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of the bin. Calmet.-A meat-offering mingled with oil, and one log of oil. Lev.

(2.) LoG, in the Jewish antiquities, is mentioned (2 Kings vi. 25.) as the fourth part of a cab. But in Leviticus the word log is often met with, and fignifies that measure of oil which lepers were to offer at the temple after they were cured of their disease.

(3.) LoG, a fea term, fignifying a small piece of timber a, Plate CCIII. fig. 1. of a triangular, fectoral, or quadrantal figure, on board a fhip, generally about a quarter of an inch thick, and 5 or 6 inches from the angular point to the circumfe rence. It is balanced by a thin plate of lead, nailed upon the arch, or circular fide, fo as to fwim perpendicularly in the water, with about two 3ds immerfed under the surface.

(4.) LOG AND LINE, or the LOG-LINE, a little cord, or line, about 150 fathoms long, faftened to the log by means of two legs ab(fig.2), one of which paffes through a hole at the corner, and is knotled on the oppofite fide, while the other leg is attached to the arch by a pin fixed into another hole, fo as to draw out occafionally. By thefe legs the log is hung in equilibrio; and the line thus annexed to it is wound round a reel fixed for that purpofe in the gallery of the fhip. This line, from the diftance of about 10, 12, or 15 fathoms off the log, has certain knots or divifions, which ought to be at least 50 feet from each other; though it was the common practice at fea not to have them above 42 feet afunder. The length of each knot ought to be the fame part of a fea-mile as half a minute is of an hour; and admitting the measurement of Mr Norwood, who makes a degree on a great circle of the earth to contain 361,200 English feet, or about 694 English ftatute miles, and, therefore, th part of it, or a nautical mile, will be 6120 feet; th of 6120, or 51 feet, fhould be the length of each knot. But because it is fafer to have the reckoning rather before the fhip than after it, therefore so feet may be taken as the proper length of each knot. The knots are fometimes made to confift only of 42 feet each, even in the prefent practice; and this method of dividing the log-line was founded on the fuppofition that 60 miles, each of 5000 English fect, made a degree; for of 5000 is 414, or, in round numbers, 42 feet. Mariners, rather than quit the old way, though known to be erroneous, ufe glaffes for half minute ones, that run but 24 or 25 feconds. They have alfo ufed a line of 45 feet to 30 feconds, or a glafs of 18 feconds to 42 feet. When this is the cafe, the diftance between the knots fhould be corrected by the following proportion; as 30 is to 50, fo is the number of feconds of the glafs to the diftance between the knots upon the line. The heat or moisture of the weather has often a confiderable effect upon the glafs, fo as to make it run flower or fafter; it fhould, therefore, be frequently tried by the pendulum in the following manner. On a round nail hang a ftring that has a musket-ball fixed to one end, carefully measuring between the centre of the ball and the ftring's loop over the peg 39 inches, being the length of a fecond pendulum; then fwing it, and count one for every time it paffes under the peg, beginning at the fe

cond time it paffes; and the number of swings made during the time the glass is running out fhows the feconds it contains. The line alfo is liable to relax and fhrink, and fhould therefore be occafionally measured. The ufe of the log and line is to keep account, and make an eftimate of the fhip's way or distance run; which is done by obferving the length of the line unwound in half a minute's time, told by a half-minute glafs; for fo many knots as run out in that time, fo many miles the fhip fails in an hour. Thus, if there be 4 knots veered out in half a minute, the ship is computed to run 4 miles an hour. No mention of this device for measuring the fhip's way occure till 1607, in an Eaft-India voyage published by Purchas; but from that time its name occurs in other voyages among his collections; and henceforward it became famous, being taken notice of both by our own authors and by foreigners; as by Gunter in 1623; Snellius in 1624; Metius in 1631; Oughtred in 1633; Herigone in 1634; Saltonftall in 1636; Norwood in 1637; Pournier in 1643; and almoft by all the fucceeding writers on navigation of every country.

(5.) LOG, HEAVING THE, is throwing it into the water on the lee-fide, letting it run till it comes without the eddy of the ship's wake; then one, holding a half-minute glafs, turns it up juft as the first knot, or the mark from which the knots begin to be reckoned, turns off the reel (fig. 3.) or paffes over the ftern. As foon as the glafs is out, the reel is ftopped, and the knots run off are told, and their parts eftimated. It is ufual to heave the log once every hour in ships of war and Eaft-Indiamen, and in all other veifels once in two hours; and if at any time of the watch the wind has increased or abated in the intervals, so as to affect the ship's velocity, the officer generally makes a fuitable allowance for it at the clofe of the watch. The log is a very precarious way of computing, and must always be corrected by experience and good fenfe; there being a great deal of uncertainty in the yawing of the fhip, going with the wind aft, or upon the quarter, in the heaving of it, by its coming home, or being drawn after the fhip; on account of the friction of the reel and lightness of the log in the course of the current, and in the ftrength of the wind, which feldom keeps the fame tenor for two hours together, which is the interval between the times for ufing the log in fhort voyages, though in longer ones they heave it every hour. Yet this is a much more exact way of computing than any other in ufe; much preferable certainly to that of the Spaniards and Portuguefe, who gueffed at the fhip's way by the running of the froth or water by the ship's fide; or to that of the Dutch, who ufed to heave a chip over-board, and to number the paces they walk on the deck while the chip swims between any two marks or bulk-heads on the fide.

(6.) LOG, THE COMPOUND. The above mentioned errors, and particularly the log's being fubject to drive with the motion which the water may have at its furface, whereas the experiment requires it to be fixed in the place where it is when the mark commencing the knots goes off the reel, have been confidered by writers, and Rra

many

- many methods have been proposed to remove, or at least to leffen them. The late M. Bouguer propofed a method, which has been thought deferving of particular attention, in the Mem. Acad. Sc. 1747; afterwards in his Treatife on Navigation, published at Paris in 1753, and fince reprinted in 1760, by the abbé De La Caille. For this purpose, take for the log a conical piece of wood, which fix to the log-line paffed through or along its axis, at about 40, 50, or 60, or mare feet, from one end; and to this end fix the diver, which is a body formed of two equal fquare pieces of tin, or of thin iron plate, fixed at right angles to one another along their diagonals; and its fize fo fitted to that of the cone that the whole may float. A cone of three inches diameter in the bafe, and of fix inches in the flant height, is propofed by M. Bouguer to suit a diver made of plates about 9 inches fquare; the interfection of the diagonals is joined to the logline, and the loop and peg fixed as in the common log. However, it has been found, that no kind of wood ufed in British dock-yards, when formed into a cone of the above dimenfions, will float a diver made of ftout tin plates, one fide of the fquare being 9 inches. Such a diver, weighing 1lb. avoirdupois, required to float it a cone of five inches diameter and twelve inches on the flant fide, fo'as the point of the cone, which was made of light fr, fhould just appear above the water. Now fuppofing one fide of fuch a fquare tin diver to be about ten inches, and made of plates only two thirds of the thickness of the former, fuch a diver would weigh, with its folder, about 20 ounces, and can be floated by a light fir cone of four inches diameter in the bafe, and ten inches in the flant height or length; and fuch a compound log might perhaps be found on trial to be affected by about as much again as that propofed by M. Bouguer; and confequently the difference between the numbers given by the common log and compound log, muft be augmented by two 3ds of itself for the neceffary correction, as below. When the compound log of Bouguer, above defcribed, is hove overboard, the diver will fink too deep to be much affected by the current or motion of water at the furface, and the log will thereby keep more fteadily in the place where it firft fell; and confequently the knots ran off the reel will show more accurately the fhip's rate of failing. As the common log is affected by the whole motion of the current, fo this compound log will feel only a part thereof, viz. fuch a part nearly as the refiftance of the cone is to the refiftance of the diver; then the refiftances of the above cone and diver are about as 1 to 5; and confequently this log will drive but one fifth part of what the common log would do; and fo the fhip's true run will be affected by one fifth only of the motion of the waters. To obtain the true rate of failing, it will be proper to heave alternately, hour and hour, the common log and this compound log; then the difference of their knots run off, augmented by its 4th' part, is the correc. tion; which, applied to the knots of the common log, will give the hip's true rate of failing at the middle time between the hours when thefe logs were hove. The correction is additive when

the compound log's run is the greatest, other. wife it is fubtractive. To find the course made good: increase the observed angle between the log-lines by one fourth part; and this gives the correction to be applied to the apparent course, or the oppofite of that shown by the common log; the correction is to be applied to the right of the apparent courfe, when the bearing of the common log is to the left of the compound log; and vice verfa, to the left, when the bearing is to the right of it. Or thus: the lengths run off both logs, together with their bearings, being known; in a card or compafs apply the knots run of, taken from a scale of equal parts along their refpective bearings, from the centre; join the ends; and in this line produced, on the fide next the compound log's length, take one fourth of the interval; then a line drawn from the end, thus produced, to the centre of the card, will show the true courfe and diftance made good. When a current, fuch as a'tide, runs to any depth, the velocity of that current may be much better afcertained by the compound log than by the common one, provided the diver does not defcend lower than the run of the current; for as thofe fhips which are deepeft immerged drive fafteft with the tide; fo the diver, by being acted on below, as well as the log on the furface, their joint motion will give the total effect of the cur rent's motion better than could be had from the motion at the furface only. Also, by fuch a com pound log, the depth to which any current russ may be cafily tried.

(7.) LOGS, OTHER KINDS OF. We have an account, in the voyage to the North Pole, p. 97. of two other logs, which were tried by capt. Phipps: one invented by Mr Ruffel, the other by Foxon both conftructed upon this principle, that a spiral, in proceeding its own length in the direction of its axis through a refifting medium, makes one revolution round the axis; if, therefore, the revolu tions of that spiral are registered, the number of times it has gone its own length through the wa ter will be known. In both thefe the motion of the fpiral in the water is communicated to the clock-work within-board, by means of a fmall line faftened at one end to the fpiral, which tows it after the fhip, and at the other to a spindle, which fets the clock work in motion. That invented by Mr Ruffel has a half spiral of two threads, made of copper, and a fmall dial with clockwork, to regifter the number of turns of the fpiral. The other log has a whole fpiral of wood with one thread, and a larger piece of clock-work with 3 dials, two of them to mark the diftance, and the other divided into knots and fathoms, to how the rate by the half minute glafs, for the conve nience of comparing it with the log. This kind of log will have the advantage of every other in smooth water and moderate weather; and it will be useful in finding the trim of a ship when alone, in furveying a coaft in a single ship, or in measur ing distances in a boat between head-lands and thoals: but it is fubject to other inconveniences, which will not render it a proper fubftitute for the common log...

(8.) LOG, THE PERPETUAL, a machine fo called by its inventor, Mr Gottlieb of Houndíditch, London. It is intended by it to keep a constant

and

and regular account of the rate of a fhip's velocity through the water; whereas the common log hitherto ufed does not indicate the variation in her velocity in the interval of heaving the log, and confequently does not ascertain the true diftance that the fhip has run in any given length of time. Fig. 1. pl. 203. reprefents the whole machine; the lower part of which, EFG, is fixed to the fide of the keel; H reprefenting only the boundary line of the hip's figure. EF are the fection of a wooden external cafe, left open at the ends KL, to admit the paffage of the water during the motion of the fhip. At M is a copper grating, placed to obftruct the entrance of any dirt, &c. into the machine. I, is a fection of a water wheel, made from 6 to 12 inches in diameter, as may be neceffary, with float boards upon its circumference, like a common water wheel, that turn by the refiftance of the water paffing through the channel LK. It turns upon a fhouldered axis, reprefented by the vertical fection at K. When the thip is in motion, the refiftance of the water through the channel LK turns round the wheel I. This wheel, by means of a pinion, is connected with and turns the rod contained in the long copper tube N. This rod, by a pinion fixed at its upper extremity, is connected with and turns upon the whole fyftem of wheels contained in the dial of the cafe ABCD. This dial, by means of the copper tube N, may be fixed to any convenient place aboard the fhip. In the front of the dial are feveral useful circular graduations, as follow: the reference by the dotted line A has an hand which is moved by the wheels within, which points out the motion of the ship in fathoms of 6 feet each. The circle at B has an hand fhowing the knots, at the rate of 48 feet for each knot; and is to be obferved with the half-minute glafs at any time. The circle at C has a fhort and long hand; the former of which points out the miles in land measure, and the latter or longer the number of knots contained in each mile, viz. 128, which is in the fame proportion to a mile as 60 minutes to the hour in the reckoning. At e, a fmall portion of a circle is feen through the front plate called the register; which fhows, in the courfe of 24 hours (if the fhip is upon one tack) the distance in miles that the has run; and in the 24 hours the mariner need take but one obfervation, as this register ferves as an useful check upon the fathoms, knots, and miles fhown upon the two other circles. f, Is a plate flowing 100 degrees or 6000 miles, and alfo acts as another regifter or check; and is ufeful in cafe of any mistake being made in obferving the distance run by the other circles. The reckoning by thefe circles, without fear of miftake, may therefore be continued to nearly 12,000 miles. A communication from this machine may eafily be made to the captain's bed-fide, where by touching a spring only, a bell in the head ABCD will found as many times in an half mipute as the fhip fails miles in an hour. Mr Gottlieb has applied this machine to the Cartaret and Weftmoreland packets. He thinks the mariner will, by this contrivance, be better enabled than heretofore to keep the veffel and his reckoning to gether; it being well known that the moft experienced navigator is too frequently erroneous in

1

this refpect, the fhip being fometimes a-head, or fometimes aftern, off their reckoning. He alfo obferves, that the conftruction of the log is fuch, that if the veflel was to be aground, ftrike a rock, or ftrip off her falfe keel, the parts would not be deranged; and further, fhould the be laid up for repairs, &c. fix months, in balf an hour after coming again into the water, the lower immerged part of the log would clear itself, and be in proper action. (1.) LOGAN, a chief among the Mingo tribe of the N. American Indians, whofe pathetic addrefs to Lord Dunmore, governor of Virginia, has been much and juftly admired. The occafion was as follows; and the authenticity of the facts and of the fpeech is unquestionable. In fpring 1774, a robbery and murder were committed on an inhabitant of the frontiers of Virginia by two Indians of the Shawanee tribe. The neighbouring whites, according to cuftom, undertook to punish this outrage in a fummary way. Colonel Crefap, a man infamous for the many murders he had committed on thofe much-injured people, collected a party, and proceeded down the Kanhaway in queft of vengeance, Unfortunately a canoe of women and children, with one man only, was feen coming from the oppofite fhore, unarmed, and unfufpecting any hoftile attack from the whites. Crefap and his party concealed themfelves on the bank of the river; and the moment the canoe reached the thore, fingled out their objects, and at one fire killed every perfon in it. This happened to be the family of LOGAN, who had long been diftinguished as a friend of the whites. This unworthy return provoked his vengeance. He accordingly fignalized himself in a war which enfued. In autumn 1774, a decifive battle was fought at the mouth of the Great Kanhaway, between the collected forces of the Shawanees, Mingoes, and Delawares, and a detachment of the Virginia militia. The Indians were defeated, and fued for peace. Logan difdained to be seen among the fuppliants; but, left the fincerity of a treaty fhould be diftrufted, from which fo dif tinguifhed a chief abfented himself, he fent by a meffenger the following speech, to be delivered to Lord Dunmore: "I appeal to any white man to fay 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 courfe of the laft 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 paffed, and faid, Logan is the friend of white men. I had even thought to have lived with you, but for the injuries of one man. Colonel Crefap, the laft fpring, in cold blood, and unprovoked, murdered all the relations of Logan, not fparing even 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 fought it; I have killed many; I have fully glutted my vengeance. For my country, I rejoice at the beams of peace; but do not harbour a thought that mine is the joy of fear. Logan never felt fear. He will not turn on his heel to fave his life. Who is there to mourn for Logan? Not one."

(2.) LOGAN, John, D.D. late a clergyman of

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the church of Scotland, author of several works of merit. He was born in Mid Lothian about 1748; ftudied divinity at the univerfity of Edinburgh, and was ordained minister of S. Leith, in 1770. In 1781, he published his Philofophy of Hiftory, the fubftance of which had been delivered in his public lectures at Edinburgh, with great approbation. He also published his Poems, which underwent a zd edition in 1732. In 1783, he wrote Rusnamede, a Tragedy, which he offered to the manager of Covent-garden theatre; but as the lord chamberlain did not relish the political sentiments difplayed in it, a licence was refufed, though it was afterwards acted at Edinburgh with much applaufe. His laft work was A Review of the Prinipal Charges against Mr Haflings; which contain ed fuch bold frokes, that Stockdale, the publisher, was tried for it, but acquitted. Mr Logan died at London, in 1788. Two vols of his Sermons were published fince his death.

(3.) LOGAN, a county of Kentucky.

(4.) LOGAN, a river of Lanarkih. which rifes among the mountains between Lefmahagoe and Muirkirk, and after running 6 miles E. falls into the Nethan, after which the united streams fall into the Clyde.

(5.) LOGAN. See STONE, NO 15. (1.) LOGARITHMIC, [from Aoyos, ratio, and aguos, number.dj. belonging to LOGARITHMS, (2.) LOGARITHMIC CURVE. See LOGARITHMS, Se&. IV.

(3.) LOGARITHMIC LINES. For many mecha. nical purposes it is convenient to have the logarithms of numbers laid down on scales, as well as the logarithmic lines and tangents; by which means, computations may be carried on by mere menfuration with compaffes. Lines of this kind are always put on the common Gunter's fcale; but as thefe inftruments must be extended to a very great length, in order to contain any confiderable quantity of numbers, it becomes an object of importance to shorten them. Such an improvement has been made by Mr William Nicholfon, and publifhed in the 77th volume of the Philof. Trans. The principles on which the conftruction of his inftruments depends are as follow: I. If two geometrical feries of numbers, having the fame common ratio, be placed in order with the terms opposite to each other, the ratio between any term in one feries and its oppofite in the other will be conftant: Thus,

2 6 18 54 162, &C.

3 9 27 81 243, &c. Then, 3.6 9 18 27 54 81 162 243, &c. where it is evident, that each of the terms in the upper feries is exactly two thirds of the correlponding one in the tower. II. The ratio of any two terms in one feries will be the fame with that between those which have an equal distance in the other. III. In all fuch geometrical feries as have the fame ratio, the property above-mentioned takes place, though we compare the terms of any feries with thofe of another; Thus,

[ 2 4 8 16 32 64, &c.
23 6 12 24 48 96, &c.
$4 8 16 32 64 128, &c.

5 10 20 40 80 160, &c.; where it is plain that 2, 4, 3, 6; alfo 2, 4, 4, 8, and 2, 4, 5,

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10, &c, have the fame ratio with that of each feries. IV. If the differences of the logarithms of the numbers be laid in order upon equidiftant pa rallel right lines, in fuch a manner, that a right line drawn across the whole fhall interfect it at divifions denoting numbers in geometrical progression: then from the condition of the arrangement, and the property of this logarithmic line, it follows, ift, That every right line fo drawn will, by its interfections, indicate a geometrical feries of numbers: 2dly, That fuch feries as are indicated by these right lines will have the fame common ratios. and, 3dly, That the feries thus indicated by two parallel right lines, fuppofed to move laterally, without changing either their mutual distance or parallelifm to themfelves, will have each the fame ratio; and in all feries indicated by fuch two lines, the ratio between an antecedent and confequent, the former taken upon one line, and the latter upon another, will be alfo the fame. The 1 of thefe propofitions is proved in the following manner: Let the lines AB, CD, EF, Plate Cill. fig. 5, reprefent parts of the logarithmic line, ar ranged according to the proportion already mentioned; and let GH be a right line paffing through the points e, c, a, denoting numbers in geometri. cal progreffion; then will any other line IK, drawn across the arrangement, likewife pals through three points ƒ, d, b, in geometrical progreilion, From one of the points of interfection f in the laft mentioned line IK, draw the line fr parallel to GH, and interfecting the arrangement in the points i, b; and the ratios of the numbers e, f, c, i, will be equal, as well as of a, k; be caufe the intervals on the logarithmic line, or dif ferences of the logarithms of thofe numbers, are equal. Again, the point f, the line id, and the line hb, are in arithmetical progreffion, denoting the differences between the logarithms of the numbers themselves; whence the quotients of the numbers are in geometrical progreffion. The zá propofition is proved in a fimilar manner. For as it was fhown that the line fg, parallel to GH, paffes through points of divifion denoting num bers in the fame continued ratio as those indicated by the line GH; it may also be shown, that the line LM, parallel to any other line IK, will pals through a series of points denoting numbers which have the fame continued ratio with those indicated by the line IK, to which it is parallel. The 3d propolition arifes from the parallelifm of the lines to their former fituation; by which means they indi cate numbers in a geometrical feries, having the fame common ratio as before: their diftance on the logarithmic line alfo remains unchanged: whence the differences between the logarithms of the oppofite numbers, and of confequence their ratios, will always be conftant. V. Suppofing now an antecedent and confequent to be given in any geometrical feries, it will always be poffible to find them, provided the line be of unlimited length. Drawing two parallel lines, then, through each of the numbers, and fuppofing the lines to move without changing their direction or parallel fituation, they will continually describe new an tecedents and confequents in the fame geometri cal feries as before. VI. Though the logarithmic line contain no greater range of numbers than

fropa

from 1 to 1o, it will not be found neceffary for the purposes of computation to repeat it. The only thing requifite is to have a flider or beam with two fixed points at the distance of the interval betwixt 1 and 10, and let a moveable point be made to range betwixt them always to indicate the antecedent; then, if the confequent fixed point fall without the rule, the other fixed point will always denote the divifion on which it would have fallen had the ruler been prolonged; and this contrivance may eafily be adapted to any arrangement of parallel lines whatever. The arrangement of right lines, however, ought always to be difpofed in fuch a manner as to occupy a right angled parallelogram, or the crofs line already mentioned ought always to be at right angles to the length of the ruler. Fig. 6. is a ruler confifting of ten parallel lines. Fig. 7. a beam compafs for measuring the intervals. B, A, C, are the parts which apply to the surface of the ruler; the middle one, A, being moveable fidewife in a groove in the piece DE, fo as always to preferve its parallelifm to the external pieces DC, which are fixed at a distance equal to the length of the ruler, and have their edges placed in fuch a manner as to form, with the parallel lines which they interfect, a ratio, which by compofition is ys; which in the prefent cafe requires them to be at right angles to the length. The piece DE is applied to the edge EG of the ruler. The edges or borders H, I, K, L, are more conveniently made of transparent horn, or tortoife-thell, than of any opaque matter. In using this ruler, apply the edge of either B or C to the confequent, and flide the piece A to the antecedent; obferving the difference between the numbers on the pieces denoting the lines they are found on: then, applying the fame edge of A to any other antecedent, the other piece B or C will interfect a consequent in the fame ratio upon that line, having the fame fituation with regard to the antecedent that the line of the former confequent had to its antecedent. But if B be the confequent piece, and fall without the ruler, the piece Ċ will thow the confequent one line lower; or if C, in like manner, fall without the ruler, then B will how the confequent one line higher."It might be convenient (fays Mr Nicholson) for the purpofe of computation, to make inftruments of this kind with 100 or more limes: but in the present inftrument, the numbers on the pieces will anfwer the fame purpose; for if a confequent fall upon a line at any given number of intervals without the ruler, it will be found on that line of the arrangement which occupies the fame number of intervals, reckoned inwards from the oppofite edge of the ruler." Fig. 8. is an inftrument on the plan of a Gunter's fcale of 28 inches long, invented by the late Mr Robertson. There is a moveable piece AB in the flider GH, acrofs which is drawn a fine ine: the flider having alfo lines CD, EF, drawn acrofs it at diftances from each other equal to the ength of the ruler AB. In using the inftrument,

the line CD or EF is to be placed at the confes quent, and the line in AB at the antecedent: then, if the piece AB be placed at any other antecedent, the fame line CD or EF will indicate its confequent in the fame ratio taken the fame way: that is, if the antecedent and confequent lie on the fame fide of the flider, all other antecedents and confequents in that ratio will be in the fame manner; and the contrary if they do not. But if the confequent line fall without the ruler, the other fixed line on the flider will fhow the confequent, but on the contrary fide of the flider to that where it would elfe have been seen by means of the first confequent line. Fig. 9. is a circular inftrument equivalent to the former; confifting of three con centric circles engraved and graduated upon a plate of an inch and a half diameter. Two legs A and B proceed from the centre, having rightlined edges in the direction of radii; and are moveable either fingly or together. In ufing the inftrument, place one of the edges at the antecedent and the other at the confequent, and fix them at the angle. Move the two legs then toge ther; and having placed the antecedent leg at any other number, the other will give the confequent one in the like pofition on the lines. If the line CD happen to lie between the legs, and B be the confequent leg, the number fought will be found one line farther from the centre than it would otherwife have been: and on the contrary, it will be found one line nearer in the like cafe, if A be the confequent leg. "This inftrument (fays Mr Nicholson), differing from that represented fig. 6. only in its circular form, and the advantages refulting from that form, the lines must be taken to fucceed each other in the fame manner laterally; fo that numbers which fall either within or without the arrangement of circles, will be found on fuch lines of the arrangement as would have occupied the vacant places, if the fucceffion of lines had been indefinitely repeated fidewife. I approve of this conftruction, as fuperior to every other which has yet occurred to me, not only in point of convenience, but likewife in the probability of being better executed; because small arcs may be graduated with very great accuracy, by divifions transferred from a larger original. The inftrument, fig. 6. may be contained conveniently in a circle of about four inches and a half diameter. The circular inftrument is a combination of the Gunter's line and the sector, with the improvements here pointed out. The property of the sector may be useful in magnifying the differences of the logarithms in the upper part of the line of fines, the middle of the tangents, and the beginning of the verfed fines. It is even poffible, as mathematicians will eafily conceive, to draw spirals, on which graduations of parts, every where equal to each other, will show the ratios of those lines by moveable radii, fimilar to thofe in this instrument."

LOGARITHMS.

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