62. On the method of computing logarithmic sines, cosines, &c.—A table of the logarithms of numbers being computed, by the methods explained above, one of logarithmic sines, &c., may easily be formed from it, by taking the logarithms of the numbers which represent the natural sines, &c.; and this, the most ready and convenient method, is that which has generally been used. But the logarithmic sines, &c., may be computed independently of a knowledge of the logarithms of natural numbers; or, with the aid of a table of logarithms, they may be obtained without reference to natural sines; and even without the aid of either the one or the the log. cosine of x=other, as we shall shortly explain. Tables of logarithmic sines, &c., are generally adapted to a radius whose logarithm is 10, but they can be more conveniently computed by assuming the radius unity; and 10 being added to the logarithms computed on that supposition, we have the logarithmic sines, &c., to the com mon tabular radius. The length of an arc of 1°, to radius unity is 01745329252, &c.; and this, multiplied by the degrees contained in any given arc, thus the length of that arc, to radius 1, and it is this length which is to be understood in what follows, when the length of an arc is said to be given. 63. It has been shown in the article FLUXIONS, in this work, that if M be 43429, &c., the modulus of the common system of logarithms, then the logarithm of r is the fluent of M + - M { + 2 2 + , or &c.; and the tabular log. cos. x = 10 17.2-8 2520' r° 4 مر the cosine is obtained without the aid of either logarithms or natural numbers. As the cosine of an arc is the sine of its complement, this last series will give the sines as well as the cosines of all arcs. Like expressions might easily be deduced for the secants and tangents of arcs; but, our object being simply to show the manner in which such expressions may be obtained, we leave it to the student to apply what we have here given to any other purpose that he may think proper. Of proportional or logistie logarithms.-These logarithms are adapted to making proportions among sexagesimals, and were particularly intended by Dr. Maskelyne, who first formed them, to facilitate the finding of the Greenwich time, corresponding to the distance of the moon from the sun or a star. If the change in the lunar distance in three hours, as given in the Nautical Almanac, be called m, and it be required to find the time r, corresponding to any other less, but given change n, in the distance, 3b n the proportion is m; n :: 3 : x = ; whence log. mlog. n = 31 - log. x. Now 3 and r being expressed in seconds of time, log. 3. log. a is the proportional logarithm of r; and similarly, log. 3h log. m is the proportional logarithm of m, and log. 3-log. n, is the proportional logarithm of n. Hence log. m - log. - n prop. log. a becomes (log. 31. m 3h or prop. log. p = prop. log. o + prop. log. n prop. log. m; so that proportions are made with these logarithms precisely as they are with common logarithms. It follows, from the construction of proportional logarithms, that the logarithm of the largest interval which they are made to comprehead is always nothing; as in those adapted to three hours, the logarithm of three hours is nothing. 66. Proportional logarithms greatly facilitate the taking of proportional points among sexagesimal quantities, whether in time or in arc; for the hours, minutes, and seconds of time may be considered as degrees, minutes, and seconds of arc. Example 1.-If the altitude of the sun increase 38' 15" in 4" 20°; what will it increase in 1" 32". Here the proportion is 4" 20. 1TM 32′ :: 38′ 15′′": 14′ 34′′, the answer; or, by proportional logarithms. 4m 20 prop. log. 1.6185 1m 32° 38′ 15′′ ditto 14' 34" ditto 2:0378 2.7104 .2.0919 Example 2.-If the change in the lunar distance in 3 be 1° 42′ 14′′, what interval of time will correspond to a change of 37′ 5′′ in the distance? 37' 5" prop. log. 1° 42′ 14′′ ditto Ans. 1h 5m 18" ditto 6861 SECT. IV.-OF CERTAIN CURVES RELATED TO 68. The discovery of logarithms suggested to mathematicians the idea of curve lines, which might have similar properties to those numbers: hence the origin of the logarithmic, or, as it has been also called, the logistic curve, the nature and properties of which we now proceed to explain. 69. Plate LOGARITHMS, fig. 11. Let CD be the logarithmic curve, and AB its base or abscissa, in which let there be taken any number of points, P, P', P", &c., so that the lines AP, AP', AP", &c. may constitute an arithmetical progression; then, if perpendiculars or ordinates, PM, P'M', P" M", &c., be drawn, meeting the curve in the points M, M', M", &c., its nature is such, that the ordinates PM, P'M', P"M", constitute a geometrical progression. Hence, and from the properties of logarithms, it appears, that the abscissas AP, AP, AP", &c., may be considered as the logarithms of their corresponding ordinates PM, P'M', P" M", &c., respectively. = 70. That we may express the relation between any abscissa and its corresponding ordinate, by means of an equation, let us put the ordinate at the point A, or AC, 1; take A p, a given portion of the abscissa, and put pm, the corresponding ordinate,= = a; take pp', pp", &c., in the abscissa, equal to one another, and to Ap; and let AP= =xx Ap; draw the ordinates p' m', p′′ m", LOG-BOARD, a sort of table, divided into several columns, containing the hours of the day and night, the direction of the winds, the course of the ship, and all the material occurrences that &c., also PM. Then, from the nature of continued proportionals, p' m' = a2, p′′ m2 = a3, &c., to PM, which will be expressed by a; hence, if we put PM=y, we have a = y for the equation of the curve. methods all the properties of the logarithmic 71. From this equation, as well as by other curve may be derived. We shall briefly mention some of the most remarkable. I. The base A B is an asymptote to the curve. II. If PM be an ordinate to the curve at M, and MQ a tangent at the same point, the subtangent PQ is a constant quantity, and equal to the modulus of the particular logarithmic system, 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 subtangent and PM-AC the difference of the ordinates. 72. There is yet another curve, the properties of which are analogous to those of logarithms; namely, the common hyperbola. Fig. 12. asymptotes, in enner of which, let the points A, Let C be its centre, and C D, C E, its A', A", A", &c., be taken, so that CA, CA, CA", CA", &c., may be continued geometrical proportionals; draw A B, A'B', A”B”, A′′E", &c., parallel to the other asymptote, meeting the curve in B, B', B", B", &c., and join C B, C B', CB", CB", &c. Then it is demonstrated, by writers on conics, that the hyperbolic sectors CB B', CB' B", C B" B", &c., are equal to each other; and that the quadrilateral spaces A B B'A', A'B' B′′A”, and to each of the sectors. Hence the sectors, A" B" B" A", &c., are also equal to one another, CBB', CBB", CBB", &c., or the quadrilateral figures A BB'A', ABB"A", ABB"A", &c., have equal differences, while their corresponding abscissas CA', CA", CA", have equal ratios to one another, viz. the ratio of CA to CA': thus latter. the former are analogous to the logarithms of the draw HG and HK parallel to the asymptotes, 73. Let H be the vertex of the hyperbola, so 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 hyperbolic area KHQP will serve to express the logarithm of CP, according to a system, the modulus of which is denoted by the area of the rhombus CKHG. If the asymptotes contain a right angle, the area of the rhombus will be = areas will express Napier's or the hyperbolic lo1, and thus the hyperbolic garithms. But any system of logarithms whatthus, if the asymptotes contain an angle of ever may be represented by hyperbolic areas; 25° 44′ 25′′ 5, the area of the rhombus will be 43429448, &c., viz. equal to the modulus of the common system of logarithms, and therefore the hyperbolic areas equal to the common logarithms. happen during the twenty-four hours, or from noon to noon; together with the latitude by observation. From this table the different officers of the ship are furnishea with materials to com pile their journals, wherein they likewise insert whatever may have been omitted, or reject what may appear superfluous in the log-board. LOG-BOOK, a book into which the contents of the log-board are daily copied at noon, together with every circumstance deserving notice that may happen to the ship, or within her cognisance, either at sea or in a harbour, &c. The intermediate divisions or watches of the logbook, containing four hours each, are usually signed by the commanding officers in ships of war or East Indiamen. See NAVIGATION. LOGIC, n. 8. LOG'ICAL, adj. LOGʻICALLY, adv. LOGICIAN, n. s. LOGOM ACHY, n. S. LOGI C. Fr. logique; Span. Port. and Lat. logica; Gr. Xoyoç. Reason; the art of reasoning. Logical is pertaining to, or skill in this art. Logician, a teacher, performer, or proficient in it: logomachy, verbal contention. The heretick complained greatly of St. Augustine, as being too full of logical subtilties. Hooker. Talk logick with acquaintance, And practise rhetorick in your common talk. Shakspeare. If a man can play the true logician, and have as well judgment as invention, he may do great matters. Bacon. By a logick that left no man any thing which he might call his own, they no more looked upon it as the case of one man, but the case of the kingdom. Clarendon. Forced terms of art did much puzzle sacred theology with distinctions, cavils, quiddities; and so transformed her to a mere kind of sophistry and logomachy. Howel. In such manner easily, without uttering any logical untruth, one may yet grievously calumniate. Barrow. Logicians use to class a proposition, As justices do criminals in prison, And in as learned authentic nonsense writ The names of all their modes and figures fit: For a logician's one that has been broke To ride and pace his reason by the book, And by their rules, and precepts, and examples, To put his wits in any kind of trammels. Butler. Those who in a logical dispute keep in general terms, would hide a fallacy. Dryden. The application of whips, racks, gibbets, gallies, dungeons, fire and faggot, in a dispute, may be looked upon as popish refinements upon the old heathen logic. Addison. Shakspeare. IIamlet. Logick is the art of using reason well in our enquiries after truth, and the communication of it to others. Watts's Logick. Add to these incentives to social life, my reputation for bookish knowledge, a certain wild logical talent, and a strength of thought, something like the rudiments of good sense; and it will not seem surprising that I was generally a welcome guest where I visited. Burns. Vociferated logic kills me quite, A noisy man is always in the right— I twirl my thumbs, fall back into my chair, Fix on the wainscot a distressful stare, And, when I hope his blunders are all out, Reply discreetly-to be sure-no doubt! Cowper. But hold, why deign I to dispute With such a scoundrel of a brute? Logic is lost upon a knave, Beattie. Let action prove the law our slave. LOGIC, accurately defined, is the art of thinking and reasoning justly: it traces the prosimple conceptions through all their different gress of our knowledge from our first and most combinations, and all those numerous deductions that result from variously comparing them one with another. The object of this science therefore is, to explain the nature of the human mind, and the proper manner of conducting its several powers, in order to attain truth and knowledge. It lays open those errors to which we are liable through inattention; and teaches us how to distinguish between truth and the appearance of it. By these means we become acquainted with the nature and power of the understanding; see what things lie within its reach; where we may attain certainty and demonstration; and when we must be contented with probability. This science has been generally divided into four parts, viz. perception, judgment, reasoning, and method; which comprehend the whole operations of the Mind. The late professor Brown resolves judgment and reason into relative suggestion. See Lectures on the Philosophy of the Human Mind, vol. iii. lec. 51. He also supplies us with an able view of the rationale of logic, which the reader will find in a condensed form at the end of this paper. PART I. OF PERCEPTION. Man is surrounded with a variety of objects, which, acting differently upon his senses, convey distinct impressions into the mind, and thereby rouse the attention and notice of the understanding. By reflecting too on what passes within us, we become sensible of the operations of our own minds, and attend to them as a new set of impressions. But in all this there is only bare consciousness. The mind, without proceeding any farther, takes notice of the impressions that are made upon it, and views things in order as they present themselves one after another. This attention of the understanding to the objects acting upon it, whereby it becomes sensible of the impressions 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 distinguished by the name of ideas. In the article METAPHYSICS, it will be shown at large how the mind, being furnished with ideas, contrives to diversify and enlarge its stock: we have here chiefly to consider the means of making known our thoughts to others; that we may not only understand how knowledge is acquired, but also in what manner it may be communicated with the greatest certainty and advantage. SECT. I.-OF WORDS, CONSIDERED AS THE SIGNS OF IDEAS. 1. Our ideas, though manifold and various, being all within our own breasts, invisible to others, cannot of themselves be made to appear. But God, designing us for society, has provided us with organs fitted to frame articulate sounds, and has given us a capacity of using these sounds as signs of internal conceptions. Hence spring words and language: for, having once fixed upon any sound to stand as the mark of an idea in the mind, custom by degrees establishes such a connexion between them, that the appearance of the idea in the understanding always brings to our remembrance the sound or name by which it is expressed; as in like manner the hearing of the sound never fails to excite the idea for which it is made to stand. And thus it is easy to conceive how a man may record his own thoughts, and bring them again into view in any succeeding period of life. For this connexion being once settled, as the same sounds will always serve to excite the same ideas; if he can but register his words in the order and disposition in which the present train of his thoughts presents itself 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 first appearance. Accordingly we find, that the inventions of writing and printing, by enabling us to fix and perpetuate such perishable things as sounds, have furnished us with the means of giving a kind of permanency to the transactions of the mind, so that they may be subjected to our review in the same manner as any other objects of nature. 2. But, besides the ability of recording our own thoughts, there is this farther advantage in the use of external signs, that they enable us to communicate our thoughts to others, and to receive information of what passes in their breasts. For any number of men, having agreed to establish the same sounds as signs of the same ideas, it is evident that the repetition of these sounds must excite the like perception in each, and create a perfect correspondence of thoughts. When, for instance, any ideas succeed one another in my mind, if the names by which I an wont to express them have been annexed by those with whom I converse to the very same set of ideas, nothing is more evident than that, by repeating those names, according to the tenor of my present conceptious, I shall raise in their minds the same course of thought that has taken possession of my own. So that we here clearly perceive how a man may communicate his sentiments, knowledge, and discoveries to others, if the language in which he converses be extensive enough to mark all the ideas and operations of his mind. But as this is not always the case, and men are often obliged to invent terms of their own to express new views and conceptions of things; it may be asked, how in these circumstances 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 raise no perceptions in our minds? To unveil this mystery, and give some insight into the foundation, growth, and improvement of language, the following observations will be found of considerable moment. 3. First, that no word can be to any man the sign of an idea, till that idea comes to have a real existence in his mind. For names being only so far intelligible as they denote known internal conceptions, where they have none such to answer them, they are plainly sounds without signification, and of course convey no instruction or knowledge. But no sooner are the ideas to which they belong raised in the understanding, than, finding it easy 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 discoveries. The first thing therefore to be considered is, how these ideas may be excited in the mind; that, being there, we may learn to connect them with their appropriated sounds, and so become capable of understanding others, when they make use of these sounds in communicating their thoughts. To comprehend this distinctly, it will be necessary to attend to the division of our ideas into the simple and complex. And first, as for our simple ideas; they originate in sensation and reflexion. If therefore any of these have as yet no being in the understanding, it is impossible by words to excite them there. A man who had never felt the sensation of heat, could not be brought to comprehend that sensation 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 senses, and bringing him within the influence 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 sign of that idea, and he thenceforth understands the meaning of the term, which, before, all the words in this world would not have been sufficient to convey into his mind. The case is the same in respect of light and colors. A man born blind, and thereby deprived of the only conveyance for ideas of this class, can never be brought to understand the names by which they are ex pressed. The reason is plain: they stand for ideas that have no existence 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 description he raised in his imagination. But it is quite otherwise in our complex notions: for these being no more than certain combinations of simple ideas, put together in various forms; if the original ideas out of which the collections are made have already found admission into the understanding, and the names serving to express them are known; it will be easy, by enumerating the several ideas concerned in the composition, and marking the order and manner in which they are united, to raise any complex conception in the mind. Thus the idea answering to the word rainbow may be readily excited in the imagination of another who has never seen the appearance itself, by barely describing the figure, largeness, position, and order of colors; if we suppose these several simple ideas, with their names, sufficiently known to him. 4. This leads to a second observation upon this subject, namely, that words standing for complex ideas are all definable, but those by which we denote simple ideas are not; for simple ideas being perceptions, which have no other enentrance into the mind than by sensation or reflection, can only be obtained by experience, from the several objects of nature proper to produce those perceptions in us. Words indeed may serve to remind us of them, if they have already found admission into the understanding, and if their connexion with the established names is known; but they can never give them their original being or existence there. Hence, when any one asks the meaning of a word denoting a simple idea, we pretend not to explain it to him by a definition, knowing that to be impossible; but, supposing him already acquainted with the idea, and only ignorant of the name by which it is called, we either mention it to him by some other name with which we presume he knows its connexion, or appeal to the object where the idea itself is found. Thus, were any man to ask the meaning of the word white, we should tell him it stood for the same idea as albus in Latin, or blanc in French; or, if we thought him a stranger to these languages, we might appeal to an object producing the idea, by saying it denoted the color we observe in snow or milk. But this is by no means a definition of the word, exciting a new idea in his understanding; but merely a contrivance to remind him of a known idea, and teach him its connexion with the established name. For if the ideas after which he enquires have never yet been raised in his mind; as suppose one, who had seen no other colors than black and white, should ask the meaning of the word scarlet; it is easy to perceive, that it would be no more possible to make him comprehend it by words, or by a definition, than to introduce the same perception into the imagination of a man born blind. The only method in this case is, to present some object, by looking at which the perception itself may be excited; and thus he will learn both the name and the idea together. 5. But how comes it to pass, that men agree in the names of their simple ideas, seeing they cannot view the perceptions in one another's minds, nor make known these perceptions by words to others? The effect is produced by experience and observation. Thus finding, for instance, that the name of heat is annexed to that sensation which men feel when they approach the fire, I make it also the sign of the sensation excited in me by such an approach, nor have any doubt but it denotes the same perception in my mind as in theirs. For we are naturally led to imagine, that the same objects operate alike upon the organs of the human body, and produce a uniformity of sensations. No man fancies that the idea raised in him by the taste of sugar, and which he calls sweetness, differs from that excited in another by the like means; or that wormwood, to whose relish he has given the epithet bitter, produces in another the sensation which he denotes by the word sweet. Presuming, therefore, upon this conformity of perceptions, when they arise from the same objects, we easily agree as to the names of our simple ideas: and if at any time, by a more narrow scrutiny into things, new ideas of this class come in our way, which we choose to express 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. 6. Being in this manner furnished with simple ideas, and the names by which they are expressed, the meaning of terms that stand for complex ideas is easily acquired, because the ideas themselves answering to these terms may be conveyed into the mind by definitions. For our complex notions are only certain combinations of simple ideas. When, therefore, these are enumerated, and the manner in which they are united into one conception explained, nothing more is wanting to raise that conception in the understanding; and thus the term denoting it comes of course to be understood. And here it is worth while to reflect a little upon the wisdom and goodness of the Deity, in thus furnishing us with the very aptest means of communicating our thoughts. For were it not so ordered, that we could thus convey our complex ideas to one another by definitions, it would in many cases be impossible to make them known at all. This is apparent in those ideas which are the proper work of the mind. For as they exist 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 description, they must lie for ever hid within our own breasts, and be confined to the narrow limits of a single mind. All the fine scenes that arise from time to time in the poet's fancy, and, by his lively painting, give such entertainment to his readers; were he destitute of this faculty of laying them open to the view of others by words and description, could not extend their influence beyond his own imagination, or give joy to any but himself. 7. There is this additional advantage, in the ability we enjoy of communicating our complex notions by definitions; that as these make by far the largest class of our ideas, and most frequently occur in the progress and improvement |