auspices of the Assyrian society, and upon his return to England the results of his researches were given in a handsome volume embellished with engravings of the sculptures and cuneiform inscriptions of Babylonia, Chaldæa, and Susiana. Subsequently he received an appointment on the staff of the geological survey of India, the operations of which were interrupted by the mutiny of 1857-'8. He died from the effects of a coup de soleil and of repeated attacks of fever caught on the banks of the Tigris and Euphrates. The specimens of ancient sculpture which he sent to the British museum are hardly inferior in interest to those excavated by Layard, and he was the reputed discoverer of the city or cemetery of Warka, supposed to be the biblical Erech. LOG, and LOG LINE, an apparatus used in connection with the half minute glass for obtaining the approximate rate of the movement of a vessel through the water. The log is a triangular or quadrangular piece of board, one side of which has a circular edge, and is weighted with lead, so as to cause the piece to sit upright when thrown into the water. It is attached by cords from its corners to the log line, which is a stout cord about 150 fathoms long, divided by knots or slips of leather into spaces called knots, and wound on a reel which revolves with freedom. Its use is called "heaving the log," and consists in dropping the wood over the stern of the vessel, with a quantity of the line sufficient to reach from the vessel to the log, at the instant the half minute glass is turned up. The reel is held up so that the line may run off freely as the vessel moves away from the log; and as the last sands run through the glass, the reel is instantly stopped. The number of knots run off in the half minute indicates the rate of motion of the vessel. This method of measurement is very inaccurate, a heavy sea sometimes throwing the log after the ship, while a head sea may carry it in the opposite direction. The glass also measures the half minute differently in damp and dry weather, and the line is liable to change its length. Various empirical allowances are made, which add but little to the correctness of the apparatus. It is not known when or by whom this contrivance was invented. Humboldt says, that in all writings on the subject, including the "Encyclopædia Britannica," he found the erroneous opinion expressed that the log was not introduced before the end of the 16th or the beginning of the 17th century, while it is certain that Pigafetta, the companion of Magellan, in the beginning of the 16th century, speaks of the log (la catena a popa) as of a well known means of measuring the course passed over. Purchas makes mention of it in 1607; but the length of a degree of the meridian not being then determined, its divisions were necessarily inaccurate. They were corrected in 1637 by Norwood. The length of a sea mile is now estimated at about 6,086.7 feet; and as the length of the knot is intended to bear the same proportion to this that half a minute bears to an hour, the measurement of the knot VOL. X.-39 is properly 51 feet. Each one is divided into 10 parts called fathoms. For glasses which run out in 28 seconds, the length of the knot should be 47.6 feet.-Numerous substitutes for the log have been contrived. The best of these is that of Massey. A box shaped like a wedge is provided with a spindle to which 4 wings are fixed spirally. With this are connected registering wheels somewhat on the plan of those of the gas meter, their object being to record the number of revolutions of the spindle. This is carried round by the motion against the water as the box is towed astern by a stout line 60 fathoms long. The box is hauled in, and the record noted whenever the course is changed; but while the ship runs full 3 knots the register is not reset except once every 24 hours. At a less rate than 3 knots its indications are uncertain from not towing horizontally. LOGAN. I. A W. co. of Va., bordering on Ky., drained by the Guyandotte and the Tug fork of Sandy river; area, 750 sq. m.; pop. in 1850, 3,620, of whom 87 were slaves. The surface is uneven and the soil generally good. Iron and coal are abundant in the highlands of the county. In 1850 it produced 1,588 bushels of wheat, 154,943 of Indian corn, 8,353 lbs. of tobacco, and 8,202 of wool. There were 6 churches, and 175 pupils attending public schools. Named from the celebrated Indian chief Logan. Capital, Arracoma. II. A S. W. co. of Ky., bordering on Tenn., and drained by branches of the Green and Cumberland rivers; area, 478 sq. m.; pop. in 1850, 16,581, of whom 5,467 were slaves. The surface, resting on cavernous limestone, is finely diversified and well timbered, and the soil fertile. It contains a number of ancient artificial mounds. The productions in 1850 were 1,103,186 bushels of Indian corn, 242,340 of oats, 2,684,767 lbs. of tobacco, and 38,001 of wool. There were 15 grist mills, 8 saw mills, 4 tanneries, 39 churches, and 740 pupils attending public schools. Named in honor of Gen. Benjamin Logan, a pioneer of Kentucky. Capital, Russellville. III. A central co. of Ohio, drained by the Miami river and its branches; area, 425 sq. m.; pop. in 1850, 19,162. The surface is moderately rolling or level, and the soil fertile. In 1850 the productions were 168,811 bushels of wheat, 665,606 of Indian corn, 97,562 of oats, 25,150 of potatoes, and 88,258 lbs. of wool. There were 18 grist mills, 30 saw mills, 2 iron founderies, 7 tanneries, 27 churches, and 7,965 pupils attending public schools. It is intersected by the Mad river and Lake Erie and the Bellefontaine and Indiana railroads. Capital, Bellefontaine. IV. A central co. of Ill., intersected by Salt creek and drained by Kickapoo and Sugar creeks; area, 529 sq. m.; pop. in 1855, 8,324. The land is level and fertile. In 1850 the productions were 839,638 bushels of Indian corn, 26,598 of wheat, 35,728 of oats, and 23,527 lbs. of wool. There were 6 grist mills and 6 saw mills. The county is intersected by the Chicago and Mississippi railroad. Capital, Mount Pulaski. LOGAN, the English name of the Indian chief Tah-gah-jute, celebrated in American revolutionary and colonial history, born about 1725, killed on the southern shore of Lake Erie in the summer of 1780. He was the son of Shikellamy, a celebrated chief of the Cayugas, who lived at Shamokin on the Susquehanna, and was called Logan from James Logan, the secretary of Pennsylvania and a firm friend of the Indians. In his early manhood he was known throughout the frontier of Virginia and Pennsylvania for his fine personal appearance, his engaging qualities, and his friendship to the whites. About 1770 he removed with his family to the banks of the Ohio, where he gave way in a measure to intemperance. In the spring of 1774 his family were massacred, it was alleged, by a party of whites led by Capt. Michael Cresap, under the pretext of retaliation for Indian murders; but it is exceedingly doubt ful whether Cresap had any connection with the transaction. Logan at once instigated a war against the scattered settlers of the far West, and for several months fearful barbarities were perpetrated upon men, women, and children. He himself took 30 scalps in the course of the war, which terminated after a severe defeat of the Indians at the mouth of the Great Kanawha. He disdained to appear among the chiefs who subsequently sued for peace, but sent by an interpreter to Lord Dunmore, the governor of Virginia, the following speech explaining his conduct, which was first published in Jefferson's "Notes on Virginia:" "I appeal to any white man to say if ever he entered Logan's cabin hungry, and he gave him not meat; if ever he came cold and naked, and he clothed him not. During the course of the last long and bloody war Logan remained idle in his cabin, an advocate for peace. Such was my love for the whites that my countrymen pointed as they passed, and said, Logan is the friend of the white men. I had even thought to have lived with you, but for the injuries of one man. Colonel Cresap, the last spring, in cold blood and unprovoked, murdered all the relations of Logan, not even sparing my women and children. There runs not a drop of my blood in the veins of any living creature. This called on me for revenge. I have sought it; I have killed many; I have fully glutted my vengeance. For my country I rejoice at the beams of peace. But do not harbor a thought that mine is the joy of fear. Logan never felt fear. He will not turn on his heel to save his life. Who is there to mourn for Logan? Not one." His habits of intemperance grew upon him after this, and while frenzied with liquor he felled his wife by a sudden blow, so that she lay to all appearance dead. He fled, and while traversing the wilderness between Detroit and Sandusky was overtaken by a party of Indians. Supposing his avengers at hand, he prepared to attack them, and was killed by his relative Todhah-dohs in self-defence. LOGAN, JAMES, an American colonial statesman and author, born in Lurgan, Ireland, Oct. 20, 1674, died at Stenton, near Philadelphia, Penn., Oct. 31, 1751. By his own efforts he acquired a knowledge of the chief ancient and modern languages, and was well informed in mathematics and various branches of natural science. In 1699, being then established in trade in Bristol, England, he accepted an invitation from William Penn to accompany him to America in the capacity of secretary. In 1701, upon the return of Penn to England, he was appointed provincial secretary, and he subsequently filled the offices of commissioner of property, chief justice, and president of the council, discharging in the last capacity the duties of governor of the province for two years after the demise of Gov. Gordon in 1736. The latter years of his life were passed at his seat called Stenton, near Philadelphia, in the pursuit of literature and science. His chief work, Experimenta et Meletemata de Plantarum Generatione (Leyden, 1739; London, translated from the Latin by Dr. Fothergill, 1747), an expansion of a paper on the growth of maize published in the "Philosophical Transactions" for 1735, was considered an important contribution to the science of botany. He was the author of two other Latin treatises of a scientific character published in Holland, of an English translation of Cicero's De Senectute, published in 1744 by Benjamin Franklin, and of Cato's "Distichs," the latter in verse; and he left a variety of papers on ethics and philology. The translation of Cicero was the first original one of a classical author printed in America, and has been called the best previous to Melmoth's. His library, numbering about 2,000 volumes, was, in conformity with his desire, presented to the city of Philadelphia, and is deposited in a separate department of the Philadelphia library under the name of the Loganian library. He was a member of the society of Friends.-GEORGE, an American statesman and philanthropist, grandson of the preceding, born at Stenton, Sept. 9, 1753, died there, April 9, 1821. He was educated in England, subsequently studied medicine in Edinburgh, where he took the degree of M.D., and after an extended tour on the continent returned in 1779 to America. For many years he devoted himself to agricultural pursuits, which he was one of the first in America to prosecute successfully in a scientific manner. He also served several terms in the Pennsylvania legislature. At the outbreak of the French revolution he embraced with enthusiasm the democratic doctrines which it inaugurated, and joined Jefferson and the republican party in opposition to the federalists. In 1798, the United States being then on the brink of a rupture with the French republic, he departed for France, principally at his own suggestion, under the idea that he might contribute to the preservation of peace. He was well received by Talleyrand and Merlin, then chief of the directory, and returned to America with the assurance of the desire of the French government to renew amicable relations with the United States. But as he had taken with him letters of introduction from Jefferson instead of passports from the state department, he was denounced by the federalists on his return as the treasonable envoy of a faction who had undertaken to institute a correspondence with a foreign and hostile power. He was coldly received by Washington and President Adams, and in the latter part of 1798 an act, known as the "Logan act," was passed by congress, making it a high misdemeanor for a private citizen to interfere in a controversy between the United States and a foreign country in the manner he had done. He was subsequently elected to the U. S. senate, of which body he remained a member from 1801 to 1807; and in 1810, urged by the same philanthropic motives which had induced him to visit France 12 years before, he went to England in the hope of preserving peace. In 1797 he published "Experiments on Gypsum" and "Rotation of Crops." LOGAN, JOHN, a Scottish clergyman and author, born in Fala, Edinburghshire, in 1748, died in London, Dec. 28, 1788. He completed his education at the university of Edinburgh. In 1768, on the recommendation of Dr. Blair, he was appointed tutor to the afterward celebrated Sir John Sinclair. He soon returned to Edinburgh, and, having obtained a license as preacher in the established church of Scotland, he was nominated a minister of the town of Leith in 1773. In 1779 he delivered in Edinburgh a course of lectures on the philosophy of history, and in the following year was an unsuccessful candidate for the professorship of history in the university. Being charged with drunkenness by his parishioners in 1785, he was under the necessity of retiring from the ministry on a small pension. He now repaired to London and devoted himself to literary composition. He was not only an eloquent orator and lecturer, but also a gifted poet. His "Ode to the Cuckoo" (1770), and his "Hymns," which have been incorporated into the psalmody of the church of Scotland, entitle him to high rank as a lyrist. A volume of his poems was published in 1781, a new edition of which in 1805 contains a life of the author. The most important of his other works are "Runnamede," a tragedy produced on the Edinburgh stage in 1783; "View of Ancient History," &c. (London, 1788); "Review of the Principal Charges against Mr. Hastings" (1788), a defence of the ex-governor-general esteemed so formidable by his enemies that they arraigned its publisher, Stockdale, for a breach of the privileges of the house of commons; and two volumes of sermons edited by Dr. Robertson in 1790-'91. LOGAN, SIR WILLIAM EDMOND, a Canadian geologist, born in Montreal in 1798. He was educated at the high school and university of Edinburgh, and in 1818 entered the mercantile office of his uncle, Mr. Hart Logan, of London, and after a time became a partner in the firm. In 1829 he went to Swansea as manager of copper smelting and coal mining operations in which his uncle was interested, but left it soon after the death of the latter in 1838. During his 7 years' residence in South Wales, Mr. Logan devoted himself to the study of the coal field of that region; and his minute and accurate maps and sections were adopted by the ordnance geological survey, and published by the government. He was the first to demonstrate that the stratum of under clay, as it is called, which always underlies coal beds, was the soil in which the coal vegetation grew. In 1841 Sir William visited the coal fields of Pennsylvania and Nova Scotia, and communicated several valuable memoirs on the subject to the geological society of London. At this time he began an examination of the older paleozoic rocks of Canada, and a geological survey of Canada having been commenced, he was placed and still continues at its head, having refused for it a very advantageous offer of a similar position in India. In the course of his investigations upon the rocks of the eastern townships, which are the continuation of those of New England, Sir William has shown that so far from being, as had been supposed, primitive azoic rocks, they are altered and crystallized palæozoic strata; a fact which, although suspected, had not hitherto been demonstrated, and which is the key to the geology of north-eastern America. He found the rocks which form the Laurentide and Adirondac mountains, previously regarded as unstratified, to be disturbed and altered sedimentary deposits of vast thickness, equal perhaps to all the hitherto known stratified rocks of the earth's crust. In 1851 he represented Canada at the great exhibition in London, and was also a commissioner from Canada at the industrial exhibition at Paris in 1855, when he received from the imperial commission the grand gold medal of honor, and from the emperor the decoration of the legion of honor. He was knighted in 1856, and in the same year was awarded by the geological society, of which he has long been a member, the Wollaston palladium medal, for his eminent services to geology. LOGANSPORT, a city and the capital of Cass co., Ind., on the Wabash river at its junction with Eel river; pop. in 1859, about 4,500. It is situated in a fertile region, for which it is an active centre of trade. In 1850 it contained 6 churches (1 Baptist, 1 Episcopal, 1 Methodist, 2 Presbyterian, and 1 Roman Catholic), 3 banks, and an academy. Two bridges cross the Wabash and one the Eel river at this place. LOGARITHMS (Gr. Xoyos, reason, and apipos, number), numbers so related to the natural numbers that the multiplication and division of the latter may be performed by addition and subtraction, and the raising to powers and the extraction of roots by very simple multiplication and division. The labor of these operations by the ordinary processes of arithmetic, when the numbers are composed of many figures, is enormous. By the use of logarithms, for the invention of which the world is indebted to John Napier of Merchiston, Scotland, this labor is greatly diminished.--The general theory of = logarithms is very simple. All numbers whatever may be regarded as the powers of some other number taken as a base. Thus, taking as a base the number 8, its successive integral powers give the series of numbers 8, 64, 512, 4,096, &c.; for 8' 8, 82 = 64, 8' 512, 8' 4,096, &c. But it is not necessary to limit the series to the integral powers. The cube root of 8='V8=8=2; the square of the cube root of 8 8 8 4. The first power of 8 multiplied by the cube root = 8X881= 16; 8 X 8 818 32, &c. Other fractional powers would give the numbers omitted in this series; so that a power of 8 could be found which would be equal to any number whatever. By taking negative powers, fractions would come into the series. In a system of logarithms of which 8 is the base, the logarithms are the exponents of the powers to which 8 must be raised to produce the number. Thus, as above, =log. of 2, # log. 4, 1 log. 8, 11 log. 16, 1= log. 36, 2 log. 64, 21 log. 128, &c. It is obvious that the base of the system may be taken to be any positive number except unity. To demonstrate the general principles of logarithms, let a represent the base of the system, m any number, and x its logarithm; then the relation between the number m and its logarithm is expressed by the equation a = m, That is, the logarithm of a number is the exponent of the power to which the base must be raised to produce the number. Let m and n be two numbers, a and y their logarithms, and a the base; then a2 = m; a2 = n. Multiply the first members of these equations together, and we have a X a2 = a*+" : That is, x+y=log. mn; or the logarithm of the product of two numbers equals the sum of the logarithms of the numbers themselves. Dividing the first of the equations above by the = = = second, we have av= or ay m , n' = mn. that is, x―y = log. or the logarithm of the quotient of one quantity divided by another is equal to the logarithm of the dividend, less the logarithm of the divisor. In the equation a*+* = mn, if we make m=n, then x=y, and we have a2 m2; 2x is then the logarithm of m2, or the logarithm of the square of a number equals twice the logarithm of the number itself. By similar reasoning it is shown that the logarithm of the cube of a number equals 3 times the logarithm of the number, &c. If we take mp, then mVpp; but log. m2 = 2 log. m = log. p. Substituting in the last equation Vp for m, it becomes 2 log. Vp = log. p, or log. Vp: log. p; i. e., the logarithm of the square root of a number equals half the logarithm of the number itself. In the same way it may be shown that the logarithm of the cube root of a number equals the logarithm of the number, and the logarithm of any root of a number equals the logarithm of the number divided by the exponent of the root.-The system of logarithms in = = common use is that proposed by Henry Briggs, professor of geometry at Oxford, soon after the publication of Napier's invention in 1614. Briggs used as the base of his system the number 10, and it was soon universally accepted, being so well adapted to the decimal notation. The logarithm of any number in this system is the exponent of the power to which the number 10 must be raised to produce the number. Thus, since (10)° 1, (10)1 = 10, (10)2 = 100, (10)3 = 1,000, (10)* = 10,000, &c., 0, 1, 2, 3, 4, &c., are the logarithms respectively of 1, 10, 100, 1,000, 10,000, &c. A number between 1 and 10 will have for its logarithm a fraction between 0 and 1. Thus the log. of 2 = 0.30103, for (10)0.50103 = 2. A number between 10 and 100 will have for logarithm a number between 1 and 2; thus the logarithm of 50= 1.69897, for (10)1.6983 50. Numbers between 100 and 1,000 will have for logarithms numbers greater than 2 and less than 3, or 2 plus a fraction; thus the log. 250-2.39794, for (10)2.39794 = 250, &c.—In order to make logarithms available for purposes of calculation, the logarithms of all numbers between convenient limits are computed and arranged in tables, the natural numbers occupying the leading or argument column, the logarithm being placed opposite in adjoining columns. Sometimes tables are arranged with the logarithms in the leading or argument column; these are called tables of anti-logarithms. For certain purposes logarithms constructed substantially according to the system originally proposed by Napier are used, and are known as Napierian, natural, or hyperbolic logarithms. In this system the base is the number 2.7182818+. These logarithms are of great use in the higher mathematics, and in the investigation of many problems in physics. The Napierian logarithm of a number is equal to the common or Briggs logarithm multiplied by 2.3025851, or divided by 0.4342945. -The early computers of logarithms carried them to 10 places of decimals; but it was soon found that 7 places were sufficient for most of the uses of astronomy, navigation, surveying. &c. In fact, 5-place logarithms are often sufficient, and, being much more convenient and portable, should be used except when very great accuracy is required. The theory and use of logarithms is now taught as a part of liberal education, and it would be well if the compilers of text books would introduce into them only 5-place decimal logarithms. They, however, often use 6-place logarithms, and make the tables of the size common to 5-place logarithms. This very much increases the labor required in using the tables, and so prevents students from acquiring the necessary facility.-An excellent collection of 5-place logarithms is that attached to "Bowditch's Navigator," and also published separately under the title of "Bowditch's Useful Tables." This contains, beside the tables of logarithms for numbers, log. sines, tangents, &c., also many auxiliary tables useful in navigation and surveying. A good collection of 5-place tables by J. Hoüel (8vo., Paris, 1858) contains also Gauss logarithms for addition and subtraction. Among tables of logarithms to 7 places of decimals may be mentioned Babbage's, which are very accurate. Taylor's tables (large 4to., London) are very valuable, but difficult to obtain. Shortrede's tables (large 8vo., Edinburgh) contain nearly all the tables required in computing; they are especially designed for military and civil engineers. The tables of Callet (8vo., Paris) are very good; they contain the logarithms of all numbers from 1 to 108,000, with log. sines, tangents, &c., beside tables of Napierian logarithms to 20 places of decimals, and short tables of common logarithms to 20 and to 61 places. For log. sines, tangents, &c., Bagay's tables (4to., Paris) are very convenient; they contain the log. sines and tangents for every second of the quadrant. A new edition of Vega's tables (8vo., Berlin, 1856), edited by Dr. Bremiker, is very convenient, and may be obtained at a very moderate price. LOGIC (Gr. Moyos, reason), the science of reasoning. More strictly and properly, it is the science of deducing ideas or conceptions one from another, and of constructing them into propositions, arguments, and systems. A wide range and great diversity of topics have, however, been included in the various treatises written under its name. Some have understood by it an account of the whole mental activity, and defined it as the art of thinking. Others have made it comprise only a knowledge of the first principles, or axioms, from which we reason. Others appear to have held it responsible for the truthfulness of all professedly logical reasoners and processes. Others again have regarded it as chiefly or exclusively an instrument of invention and discovery, and worthless except for the attainment of some new truth. It is now generally held that logic assumes certain first principles or axioms, from which as premises to reason; that it is concerned with the form only of reasoning or argument, and not at all with the subject matter; that it is and of necessity must be a purely a priori science, and moreover a hypothetical science, since it neither assumes nor proves as such the reality of any thing, does not assert that any objects corresponding to our conceptions do really exist, but only gives results and conclusions, based on premises, and true provided the premises be true. Logic is thus limited to the method of reasoning: Though commonly regarded as consisting of two parts, analytics and method, it is essentially a constructive science; it explains the way in which theories and systems are constructed from our primary ideas of objects, and it proves and tests, not their truth, but their legitimacy as deductions. In this view, it presupposes psychology, which is a sort of natural history of thought, and it is preliminary and prerequisite to ontology, the science of being.Logic begins with ideas. Our ideas of objects are complex wholes, and may be analyzed into conceptions of the known properties of objects. Thus, snow is represented by its properties of whiteness, coldness, &c., and an orange by its Universal affirmative: All S is P. For convenience these propositions are desig nated by the 4 first vowels; thus: A, universal affirmative; E, universal negative; I, particular these 4 propositions in all possible ways of 3 in a affirmative; O, particular negative. Combining set, we obtain 64 sets, which are called moods. Of these moods, however, only 11 are found to give valid conclusions, viz.: AAA, AAI, AEE, AEO, AII, AOO, EAE, EAO, EIO, IAI, and OAO. It is found also that the position of the middle term is of essential importance, for let. the mood AAA be written thus: "All M is P; all S is M; therefore all S is P;" and it is evident at once that if M is included in the class P, and S is included in the class M, then S must be included in P also. But if the same mood be written: "All P is M; all S is M," then it does not follow that S is included in P; for men are animals, and horses are animals, but men are not therefore horses. Every mood of the syllogism thus has what are termed figures, of which there are four. In the 1st figure, the middle term is the subject of the major premise middle term is the predicate of both premises; and the predicate of the minor; in the 2d, the and in the 4th, it is the predicate of the major in the 3d, it is the subject of both premises; premise and the subject of the minor. The 11 syllogisms, of which, however, only 19 are found moods each having 4 figures would give 44 by examination to be distinct and valid. These are designated by the capital vowels in the following mnemonic hexameters: BArbArA, cElArEnt, DArII, fErIOque, prioris: |