dx 2n 2n The two first terms d+ of this series, being sufficient for the construction of a canon of logarithms, even to 14 places of figures, provided the number, whose logarithm is to be found, be less than 1000; which cannot be very troublesome, because x is either 1 or 2: yet it is not necessary to interpolate all the places by help of this rule, since the logarithms of numbers, which are produced by the multiplication or division of the number last found, may be obtained by the numbers whose logarithms were had before, by the addition or subtraction of their logarithms. Moreover, by the difference of their logarithms, and by their second and third differences, if necessary, the void places may be supplied more expeditiously, the rule beforegoing being to be applied only where the continuation of some full places is wanted, in order to obtain these differences. By the same method rules may be found for the intercalation of logarithms, when of three numbers the logarithm of the lesser and of the middle number are given, or of the middle number and the greater; and this although the numbers should not be in arithmetical progression. Also by pursuing the steps of this method, rules may be easily discovered for the construction of artificial sines and tangents, without the help of the natural tables. Thus far the great Newton, who says, in one of his letters to M. Leibnitz, that he was so much delighted with the construction of logarithms, at his first setting out in those studies, that he was ashamed to tell to how many places of figures he had carried them at that time: and this was before the year 1666; because, he says, the plague made him lay aside those studies, and think of other things. Dr. Keil, in his Treatise of Logarithms, at the end of his Commandine's Euclid, gives a series, by means of which may be found easily and expeditiously the logarithms of large numbers. Thus, let z be an odd number, whose logarithm is sought: then shall the numbers 2-1 and z+1 be the difference of the logarithms will be had, even, and accordingly their logarithms, and which let be called y. Therefore, also the logarithm of a number, which is a geometrical mean between 2-1 and 2+1, will be given, riz. equal to half the sum of the logarithms. Now the series y ×+ 42' 1 4 &c. shall be 1 181 13 + + 24 z3 15120 27 25200 equal to the logarithm of the ratio, which the geometrical mean between the numbers %-1 and z+1, has to the arithmetical mean, viz. to the number z. If the number. exceeds 1000, the first term of the series, viz. Y , is sufficient for producing the logarithm to 13 or 14 places of figures, and the second term will give the logarithm to 20 places of figures. But if z be greater than 10000, the first term will exhibit the logarithm to 18 places of figures: and so this series is of great use in filling up the chiliads omitted by Mr. Briggs. For example, it is required to find the logarithm of 20001: the logarithm of 20000 is the same as the logarithm of 2, with the index 4 prefixed to it; and the difference of the logarithms of 20000 and 20001, is the same as the difference of the logarithms of the numbers 10000 and 10001, viz. 0.000043427%, &c. And if this difference be divided by 4 z, or 80004, the quotient shall be y 42 0.000000000542813; and if the logarithm of the geometrical mean, viz. 4.301051709302416 be added to the quotient, the sum will be 4.301051709845250 = the logarithm of 20001. Wherefore it is manifest that to have the logarithm to 14 places of figures, there is no necessity of continuing out the quotient beyond 6 places of figures. But if you have a mind to have the logarithm to 10 places of figures only, the two first figures are enough. And if the logarithms of the numbers above 20000 are to be found by this way, the labour of doing them will mostly consist in setting down the numbers. This series is easily deduced from the consideration of the hyperbolic spaces aforesaid. The first figure of every logarithm towards the left hand, which is separated from the rest by a point, is called the index of that logarithm; because it points out the highest or remotest place of that number from the place of unity in the infinite scale of proportionals towards the left hand: thus, if the index of the logarithm be 1, it shows that its highest place towards the left hand is the tenth place from unity; and therefore all logarithms which have 1 for their index, will be found between the tenth and hundredth place, in the order of numbers. And for the same reason all logarithms which have 2 for their index,will be found between the hundredth and thousandth place in the order of numbers, &c. Whence universally the index or characteristic of any logarithm is always less by one than the number of figures in whole numbers, which answer to the given logarithm; and, in decimals, the index is negative. As all systems of logarithms whatever are composed of similar quantities, it will be easy to form, from any system of logarithms, another system in any given ratio; and consequently to reduce one table of logarithms into another of any given form. For as any one logarithm in the given form is to its correspondent logarithm in another form, so is any other logarithm in the given form to its correspondent logarithm in the re, quired form; and hence we may reduce the logarithms of Lord Neper into the form of Briggs's, and contrarywise, For as 2.302585092, &c. Lord Neper's logarithm of 10, is to 1.0000000000, Mr. Briggs's logarithm of 10; so is any other logarithm in Lord Neper's form to the correspondent tabular logarithm in Mr. Briggs's form: and because the two first numbers constantly remain the same; if Lord Neper's logarithm of any one number be divided by 2.302585, &c. or multiplied by .4342944, &c. the ratio of 1.0000, &c. to 2.50258, &c. as is found by dividing 1.00000, &c. by 2.30258, &c. the quotient in the former, and the product in the latter, will give the correspondent logarithm in Briggs's form, and the contrary. And, after the same manner, the ratio of natural logarithms to that of Briggs's will be found=868588963806. The use and application of LOGARITHMS. It is evident, from what has been said of the construction of logarithms, that addition of logarithms must be the same thing as multiplication in common arithmetic; and subtraction in logarithms the same as division; therefore, in multiplication by logarithms, add the logarithms of the multiplicand and multiplier together, their sum is the logarithm of the product. num. logarithms. Example. Multiplicand.. 8.5 0.9294189 1.0000000 Multiplier..... 10 Product....... 85 1.9294189 And in division, subtract the logarithm of the divisor from the logarithm of the dividend, the remainder is the logarithm of the quotient. num. logarithms. Example. Dividend.. 9712.8 3.9873444 Divisor.... 456 2.6589648 Quotient.. 21.3 1.3283796 LOGARITHM, to find the complement of a. Begin at the left hand, and write down what each figure wants of 9, only what the last significant figure wants of 10; so the complement of the logarithm of 456, viz. 2.6589648, is 7.3410352. In the role of three. Add the logarithms of the second and third terms together, and from the sum subtract the logarithm of the first, the remainder is the logarithm of the fourth. Or, instead of subtracting a logarithm, add its complement, and the result will be the same. LOGARITHMS, to raise powers by. Multiply the logarithm of the number given by the index of the power required, the product will be the logarithm of the power sought. Example. Let the cube of 32 be required by logarithms. The logarithm of 32 1.5051500, which multiplied by 3, is 4.5154500, the logarithm of 32768, the cube of 32. But in raising powers, viz. squaring, cubing, &c. of any decimal fraction by logarithms, it must be observed, that the first significant figure of the power be put so many places below the place of units, as the index of its logarithm wants of 10, 100, &c. multiplied by the index of the power. LOGARITHMS, to extract the roots of powers by. Divide the logarithm of the number by the index of the power, the quotient is the logarithm of the root To find mean proportionals between any Example. Let three mean proportionals be sought, between 106 and 100. Logarithm of 106= 2.0253059 Divide by 4)0.0253059(0.0063264.75 Logarithm of the least term 100 addded LOGIC, the art of reasoning. As the necessities of our existence oblige us to think, and to arrange our thoughts in such a manner as may enable us to communicate with each other, we are habitually impelled towards a conclusion that it is unnecessary to teach reasoning as an art. It is hardly needful to combat this notion by arguments which will easily occur to most men of reflection; and indeed the contrary per suasion was so prevalent in the middle ages, that men seem to have been more occupied with the art, than with the proper use of it. In order to reason well, it is necessary that the nature of our perceptions and ideas, and the notions or conclusions we draw from them, should be well understood. Logic, therefore, is a science of extensive occupation; which has its beginning in the constitution of things, and the processes of the human intellect, and its practical termination in the structure, use, and applica. tion of language. Its objects are no less than the universal acquisition of knowledge, and that mutual communication which con. stitutes a large part of the employment, and is the most distinguishing character of man. The impressions made by external objects upon the senses, are called sensations or ideas of sensation. See IDEOLOGY. The recollection or remembrance of those sensations are simply called ideas. The general notions which are produced in the mind by reflecting upon ideas, have been called 2.0000000 2.0063264.75 2.0126529.5 ideas of reflection; but as they all grow out Logical writers divide ideas into simple Abstraction, or the leaving out parts of ideas or notions; generalization, or classing things together, as possessing the remaining distinctive characters; composition, or the re-assumption of some of the abstracted or 1 rejected ideas, are the voluntary acts of the mind, adopted in order to facilitate the useful process of Comparison. Thus we may abstract from bodies all ideas but those of structure, and divide them into organized and unorganized; or we may take the organized bodies, and call them animals and vegetables; or we may attend to their place of existence, and call them terrestrial, aquatic, volatile, and the like; and many of our most useful propositions will, thus, in all our mental operations, continue equally general and abstracted. In the scientific arrangement of natural objects, philosophers have pursued the course of abstraction, until by rejecting all the ideas capable of affording the distinctive characters of individuals, they arrived at an hypothetical being called substance. Much has been written concerning it; but it will perhaps be attended with the least obscurity to say, that it is supposed to be an independent existence which serves as the basis or support to those properties which are perceived by our senses; or, in the words of logicians, it is the subject of modes and accidents. The modes of substance are those distinguishable objects of sense which might, if separate, produce simple ideas. Thus, softness, fragrance, yellowness, and acidity, are among the modes which co-exist in the subject or substance, lemon. Many distinctions are made in modes. They are called essential or accidental, absolute or relative, &c. The moderns appear to use the words properties of bodies, and powers and laws of nature, with much more distinctness than the earlier logicians did their modes and accidents. Words are intended to be the signs of things, but are very far from being so. If our ideas were adequate representations of the things which cause them, which they are not; if they were not of necessity mutilated by abstraction, and there were not a continual exertion in language to emulate the rapidity of thought, then might words obtain the supposed resemblance. But the boasted extent and perspicuity of the intellect of man proceeds but a little beyond the signs and tones of those inferior animals who are supposed to have no power of conversing. And even if we could vanquish the insuperable difficulties which impede our clear mutual communication, what are the grounds of our knowledge? they are very limited and often fallacious. Knowledge consists in the determination of those modes of surrounding beings which are taken to be permanent, and of those which are observed to vary. The former are chiefly of the nature of quantity and position, and the latter seem resolvable into motion. Mathematical science appears to comprehend the whole of the first; and the latter, which embraces by far the greater part of what concerns our existence and wellbeing, is included in those histories of events upon which we establish our principles of cause and effect. Abstraction, or analysis, can give us very clear notions of the subjects of mathematics; and in these alone it is that we find absolute proof or demonstration. But in all the rest of our knowledge the facts are complex, obscure, and of uncertaine vidence; and the principal, nay the only ground, of our reliance upon our doctrines respecting them, is that our predictions are in many instances verified. Words being constructed and established by mere usage, are not only inadequate and contracted in their use, but equivocal and synonimous; that is to say, one word may be used to denote several distinct and different things; as when we speak of a beam of light, a beam of timber, or the beam of a pair of scales; or, on the contrary, as when we speak of an house, an habitation, or a residence. It must be admitted, however, that there are few synonymes in the practice of those who are masters of a language; because few words are consecrated by usage to precisely the same meaning. Many acute and useful disquisitions have been written upon language and universal grammar. See LANGUAGE. Since our idea of a thing must be composed or made up of all the simple ideas which that thing can produce by our perceptions, and this will for the most part be inadequate; the word, denomination, or name of a thing, must be the sign of that idea, liable to such additional error as may arise from any improper use that may be made of it. And as by abstraction we generalize our ideas and notions, and afterwards comprehend and compare them at our pleasure; so in the construction of lan. guage a like order is followed in words. Thus we may arrange things, from their similitude, under classes more or less abstracted as to their modes, calling these classes by the names of genera and species. And in the names of things, we shall have not only to regard this arrangement; but likewise the appropriation and correct use of the denomination itself. If we had terms for all simple ideas,and were to enumerate in due order all the simple ideas subsisting in a thing, that enumeration would constitute what is called a definition of the thing; and simple ideas would be, as in strictness they are, undefinable. But since all our sensations are complex, the relations of simple ideas with regard to each other, as residing in the same subject, will afford the means of indicating them. Thus, light is that by which the organ of vision is acted upon, and the word is therefore defined or indicated from that organ. Colour is a mode of light perhaps too simple to be defined, but clearly indicable from any natural subject in which it may subsist; as, for example, green is the colour of grass, red is the colour of a rose, and yellow the colour of an orange. Thus, then, the nature of terms, or words, is fixed by definition; a thing for the most part of extreme difficulty, as, from our ignorance of things, and the complexity of the objects comprehended by usage under any term, it can in few cases be done. The arrangement of things is by genera, where the same class of beings agree in a few attributes only; and by species, where they agree in more; and these genera and species may be subordinate to each other in numerous pairs, the genus immediately above each species being called the proximate genus. And from this ordinary arrangement logicians obtain a ready method of defining from the specific differ ence, which, though certainly much less adequate than those of the mathematicians, is nevertheless very useful. That is to say, the genus and the specific difference is held to constitute the definition of the species. Thus, if the words, 1. animal; 2. four-footed ; 3. graminivorous; and, 4. fleece-bearing, be the arrangement of certain beings pos. sessing life, we should define the first genus from the only character left by the abstraction, namely, that it is a being possessing life; ans the first species would be admitted to be well defined by the words four-footed animal (named quadruped); the second, by the words graminivorous quadruped (named cattle); and the third by the words fleece-bearing cattle (named sheep): or we might less conveniently go through the whole series, and call the sheep a fleecebearing, graminivorous, four-footed animal. Logicians also avail themselves in defining, where practicable, of some striking attribute called the essence of a thing. Thus, under the genus, measure, the species bushel, peck, quartern, &c. are essentially distin guished by the respective magnitudes which are capable of being numerically expressed. All our knowledge is contained in propositions, and every proposition consists of three parts. Thus in the proposition, "Snow is white," there are three parts or terms, snow, which is called the subject; is which is called the copula; and white, which is called the predicate. If the proposition agree with the nature of things it is true, if not it is false. All propositions are reducible to this form, though both the subject and predicate may be expressed by many words; but the copula will always be some inflexion of the verb to be, with the word not if the proposition be negative. Propositions which contain either a plurality of predicates or of subjects, or which manifest a compounded nature in either, have been called compound propositions. In the first, however, the proposition seems merely to be a number of propositions conjoined, &c.; in the latter, the form of words may be considered as forming a definition of the words or terms. Thus, "John and Thomas departed," includes the propositions, "John was departing, and Thomas was departing." And again the proposition, "Water frozen in flakes as it falls from the atmosphere is coloured like the powder of pure dry salt," is evidently the same proposition as was first given, excepting that it contains a definition of the word snow taken from its formation, and of the word whiteness from a substance of which it is one of the modes. Our limits will not permit us to enter into the form of propositions from which they are denominated copulative, casual, relative, or disjunctive or modal; as where a proposition itself becomes the subject, or positive, or negative, and so forth. These distinctions are in few cases useful, and in many tedious, trifiling, and deceptive. Truth is determined either intuitively; as when the relation between the predicate and its subject is immediately seen and admitted. So "the whole is equal to all its parts:"-and these simple truths are called axioms :— Or else it is determined demonstratively; so the proposition, "the opposite angles made by right lines crossing each other are equal," is not intuitive, but requires to be demonstrated by a succession of axioms connected together: Or lastly it is determined analogically; upon the probability that what has happened will, in like circumstances, happen |