatiguing care of their business; but, even after this release, he was constantly consulted and appealed to in difficulty. And the quiet and

good government of the province, for a number of years, was due to his prudence and experience. He lived about 20 years at Stenton, enjoying literary leisure, corresponding with eminent men in various countries, and engaged in collecting that library which he bequeathed to the public. He was also the author of several learned works. His Experimenta Meletemata de Plantarum Generatione entitles its author to be ranked among the earliest improvers of botany. It was written in 1739. He corresponded with the great Swedish botanist. The aborigines, of whose relations with the government of Pennsylvania he had the chief management, paid an affecting tribute to his worth, when, in his` old age, they entreated his attendance, on their behalf, at a treaty held in Philadelphia, 1742, where they publicly testified by their chief, Cannassatego, their satisfaction for his services, calling him a wise and good man, and expressing their hope that, when his soul ascended to God, one just like him might be found for the good of the province, and their benefit. He was a man of uncommon natural and acquired abilities, of great wisdom, moderation and prudence; well acquainted with the world and mankind, as well as with books; of unblemished morals, and inflexible integrity. He died at Stenton, near Philadelphia, Oct. 31, 1751, having just completed his 77th year.

LOGAN, George, doctor, son of William and grandson of James Logan, was born at Stenton, near Philadelphia, Sept. 9, 1753. He was sent to England for his education when very young, and, on his return, served an apprenticeship with John Reynolds, merchant of Philadelphia. He had early a great desire to study medicine, which he undertook after he had attained the years of manhood. After spending three years at the medical school of Edinburgh, he travelled through France, Italy, Germany and Holland, and returned to his own country in 1779. Here he applied himself to agriculture with success, and was one of the first who made experiments with gypsum as a manure. was, in a few years, elected to the legislature, and served in several sessions. His character, as a representative, was marked by strict integrity, and an adherence to what he believed to be the public benefit. The public mind being much agitated, on account of the French revolution, and the violent ascendency of party spirit, and the nation standing on the brink of a war with France, he embarked for that country in June, 1798, in order to try to prevent such

He

an issue. For this step he was denounced as a parricide to his country, and loaded with the utmost abuse. But he succeeded in his intentions. Upon his arrival at Hamburg, he found that all entrance into the French territory was interdicted to American citizens; yet, by the friendly interference of Lafayette in his favor, he obtained a passport from the French chargé d'affaires, and proceeded to Paris, where he heard that Elbridge Gerry (q. v.), the last of our commissioners, had left that city for the U. States, that an embargo had been laid on all our shipping in French ports, that several hundreds of our seamen were confined in French prisons, and that all negotiation was at an end. Finding that he could not get introduced to the chief director, Merlin, then the highest functionary in France, by means of Talleyrand,-who, nevertheless, received doctor Logan himself with courtier-like complacency, and used every art to sound what was his message or intentions, in vain,-doctor Logan introduced himself to M. Schimmelpennick, the Batavian minister, who presented him to Merlin, by whom he was very cordially received. In the visits which he made him, he succeeded in convincing the director of the impolicy of the measures pursued by France towards this country, and, finally, obtained a decree, raising the embargo, and liberating our seamen, and giving, through the American consul-general, assurances to our government that they desired to renew their former amity and friendship with the U. States. He returned to the U. States in 1798, and published, in the Aurora of Jan. 12 (date of his Letter to the Public), 1799, a justification of himself, most decidedly repelling the charge of having been sent to France by a faction, &c. Directly after his return, the law familiarly called Logan's law, was enacted by congress, and a negotiation was entered upon which terminated in a peace with France. Mr. Logan sat in the seventh and eighth congresses, from December, 1801, to March, 1807, as senator from Pennsylvania, and might have continued longer in that station, but he declined a reelection. In 1810, he visited England, with the same philanthropic desire of preserving peace between the two countries. Here, though he failed in effecting the good which he had so much at heart, yet his reception, by men of the highest respectability of both parties, was highly flattering. He was exceedingly grieved at the war which followed. His health gradually declined for some years, and he died April 9, 1821.

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LOGARITHM (from the Greek λoyos, proportion, and àpiðμós, number). “The logarithms of numbers are the exponents of the different powers to which a constant number must be raised, in order to be equal to those numbers; the principles, therefore, which apply to exponents in general, apply to logarithms." To constitute a logarithm, it is necessary that the exponent should refer to a system or series. These exponents, therefore, constitute a series of numbers in arithmetical proportion, corresponding to as many others in geometrical proportion. Take, for instance, the series 10=10; 102=100; 103 = 1000; 10: 10,000 then we have the logarithm of 10=1; logarithm, 100=2; logarithm, 1000=3; logarithm, 10,000=4, &c. Perhaps the definition of a logarithm may be more scientifically expressed thus: Logarithm is a mathematical term for a number by which the magnitude of a certain numerical ratio is expressed in reference to a fundamental ratio. The value of a ratio becomes known to us by the comparison of two numbers, and is expressed by a number called the quotient of the ratio; for instance, 12:4 is expressed by 3, or 18:9 by 2; 3 and 2 being called the quotients of the two proportions, 12:4 and 18:9. If we now imagine a series of proportions, which have all the same value or quotient, as, for instance, 1 to 3, 3 to 9, 9 to 27, 27 to 81, &c. (in which 9 and 3, 27 and 9, 81 and 27, are in the same ratio as 3 and 1), and if we at the same time adopt the ratio 3 to 1, as the fundamental ratio (or the unit of these ratios), then 9 to 1 is the double of this ratio, 27 to 1 the triple, 81 to 1 the quadruple, and so on. The numbers 1, 2, 3, 4, which indicate the value of such ratios, in respect to the fundamental ratio, are called logarithms. If, therefore, in this case, 1 is the logarithm of 3, 2 must be the logarithm of 9, 3 of 27, 4 of 81, &c. If we adopt, however, the ratio of 4:1 as the fundamental one, and hence 1 as the logarithm of 4, then 2 would be the logarithm of 16, 3 of 64, &c. The logarithms of the numbers which lie between, must be fractions, and are to be calculated and put in a table. A table of logarithms, made according to an assumed basis or fundamental ratio, of all numbers to a certain limit, is called a logarithmic system. The most common, at present, is that of Briggs, in which the fundamental basis is 10 to 1; hence I is the logarithm of 10, 2 of 100, 3 of 1000, 4 of 10,000, &c. It is evident that all logarithms of numbers between 1 and 10, must be more

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than 0, yet less than 1, i. e. a fraction; thus the logarithm of 6 is 0.7781513. In the same way, the logarithms of the numbers between 10 and 100 must be more than 1, but less than 2, &c.; thus the logarithm of 95 is = 1.9777236. All logarithms of the numbers between 0, 10, 100, 1000, &c., are arranged in tables, the use of which, particularly in calculations with large numbers, is very great. The process is simple and easy. If there are numbers to be multiplied, we only have to add the logarithms; if the numbers are to be divided, the logarithms are merely to be subtracted; if numbers are to be raised to powers, their logarithms are multiplied; if roots are to be extracted, the logarithms are merely to be divided by the exponent of the root. In a table of logarithms, the integer figure is called the index or characteristic. The decimals are called, by the Germans and Italians, the mantissa. In general, the logarithms of the system in which 1 indicates 10, are called common or Briggs's logarithms. The properties of logarithms, and some of their uses, were taken notice of by Stiefel or Stifėlius, a German clergyman, who wrote as early as 1530; but the use of them in trigonometry was discovered by John Napier, a Scotch baron, and made known by him in a work published at Edinburgh, in 1614. Logarithmic tables are of great value, not only to mathematicians, but to all who have to make calculations with large numbers. The best logarithmical tables are those of Vega (q. v.) and of Callet. The former are calculated with 10 decimals.* Logarithms are of incalculable importance in trigonometry and in astronomy. Vega's edition of Vlacq's tables contains a trigonometrical table of the common logarithms of the radius or log. sin. tot. 10.0000000, which gives the logarithms of sines, arcs, co-sines, tangents and co-tangents for each second of the two first and two last degrees, and for each ten seconds of the rest of the quadrant. Under Napier's direction, B. Ürsinius first gave the logarithm of the sines of the angles from 10 to 10 seconds, the logarithm of the tangents, which are the differences of the logarithms of each sine and co-sine, together with the natural sine for a radius of 100,000,000 parts. Kepler turned his attention particularly upon the invention of Napier, and gave a new theory and

*Logarithmic and Trigonometric Tables have lately been published by F. R. Hassler (New York, 1830); and Mathematical Tables, comprising Logarithms of Numbers, &c. (Boston, 1830). The English tables are too numerous to mention.

new tables. Briggs was also conspicuous in the construction of tables. Mercator shows a new way for calculating the logarithms easily and accurately. Newton, Leibnitz, Halley, Euler, L'Huillier, and others, perfected the system much, by applying to it the binomial theorem and differential calculus. The names of Vlacq, Sherwin, Gardiner, Hutton, Taylor, Callet, and others, deserve to be honorably mentioned. The edition of Vlacq, within a few years, by Vega, is particularly valuable. During the French revolution, when all measures were founded on the decimal division, new tables of the trigonometrical lines and their logarithms became necessary. The director of the bureau du catastre, M. Prony, was ordered, by government, to have tables calculated, which were to be not only extremely accurate, but to exceed all other tables in magnitude. This colossal work, for which the first mathematicians supplied the formulas and the methods for using the differences in the calculations, was executed, but the depreciation of the paper money prevented its publication. The tables would have occupied 1200 folio_pages. (Notices sur les grandes Tables Logarithmiques et Trigonométriques, calculés au Bureau du Catastre à Paris, an IX.)

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LOGAU, Frederic, baron of; an epigrammatist, born in Silesia, 1604, and died in 1655. He early showed poetical talents, but, at a later period, his avocations appear to have prevented him from attempting any large poems, and his poetical productions were confined to short pieces and epigrams. He published a selection of 200 epigrams, which were so well received, as to induce him (probably in 1654) to publish a new collection of 3000. contemporary of Opitz, he followed in the steps of his great predecessor, and often expresses himself with as much vigor. Many of his epigrams are original and happy, and are the more striking as this department has been little cultivated by German writers. Logau is particularly original in the gnome, and truly poetical in a form which is now become foreign to poetry. Ramler and Lessing, who edited a collection of his epigrams in 1759, revived his reputation. After Lessing's 'death, Ramler republished the collection, in 1791. Select poems of Logau are contained in W. Müller's Bibliothek deutscher Dichter des 17 Jahrh. (Library of the German Poets of the seventeenth Century, volume vi, Leipsic, 1824).

LOGGE DI RAFFAELLO; part of the

Vatican, and one of those beautiful scenes to be found nowhere but in Rome. Leo X had these logge or arcades built under the direction of the immortal Raphael. There are three stories which enclose a court called il Cortile di S. Dama80. The middle story is the most celebrated. It is formed by thirteen arches, and the vault of each contains four paintings in fresco, representing scenes from the Old Testament, and executed by Giulio Romano, Pierin dal Vaga, Pellegrino da Modena, Polidoro, and Maturino da Caravaggio, and others, after cartoons prepared by the great Raphael himself. The number of these exquisite pictures is fifty-two; the arches and pilasters are adorned with grotesque paintings, executed by Giovanni da Udine, so famous in this branch, also under the direction of Raphael.

LOGIC (Moyen, i. e. inornμn); the science of the laws of thought, and the correct connexion of ideas. It is not certain, however, whether the name was derived originally from thought or from language, because both may be designated by Xoyos, i. e. reason and word. In German, this science has also been called Denk-Lehre, or Verstandes-Lehre (rule of thinking, or rule of the understanding), because logic strives to represent, in a scientific way, those laws which the understanding is bound to follow in thinking, and without the observance of which, no correct conclusions are possible. Logic is valuable, not only as affording rules for the practical use of the understanding, but also as a science preparatory to, all other sciences, particularly mental philosophy, as it affords the rules for giving scientific connexion to all knowledge, the laws of thinking determining the character of scientific arrangement. But, inasmuch as the laws of logic can only determine the form of our knowledge, but can by no means teach us how to obtain the materials of knowledge, and gain a clear insight into things (which is the business of mental philosophy, properly so called), in so far logic has been, of late, separated from intellectual philosophy. But if, as is not unfrequently done, all sciences are divided into the historical (those which proceed from experience, as history, natural philosophy, medicine, &c.) and the philosophical (the subjects of which do not fall within the domain of experience), logic is a philosophical science, because the laws of the connexion of thoughts and ideas are founded in reason itself, and not in experience, and the sub

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jects of logic are, therefore, capable of a demonstrative certainty beyond those of any other philosophical science. Logic has not unfrequently been overvalued, particularly by the ancient philosophers. It should be always kept in mind, that the most systematic order, alone, does not render assertions truth. The province of logic has been enlarged or restricted by different philosophers. Among the ancients, logic was made to include the deeper philosophical investigation of the general characteristics of truth, or the essential conditions of the truth of our knowledge, which some modern philosophers have referred to metaphysics. Logic may be divided into the pure and the applied; the former treats of the general laws and operations of thought (conceiving, judging, concluding), and their products (notion, judgment, conclusion). Applied logic treats of thought under particular and special relations, which are to be taken into consideration in applying the general laws of thought, viz. the connexions of thought with other operations of the mind, and the impediments and limitations which it thereby experiences, as, also, the means of counteracting them. For the first scientific treatment of logic, we are to look to the Greeks. Zeno of Elea is called the father of logic and dialectics; but it was then treated with particular reference to the art of disputation, and soon degenerated into the minister of sophistry. The sophists and the Megarean school (founded by Euclid of Megara) greatly developed this art. The latter, therefore, became known under the name of the heuristic or dialectic school, and is famous for the invention of several sophisms. The first attempt to represent the forms of thinking, in abstracto, on a wide scale, and in a purely scientific manner, was made by Aristotle. His logical writings were called, by later ages, organon, and for almost two thousand years after him maintained authority in the schools of the philosophers. His investigations were directed, at the same time, to the criterion of truth, in which path Epicurus, Zeno, the founder of the stoic school, Chrysippus and others followed him. Logic, or dialectics, enjoyed great esteem in later times, particularly in the middle ages, so that it was considered almost as the spring of all science, and was taught as a liberal art from the eighth century. The triumph of logic was the scholastic philosophy (which was but a new form of the ancient sophistry); and theology, particularly, became filled with verbal

subtilties. Raymundus Lullus strove to give logic another form. The scholastics were attacked by Campanella, Gassendi, Peter Ramus (Pierre de la Ramée), Bacon and others with well-founded objections. Descartes and Malebranche again confounded logic and metaphysics. Locke, Leibnitz and Wolf, Tchirnhausen, Thomasius, Crusius, Ploucquet, Lambert (in his New Organon), Reimarus and others, have rendered great service to modern logic. Kant, Fichte, Schelling, Hegel, have maintained very various opinions on the subject. Whateley's Treatise on Logic, first published in the Encyclopædia Metropolitana, and since in a separate volume, is one of the best treatises, in English, on the subject.

LOGIER, John Bernard, descended from a family of French refugees, was born in 1780, at Kaiserslautern in the Palatinate, where his grandfather was organist. His father was appointed, in 1796, violinist in the chapel of the elector of Hesse-Cassel. When the subject of this article was ten years old, he played the flute, then his favorite instrument, at a public concert. His parents having died, his guardian endeavored to dissuade him from cultivating music, and he accepted the offer of an Englishman to accompany him to England, in 1805. De Griffe instructed him on the piano-forte. He received an appointment in the band of a regiment, composed several pieces for the band, and gave instruction on the piano-forte, which led to his attempts to simplify the manner of teaching. He was appointed organist in Westport, Ireland, the regiment having been disbanded in consequence of peace. Wishing to teach his daughter, then seven years old, to play the organ in his absence, and finding her hand defy all his endeavors, he was led to think of some contrivance for giving it the necessary flexibility. The result was his valuable chiroplast (former of the hand), which was completely successful. In 1814, he began to teach his system more generally in Dublin. In 1817, Mr. Logier went to London to have his system examined by the philharmonic society. Although the result of the examination was not favorable, the system became very popular. In 1821, the Prussian government sent an agent to London to inquire into its merits, and Mr. Logier was soon after invited by the same government to introduce it in Berlin, whither he went in 1822, and, at the end of five months, received an order from the king to instruct twenty persons so that they might spread his method throughout

Prussia. It was introduced into Leipsic, and many other places of Germany. Its peculiarity consists in giving instruction to many pupils at the same time, and, though open to the objection to which all systems are exposed, that they cannot produce genius, its success sufficiently shows not only its practicability, but also its advantages.

LOGOS (Greek, λóyos, from Aéyev, to speak) has a great variety of meanings: 1. language, speech in general; hence, 2. every manifestation of the reason and understanding by language, so that it has the meanings of oration, eloquence, conversation, address, also of the right and opportunity of speaking, &c. Language being peculiar to man, as a reasonable being, and speech presupposing thought, logos signifies, 3. reason, the faculty of thinking in general; 4. every thing which is a production of the latter, as notions, conceptions, demonstration, calculation, explanation, condition and relation, nay, even wisdom and logic. Thus logos has the meaning both of ratio and oratio.* In Christian theology, the word Móyos, as used in certain passages in the Scriptures, has been the source of continual disputes ever since the third century of our era. The passage in the Bible which chiefly gives rise to this discussion, is the opening of the gospel of St. John:-" In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him, and without him was not any thing made that was made," &c. In the Greek text, the expression here translated Word (le verbe, das wort, &c.) is Xoyos. What is here to be understood by

óyos, what is its essential character, whether it is a person of the Deity or not, the creative intellect of God, or the Son, through whom he created, or the divine truth which was to be revealed, &c.—this work is not the proper place to examine, nor will our limits permit us even to enumerate the different opinions which have been entertained on this interesting point of Christian metaphysics. We can refer the reader to no better source of information than the General History of Christianity and the Church (in German), by Augustus Neander, Hamburg, 1827 et seq.

A slight study of cultivated languages will show how generally the word signifying speech, or some word derived from the original verb to speak, has acquired a very extended meaning; Adyos from Aryav. Emer and Deber, signifying as the Latin res, from the Greek tw, I speak, word, are the most generic terms in the Oriental languages.