(62.) Pl. CCIII. fig. 11. Let CD be the logarithmic curve, and AB its bafe or abfciffa, in which let there be taken any number of points, P, P, P", &c. fo that the lines AP, AP, AP", &c. may conftitute an arithmetical progreffion; then, if perpendiculars or ordinates, PM, PM', P"M", &c. be drawn, meeting the curve in the points M, M', M", &c. its nature is fuch, that the ordinates PM, P'M', P"M", conftitute a geometrical progreffion. Hence, and from the properties of logarithms, it appears, that the abfciffas AP, AP, AP", &c. may be confidered as the logarithms of their correfponding ordinates PM, P'M', P"M", &c. refpectively.

(63.) That we may express the relation between any abfciffa and its correfponding ordinate, by means of an equation, let us put the ordinate at the point A, or AC, 1; take Ap, a given portion of the abfciffa, and put pm, the correfponding ordinate, a; take pp', pp", &c. in the abfciffa, equal to one another and to Ap; and let APxx Ap; draw the ordinates p'm', p'm", &c. alfo PM. Then, from the nature of continued proportionals p'm' = a2, p''m'a3, &c. to PM, which will be expreffed by a*; hence, if we put PMy, we have a y for the equation of the curve.

30

(64.) From this equation, as well as by other methods, all the properties of the logarithmic curve may be derived. We fhall briefly mention fome of the most remarkable.

I. The base AB is an afymptote to the curve. II. If PM be an ordinate to the curve at M, and MQ a tangent at the fame point, the fubtangent PQ is a conftant quantity, and equal to the modulus of the particular logarithmic fyftem, to which the curve belongs.

III. The curvilineal space, comprehended between any two ordinates, AC, PM, is equal to the rectangle contained by PQ the fubtangent and PM-AC the difference of the ordinates.

(65.) There is yet another curve, the properties of which are analogous to thofe of logarithms; namely, the common hyperbola. Plate CCIII. fig. 12. Let C be its center, and CD, CE, its afymptotes, in either of which, let the points A, A', A", A", &c. be taken, fo that CA, CA', CA', CA", &c. may be continued geometrical propor. tionals; draw AB, A'B', A"B", AB", &c. parallel to the other asymptote, meeting the curve in B, B', B", B", &c. and join CB, CB, CB, CB"", &c. Then it is demonftrated by writers on conics, that the hyperbolic fectors CB B, CB' B", CB" B"", &c. are equal to each other; and that the quadrilateral spaces ABB'A', A'B'B'A", AB"B"A", &c. are alfo equal to one another, and to each of the fectors. Hence the fectors, CBB', CBB", CBB",, &c. or the quadrilateral figures ABB'A', ABB"A", ABB′′Á", &c. have equal differences, while their corresponding ab fciflas CA', CA", CA", have equal ratios to one another, viz. the ratio of CA to CA': Thus the former are analogous to the logarithms of the latter.

(66.) Let H be the vertex of the hyperbola, draw HG and HK parallel to the afymptotes, fo as to form the rhombus HGCK; then, putting CK =

1, if CP denote any number whatever, and PQ be drawn parallel to the other asymptote, the hy perbolic area KHQP will ferve to exprefs the logarithm of CP, according to a fyftem, the modu lus of which is denoted by the area of the rhombus CKHG. If the afymptotes contain a right angle, the area of the rhombus will be 1, and thus the hyperbolic areas will exprefs NAPIER'S or the hyperbolic logarithms. But any fyftem of logarithms whatever may be reprefented by byperbolic areas; thus, if the afymptotes contain an angle of 25° 44' 25" 5, the area of the rhombus will be 43429448, &c. viz. equal to the modulus of the common fyftem of logarithms, and therefore the hyperbolic areas equal to the common logarithms.

LOG

LOG-BOARD, a fort of table, divided into feveral columns, containing the hours of the day and night, the direction of the winds, the courfe of the fhip, and all the material occurrences that happen during the 24 hours, or from noon to noon; together with the latitude by obfervation. From this table the different officers of the fhip are furnished with materials to compile their journals, wherein they likewife infert whatever may have been omitted, or reject what may appear fuperfluous in the log-board.

LOG-BOOK, a book into which the contents of the log-board is daily copied at noon, together with every circumftance deferving notice that may happen to the fhip, or within her cognifance, either at fea or in a harbour, &c. The intermediate divifions or watches of the log-book, containing four hours each, are ufually figned by the commanding officers in fhips of war or Eaft IndiaSee NAVIGATION. LOGEFOUGEROUSE, a town of France, in

men.

-LOG

the dep. of the Vendee; 3 miles S. of Chataigneraye.

LOGENWASSER, a river of Silefia, in Naiffe. *LOGGATS. n. f. Loggats is the ancient name of a play or game, which is one of the unlawful games enumerated in the 33d ftatute of Henry VIII. It is the fame which is now called kittle-pins, in which boys often make use of bones inftead of wooden pins, throwing at them with another bone inftead of bowling. Hanmer.-Did thefe bones coft no more the breeding, but to play at leggats with them? Shak.

* LOGGERHEAD. n. f. [logge, Dutch, Stupid, and head; or rather from log, a heavy motionless mafs, as blackhead.] A dolt; a blockhead; a thickfcull.

*LOGGERHEADED. adj. [from loggerhead.] Dull; ftupid; doltish-You loggerheaded and unpolish'd groom, what! no attendance. Shak.

LOGGERHEADS, TO FALL TO. To fcuffle; LOGGERHEADS, TO GO TO. to fight without weapons.-A couple of travellers that took up an afs, fell to loggerheads which should be his mafter. L'Etrange.

(1.-6.) LOGGIE. See LOGIE.

(7.) LOGGIE EASTER, a parish of Scotland, in

the counties of Rofs and Cromarty, about 7 miles long and 2 broad, 4 miles from the town of Tain. The foil is various, but fertile; yet a confiderable part of it is not cultivated. The furface is hilly in the middle. Oats, barley, peas, and potatoes, are the chief crops. The population in 1791 was 1125; increafe 275, fince 1755; the number of horfes was 600; and of black cattle about 1500, befides, fome sheep.

LOG-HILL, a town of Ireland, in Limerick.

DEFINITIONS, and DIVISION, of LOGIC.. LOGIC is defined by Dr Johnfon in the following manner:

LOGICK. n. f. [logique, French; logica, Latin. from yos.] The art of reafoning. One of the feven sciences.-Logick is the art of ufing reafon well in our inquiries after truth, and the communication of it to others. Watts.

Talk logick with acquaintance, And practife rhetorick in your common talk. Shak. -By a logick that left no man any thing which he might call his own, they no more looked upon it as the cafe of one man, but the cafe of the king. dom. Clarendon

Here foam'd rebellious logick, cagg'd and bound,

There ftript fair rhetorick languifh'd on the ground. Pope. LOGIC is, by others, more accurately defined, the art of thinking and reasoning justly, or the fcience or hiftory of the human mind; as it traces the progrefs of our knowledge from our firft and moft fimple conceptions through all their different combinations, and all thofe numerous deductions that refult from variously comparing them one with another.

The object of this fcience therefore is, To explain the nature of the human mind, and the proper manner of conducting its feveral powers, in order to attain truth and knowledge. It lays open thofe errors which we are apt through inattention to run into; and teaches us how to diftinguish between truth, and the appearance of it. By thefe means we become acquainted with the nature and power of the understanding; fee what things lie within its reach; where we may attain certainty and demonstration; and when we must be contented with probability.

This fcience is generally divided into four parts, viz. PERCEPTION, JUDGMENT, REASONING, and METHOD; which comprehend the whole fenfations and operations of the human mind,

1

PART I.

OF PERCEPTION.

MAN is furrounded with a variety of objects, which acting differently upon his fenfes, convey diftin&t impreffions into the mind, and thereby roufe the attention and notice of the understanding. By reflecting too on what paffes within us,

C.

we become fenfible of the operations of our own minds, and attend to them as a new set of im

preffions. But in all this there is only bare conSCIOUSNESS. The mind, without proceeding any farther, takes notice of the impreffions that are made upon it, and views things in order as they prefent themfelves one after another. This attention of the understanding to the objects acting upon it, whereby it becomes fenfible of the impref fions they make, is called by logicians PERCEPTION; and the notices themselves, as they exist in the mind, and are there treasured up to be the materials of thinking and knowledge, are diftinguished by the name of IDEAS. In the article METAPHYSICS, it will be fhown at large how the mind, being furnished with ideas, contrives to diverfify and enlarge its ftock: we have here chiefly to confider the means of making known our thoughts to others; that we may not only underRand how knowledge is acquired, but also in what manner it may be communicated with the greatest certainty and advantage.

SECT. I. Of WORDS, confidered as the SIGNS of our IDEAS.

I. OUR ideas, though manifold and various, are nevertheless all within our own breasts, invifible to others, nor can of themselves be made to appear. But God, defigning us for society, and to have fellowship with those of our kind, has provided us with organs fitted to frame articulate founds, and given us alfo a capacity of ufing thofe founds as figns of internal conceptions. Hence fpring wORDS and LANGUAGE: for, having once pitched upon any found to ftand as the mark of an idea in the mind, cuftom by degrees establishes such a connection between them, that the appearance of the idea in the understanding always brings to our remembrance the found or name by which it is expreffed; as in like manner the hearing of the found never fails to excite the idea for which it is made to ftand. And thus it is eafy to conceive how a man may record his own thoughts, and bring them again into view in any fucceeding period of life. For this connection being once fettled, as the fame founds will always ferve to excite the fame ideas; if he can but regifter his words in the order and difpofition in which the prefent train of his thoughts prefent themselves to his imagination, it is evident he will be able to recal his thoughts at pleasure, and that too in the very manner of their firft appearance. Accordingly we find, that the inventions of wri

ting and printing, by enabling us to fix and perpetuate fuch perishable things as founds, have furnifhed us with the means of giving a kind of permanency to the tranfactions of the mind, infomuch that they may be in the fame manner fubjected to our review as any other objects of na

ture.

II. But, befides the ability of recording our own thoughts, there is this farther advantage in the ufe of external figns, that they enable us to communicate our thoughts to others, and also to receive information of what paffes in their breafts. For any number of men, having agreed to establish the fame founds as figns of the fame ideas, it is apparent that the repetition of these founds muft excite the like perceptions in each, and create a perfect correfpondence of thoughts. When, for instance, any train of ideas fucceed one another in my mind, if the names by which I am wont to express them have been annexed by thofe with whom I converfe to the very fame fet of ideas, nothing is more evident, than that, by repeating thofe naines according to the tenor of my prefent conceptions, I shall raife in their minds the fame courfe of thought that has taken poffeffion of my own. For by barely attending to what paffes within ther felves upon hearing the founds which I repeat, they will alto become acquainted with the ideas in my understanding, and have them in a tnanner laid before their view. So that we here clearly perceive how a man may communicate his fentiments, knowledge, and difcoveries to others, if the language in which he converfes be extenfive enough to mark all the ideas and tranf. actions of his mind. But as this is not always the cafe, and men are often obliged to invent terms of their own to exprefs new views and conceptions of things; it may be afked, how in thefe circumftances we can become acquainted with the thoughts of another, when he makes use of words to which we have never annexed any ideas, and which of course can raife no perceptions in our minds? To unveil this myftery, and give fome infight into the foundation, growth, and improve ment of language, the following obfervations will be found of confiderable moment.

III. First, that no word can be to any man the fign of an idea, till that idea comes to have a real exiftence in his mind. For names being only fo far intelligible as they denote known internal conceptions, where they have none fuch to anfwer them, they are plainly founds without fignification, and of courfe convey no inftruction or knowledge. But no fooner are the ideas to which they belong raised in the understanding, than, finding it eafy to connect them with the established names, we can join in any agreement of this kind made by others, and thereby enjoy the benefit of their difcoveries. The first thing therefore to be confidered is, how thefe ideas may be conveyed into the mind; that being there, we may learn to connect them with their appropriated founds, and fo become capable of understanding others, when they make use of these founds in laying open and communicating their thoughts. To comprehend this diftinctly, it will be neceffary to attend to the divifion of our ideas into the fimple and complex: (fee METAPHYSICS.) And first, as for our fimple

ideas; they cannot find admiffion into the mind but by the two original fountains of knowledge, fenfation and reflection. If therefore any of thefe have as yet no being in the understanding, it is impoffible by words or a defcription to excite them there. A man who had never felt the fenfation of HEAT, could not be brought to comprehend that fenfation by any thing we might say to explain it. If we would really produce the idea in him, it must be by applying the proper object to his fenfes, and bringing him within the influ ence of a hot body. When this is done, and experience has taught him the perception to which men have annexed the name heat, it then becomes to him the fign of that idea, and he thenceforth underftands the meaning of the term, which, before, all the words in this world would not have been fufficient to convey into his mind. The cafe is the fame in respect of light and colours. A man born blind, and thereby deprived of the only conveyance for ideas of this clafs, can never be brought to understand the names by which they are expreffed. The reafon is plain: they ftand for ideas that have no exiftence in his mind; and as the organ appropriated to their reception is wanting, all other contrivances are vain, nor can they by any force or defcription be raised in his imagination. But it is quite otherwife in our complex notions: for thefe being no more than certain combinations of fimple ideas, put together in various forms; if the original ideas out of which the collections are made have already got admiffion into the understanding, and the names ferving to exprefs them are known; it will be eafy, by enumerating the feveral ideas concerned in the compofition, and marking the order and manner in which they are united, to raise any complex conception in the mind. Thus the idea anfwering to the word rainbow may be readily excited in the imagination of another who has never feen the appearance itself, by barely defcribing the figure, largenefs, pofition, and order of colours; if we fuppofe these feveral fimple ideas, with their names, fufficiently known to him.

IV. This leads to a ad obfervation upon this fubject, namely, That words flanding for complex ideas are all definable, but thofe by which we denote fimple ideas are not; for fimple ideas being fecondary perceptions, which have no other entrance into the mind than by fenfation or reflection, can only be got by experience, from the feveral objects of nature proper to produce those perceptions in us. Words indeed may ferve to remind us of them, if they have already found admiffion into the understanding, and their connection with the established names is known; but they can never give them their original being and exiftence there. Hence, when any one afks the meaning of a word denoting a fimple idea, we pretend not to explain it to him by a definition, knowing that to be impoffible; but, fuppofing him already acquainted with the idea, and only ignorant of the name by which it is called, we either mention it to him by fome other name with which we prefume he knows its connection, or appeal to the object where the idea itself is found. Thus, were any one to afk the meaning of the

word.

word white, we should tell him it food for the fame idea as albus in Latin, or blanc in French; or, if we thought him a ftranger to thefe languages, we might appeal to an object producing the idea, by faying it denoted the colour we obferve in fnow or milk. But this is by no means a definition of the word, exciting a new idea in his underftanding; but merely a contrivance to remind him of a known idea, and teach him its connection with the established name. For if the ideas after which he inquires have never yet been raifed in his mind; as fuppofe one who had feen no other colours than black and white, fhould ask the meaning of the word fearlet; it is eafy to perceive, that it would be no more poffible to make him comprehend it by words, or a definition, than to introduce the fame perception into the imagination of a man born blind. The only method in this cafe is, to prefent fome object, by looking at which the perception itfelf may be excited; and thus he will learn both the name and the idea together.

V. But how comes it to pafs, that men agree in the names of their fimple ideas, feeing they can not view the perceptions in one another's minds, nor make known thefe perceptions by words to others? The effect is produced by experience and obfervation. Thus finding, for inftance, that the name of heat is annexed to that fenfation which men feel when they approach the fire, I make it alfo the fign of the fenfation excited in me by fuch an approach, nor have any doubt but it denotes the fame perception in my mind as in theirs. For we are naturally led to imagine, that the fame objects operate alike upon the organs of the human body, and produce an uniformity of fenfations. No man fancies that the idea raised in him by the tafte of fugar, and which he calls fweetness, differs from that excited in another by the like means; or that wormwood, to whofe relish he has given the epithet bitter, produces in another the fenfation which he denotes by the word feet. Prefuming therefore upon this conformity of perceptions, when they arife from the fame objects, we eafily agree as to the names of our fimple ideas: and if at any time, by a more narrow fcrutiny into things, new ideas of this clafs come in our way, which we choose to exprefs by terms of our own invention; these names are explained, not by a definition, but by referring to the objects whence the ideas themselves may be obtained.

VI. Being in this manner furnished with fimple ideas, and the names by which they are expreffed; the meaning of terms that ftand for complex ideas is eafily got, because the ideas themselves anfwering to thefe terms may be conveyed into the mind by definitions. For our complex notions are only certain combinations of fimple ideas. When therefore thefe are enumerated, and the manner in which they are united into one conception explained, nothing more is wanting to raife that conception in the understanding; and thus the term denoting it comes of courfe to be understood. And here it is worth while to reflect a little upon the wildom and goodness of the Deity, in thus furnishing us with the very aptest means of communicating our thoughts. For were it not fo ordered, that we could thus convey our

complex ideas to one another by definitions, it would in many cafes be impoffible to make them known at all. This is apparent in thofe ideas which are the proper work of the mind. For as. they exift only in the understanding, and have no real objects in nature, in conformity to which they are framed; if we could not make them known by defcription, they muft lie for ever hid within our own breafts, and be confined to the narrow limits of a fingle mind. All the fine fcenes that arife from time to time in the poet's fancy, and, by his lively painting, give fuch entertainment to his readers; were he deftitute of this faculty of laying them open to the view of others by words and defcription, could not extend their influence beyond his own imagination, or give joy to any but himfelf."

VII. There is this additional advantage, in the ability we enjoy of communicating our complex notions by definitions; that as thefe make by far the largest class of our ideas, and most frequently occur in the progrefs and improvement of knowledge, fo they are by these means imparted with the greateft readiness, than which nothing would tend more to the increase and spreading of science: for a definition is foon perufed; and if the terms of it are well understood, the idea itself finds an eafy admiffion into the mind. Whereas in fimple perceptions, where we are referred to the objects producing them, if thefe cannot be come at, as is fometimes the cafe, the names by which they are expreffed muft remain empty founds. But new ideas of this clafs occurring very rarely in the fciences, they feldom create any great ob. ftruction. It is otherwife with our complex notions: for every step we take, leading us into new combinations and views of things, it becomes neceffary to explain thefe to others, before they can be made acquainted with our discoveries: and as the manner of defining is eafy, requiring no apparatus but words, which are always ready, and at hand, we can with the lefs difficulty remove fuch obftacles as might arife from terms of our own invention, when they are made to ftand for new complex ideas fuggefted to the mind by fome prefent train of thinking. And thus at laft we are let into the mystery hinted at in the beginning of this fection, viz. how we may become acquainted with the thoughts of another, when he makes ufe of words to which we have as yet joined no ideas. The answer is obvious from what has been already faid. If the terms denote fimple perceptions, he muft refer us to these objects of nature whence the perceptions themselves are to be obtained; but, if they ftand for complex ideas, their meaning may be explained by a definition.

SECT. II. Of DEFINITIONS.

I. A definition is the unfolding of some conceptions of the mind, anfwering to the word or term made ufe of as the fign of it. Now as, in exhibiting any idea to another, it is neceffary that the defcription be fuch as may excite that precife idea in his mind; it is plain that definitions, properly fpeaking, are not arbitrary, but confined to the representing of certain determinate settled notions, fuch namely as are annexed by the speaker or writer to the words he ufes. As nevertheless it is

univerfally

univerfally allowed, that the fignification of words is perfectly arbitrary, and not the effect of any natural and neceffary connection between them and the ideas for which they ftand; fome may perhaps wonder why definitions are not so too. To unravel this difficulty, and fhew diftinctly what is and what is not arbitrary in fpeech, we muft carefully diftinguish between the connection of our words and ideas, and the unfolding of the ideas themfelves.

II. Firft, as to the connection of our words and ideas; this, it is plain, is a purely arbitrary inftitution. When, for inftance, we have in our minds the idea of any particular fpecies of metals, the calling it by the name gold is an effect of the voluntary choice of men fpeaking the fame language, and not of any peculiar aptnefs in that found to exprefs that idea. Other nations we find make ufe of different founds, and with the fame effect. Thus aurum denotes that idea in Latin, and or in French; and even the word gold itfelf would have as well ferved to exprefs the idea of that metal which we call filver, had cuftom in the beginning established it.

III. But although we are thus entirely at liberty in connecting any idea with any found, yet it is quite otherwise in unfolding the ideas themfelves. For every idea having a precife appearance of its own, by which it is diftinguished from every other idea; it is manifeft, that in laying it open to others, we must study such a description as shall exhibit that peculiar appearance. When we have formed the idea of a figure bounded by 4 equal fides, joined together at right angles, we may exprefs that idea by any found, and call it either a fquare or a triangle. But whichever of thefe names we ufe, fo long as the idea is the fame, the description by which we would fignify it to another must be fo too. Let it be called Square or triangle, it is ftill a figure having 4 equal fides, and all its angles right ones. Hence we clearly fee what is and what is not arbitrary in the use of words. The eftablishing any found as the mark of fome deter. minate idea in the mind, is the effect of free choice, and a voluntary combination among men: and as different nations make use of different founds to denote the fame ideas, hence proceeds all that variety of languages which we meet with in the world. But when a connection between our ideas and words is once fettled, the unfolding of the idea answering to any word, which properly conftitutes a definition, is by no means an arbitrary thing for here we are bound to exhibit that precife conception which either the use of language, or our own particular choice, hath annexed to the term we ufe.

IV. Thus it appears, that definitions, confidered as descriptions of ideas in the mind, are fteady and invariable, being bounded to the reprefentation of these precife ideas. But then, in the application of definitions to particular names, we are altogether left to our own free choice; becaufe, as the connecting of any idea with any found is a perfectly arbitrary inftitution, the applying the defcription of that idea to that found muft be fo too. When therefore logicians tell us that the definition of the name is arbitrary, they mean no more than this; that as different ideas may be connect

ed with any term, according to the good pleasure of him that ufes it; in like manner may different descriptions be applied to the term, fuitable to the ideas fo connected. But this connection being fettled, and the term confidered as the fign of fome fixed idea in the understanding, we are no longer left to arbitrary explications, but must study fuch a defcription as correfponds with that precife idea. Now this alone, according to what has been before laid down, ought to be accounted a definition. What feems to have occafioned no fmall confufion in this matter, is, that many explanations of words, where no idea is unfolded, but merely the connection between fome word and idea afferted, have yet been dignified with the name of definitions. Thus, when we say that a clock is an instrument by which we measure time: that is by fome called a definition; and yet it is plain that we are previoufly supposed to have an idea of this inftrument, and only taught that the word clock ferves in common language to denote that idea. By this rule all explications of words in our dictionaries will be definitions, nay, the names of even fimple ideas may be thus defined. White, we may fay, is the colour we observe in fnow or milk; heat the fenfation produced by ap proaching the fire; and fo in innumerable other inftances. But thefe, and all others of the like kind, are by no means definitions, exciting new ideas in the understanding, but merely contrivances to remind us of known ideas, and teach their connection with the established names.

V. But now, in definitions properly so called, we firft confider the term we ufe as the fign of fome inward conception, either annexed to it by cuf tom, or our own free choice; and then the bufinefs of the definition is to unfold and explain that idea. As therefore the whole art lies in giving juft and true copies of our ideas; a definition is then faid to be perfect, when it ferves diftinctly to excite the idea defcribed in the mind of another, even fuppofing him before wholly unacquainted with it. This point fettled, let us next inquire what thofe ideas are which are capable of being thus unfolded? In the first place, it is evident, that all our fimple ideas are neceffarily excluded. We have seen that experience alone is to be confulted here, infomuch that if either the objects whence they are derived come not in our way, or the avenues appointed by nature for their reception are wanting, no defcription is fufficient to convey them into the mind. But where the understanding is already fupplied with these original and primitive conceptions, as they may be united together in an infinity of different forms; fo may all their feveral combinations be distinctly laid open, by enumerating the fimple ideas concerned in the various collections, and tracing the order and manner in which they are linked one to another. Now these combinations of fimple notices conftitute what we call our complex notions; whence it is evident, that complex ideas, and thofe alone, admit of that kind of defcription which goes by the name of a definition.

VI. Definitions, then, are pictures or representations of our ideas; and as these representations are then only poffible when the ideas themselves are complex, it is obvious, that definitions cannot