Page:The American Cyclopædia (1879) Volume X.djvu/584

578- LOGARITHMS LOGIC used, and are known as Napierian, natural, or hyperbolic logarithms. In this system the base is the number 2-7182818 + . These logarithms are of great use in the higher mathematies, and in the investigation of many problems in phys- ics. The Napierian logarithm of a number is equal to the common or Briggs logarithm mul- tiplied by 2-3025851, or divided by 0-4342945. The early computers of logarithms carried them to ten places of decimals; but it was soon found that seven places were sufficient for most of the uses of astronomy, navigation, surveying, &c. In fact, five-place logarithms are often sufficient, and, being much more con- venient and portable, should be used except when very great accuracy is required. The theory of logarithms is now taught as a part of liberal education, and is explained in all the treatises on algebra used in our high schools and colleges. Tables of logarithms are always preceded by directions how to use them. For this purpose no knowledge of the theory is required, an acquaintance with the rules of arithmetic being all that is necessary. An ex- cellent collection of five-place logarithms is that attached to " Bowditch's Navigator," and also published separately under the title of "Bowditch's Useful Tables." This contains, besides the tables of logarithms for numbers, log. sines, tangents, &c., also many auxiliary tables useful in navigation and surveying. A good collection of five-place tables by J. Hoiiel (8vo, Paris, 1858) contains also Gauss loga- rithms, so called from the name of their in- ventor. They are numbers by means of which, when the logarithms of two numbers are known, the logarithm of their sum or differ- ence can be found without knowing the num- bers themselves. Thus, suppose we have the log. of a and the log. of 5, but do not know what numbers correspond to these logarithms, and we wish to know the log. of (a + l> With the ordinary tables we should first have to find in the table the number corresponding to log. a, then the number corresponding to log. &, then add the two numbers together and find from the table the log. (a + 5). This process requires three references to the table and one addition. By means of a table of Gauss loga- rithms the same result is reached by one refer- ence to the table and two additions. Among tables of logarithms to seven places of deci- mals may be mentioned Babbage's, which are very accurate. Taylor's tables (large 4to, Lon- don) are very valuable, but difficult to obtain. Shortrede's tables (large 8vo, Edinburgh) con- tain nearly all the tables required in compu- ting ; they are especially designed for military and civil engineers. The tables of Oallet (8vo, Paris) are very good; they contain the loga- rithms of all numbers from 1 to 108,000, with log. sines, tangents, &c., besides tables of Na- pierian logarithms to 20 places of decimals, and short tables of common logarithms to 20 and to 61 places. For log. sines, tangents, &c., Bagay's tables (4to, Paris) are very conve- nient ; they contain log. sines and tangents for every second of the quadrant. Of the tables recently published the most valuable are those of L. Schron, Siebenstellige Logarithmen (12th ed., Brunswick, 1873) ; Bruhns, " A New Man- ual of Logarithms to Seven Places of Deci- mals" (Leipsic, 1870), very beautifully print- ed ; and G. von Vega, " Logarithmic Tables of Numbers and Trigonometrical Functions, re- vised and corrected by Bremiker " (55th ed., Berlin, 1873), which is very accurate and con- venient. The logarithms are given to seven places of decimals. LOGIC (Gr. A<5yof, reason), the science of rea- soning. More strictly and properly, it is the science of deducing ideas or conceptions one from another, and of constructing them into propositions, arguments, and systems. A wide range and great diversity of topics have, how- ever, been included in the various treatises written under the name. Some have under- stood by it an account of the whole mental ac- tivity, and defined it as the art of thinking. Others have made it comprise only a knowl- edge of the first principles, or axioms, from which we reason. Others appear to have held it responsible for the truthfulness of all pro- fessedly logical reasonings and processes. Oth- ers again have regarded it as chiefly or exclu- sively an instrument of invention and discov- ery, and worthless except for the attainment of some new truth. It is now generally held that logic assumes certain first principles or axioms, from which as premises to reason; that it is concerned with the form only of rea- soning or argument, and not at all with the subject matter ; that it is and of necessity must be a purely a priori science, and moreover a hypothetical science, since it neither assumes nor proves as such the reality of anything, does not assert that any objects corresponding to our conceptions do really exist, but only gives results and conclusions based on prem- ises, which are true provided the premises are true. Logic is thus limited to the method of reasoning. Though commonly regarded as consisting of two parts, analytics and method, it is essentially a constructive science ; it ex- plains the way in which theories and systems are constructed from our primary ideas of ob- jects, and it proves and tests, not their truth, but their legitimacy as deductions. In this view it presupposes psychology, which is a sort of natural history of thought, and it is pre- liminary and prerequisite to ontology, the sci- ence of being. Logic begins with ideas. Our ideas of objects are complex wholes, and may be analyzed into conceptions of the known properties of objects. Thus, snow is repre- sented by its properties of whiteness, coldness, &c., and an orange by its color, shape, &c. These properties, or rattier the terms describing them, become predicates which we may affirm of the object. Thus, having analyzed our idea of an orange, we obtain the properties of roundness, &c., and hence may say, "The or-