PREFACE. THE present volume of the "CIRCLE OF THE SCIENCES" contains a series of Treatises upon Elementary Mathematics. In this we comprise Arithmetic-including Algebra, Geometry, and Trigonometry; and of each of these we propose to say a few words. Arithmetic is the science of numbers, and therefore treats of the various combinations of which numbers are susceptible, and of the relations existing between symbols that express numbers. The symbols employed to denote numbers may be either particular, as 5.6.7..., each of which expresses a certain determinate number; or general, as a. b. c.... each of which may express numbers, but not necessarily the same number in two different operations. In the former case the science is termed Arithmetic, in the restricted sense which is generally given to that word; in the latter case it is termed Algebra, or, more accurately, Arithmetical Algebra. Considered speculatively, the latter is antecedent to the former, the rules of Arithmetic being founded, as special cases, on the demonstrations of Algebra; and in our present volume the treatises on Arithmetic and Algebra may, in some degree, be considered as of corresponding scope; the one treating of certain classes of questions concerning numbers, by means of rules and particular symbols; the latter treating of similar classes of questions, by means of demonstrations and general symbols. The science of Algebra admits of many developments besides those contained in the treatises above referred to. Some account of these generalizations will be found in the treatise on Series and Logarithms, of which a word must be said. This treatise consists of two parts,-an algebraical part, discussing the properties of certain Series, and an arithmetical part, containing an exposition of the mode of calculating Logarithms, and rules for using them when calculated. In this treatise the speculative order is observed, the rules of the arithmetic of Logarithms being deduced from algebraical demonstrations. The same order could not be observed in the earlier part of the volume, since a man's mind cannot entertain speculative views of the science of numbers till it has obtained some familiarity with their more elementary combinations; so that it is necessary to teach Arithmetic empirically. It will be observed, that in discussing the question of Series it has been no further dealt with than was necessary for treating the subject of Logarithms with sufficient fulness. This restriction was necessary, to prevent the discussion of the nature of Series from running to an interminable length; for, strictly speaking, the subject of Series is endless. M. De Morgan very justly observes, that "The theory of Series is both difficult and incomplete. So far from being an isolated branch of Algebra, it is an infinite subject, in which every question answered will point out questions to ask." In regard, then, to the arithmetical part of this volume, we may say that it contains Fractional, Decimal, and Logarithmic Arithmetic, with a full description of the principles on which they rest. The treatise of Geometry is founded on Euclid's Elements; the first Four and the Sixth Books, together with the Tract on Planes, being very nearly the same as in Euclid's treatise. The Fifth Book of Euclid's Elements has been replaced by a tract on Proportion, the object of which Mr. Young has fully explained. It is to be observed that Arithmetic, and the science of Geometry as treated by Euclid, differ essentially; since the conception of equality employed in Geometry is quite different from the conception of equality employed in Arithmetic. In the former case two magnitudes are considered equal which can be so adjusted as to coincide, or fill the same space; in the latter, two magnitudes are considered as equal which contain the same number of units. In consequence, these two sciences admit of entirely independent development. To what extent this independent development might be carried is doubtful; but certainly to a far greater extent than is usual in our treatises on Geometry; as it is found that the more difficult questions in Geometry are solved with greater ease by employing algebraical symbols to represent geometrical magnitudes, than by the process of reasoning conducted after Euclid's method. In the present volume the tract on Proportion, like most of the propositions in the treatise on Mensuration, is an instance of this application of Algebra to the discussion of geometrical questions. The difference between Geometry and Algebra may be very conveniently expressed by saying, that Geometry is a science of construction; Algebra (or Arithmetic) of calculation. Thus, when certain sides and angles of a triangle are given, we may by rule and compass construct that triangle; whereas, if we treated the same question algebraically, we should, after representing the sides and angles numerically, calculate the remaining sides and angles. And it is to be observed, that a determination by the latter means is very far more accurate than by the former. In obtaining the means for such process of calculation there are considerable difficulties to be overcome in arriving at a satisfactory mode of measuring angles. The mode actually adopted is to measure angles by certain straight lines, or ratios of straight lines, related in a certain fixed manner to the angles. These are called the sines, tangents, &c. of the angles. Their relations and axes are discussed in the treatise on Plane Trigonometry, which contains:—1. A view of the relations between the sines, &c., of the same and of different angles, which part of the subject is often called the arithmetic of sines, and the expressions for the chief relations between the sides and angles of triangles :— 2. A somewhat full view is given of the mode of constructing tables that give for fixed intervals, e.g. of l′ or of 10", the sines, &c., of every angle from 0° up to 90°, and the logarithms of those sines, &c; this part of the subject being completely analogous to the treatise on Series and Logarithms :-3. A full account of the application of the formulas of 1, by means of tables calculated as explained in 2, to the actual calculation, from certain data, of the sides and angles of triangles. The treatise on Mensuration may be regarded as an Appendix to that on Plane Trigonometry. It contains a considerable variety of questions on the determination of heights and distances, of areas, of surfaces, and contents of solids-of nearly all such as are likely to occur in practice. The results in this treatise are always given as formulas, which, it is thought, are more easily remembered, and more readily reduced to numbers than rules—though in several cases, where a rule seemed to possess any advantage, it is stated, as well as the formula on which it is founded; e. g., in the case of the area of a surface bounded by an irregular curve. The treatises on Spherical Geometry, Spherical Trigonometry, and Practical Geometry, scarcely require any special mention,-the authors of the respective articles having stated everything that appeared essential. The present volume, treating only of Elementary Mathematics, contains no general discussion of questions involving the idea of a limit, or any general investigation of the properties of conic sections; however, in a few cases it has been found necessary to overstep the boundary thus imposed on the writers. Wherever any reasoning about a limiting value is introduced, it is of so simple a kind as to present no serious difficulty. Moreover, the general conception of a limit has been stated explicitly in page 330. Certain questions depending on conic sections are sometimes of practical importance ; and some of the determinations in the treatise on Mensuration, presume a knowledge of their more elementary properties. These properties have been proved at the end of Mr. Jardine's Treatise on Practical Geometry, where they occur as demonstrations of the constructions for which he gives rules. They also supply everything that is required for the complete understanding of pages 376 and 377, and of pages 398 and 399. Before bringing our Preface to an end, we must offer a few words of advice to our readers who are inclined to study Mathematics earnestly. The reader will find several hints given in the body of the work which he will find useful, as in the introduction to Arithmetic (page 1), the remarks on Euclid's First Book (page 68), and elsewhere. We may here observe, that those parts of the subjects which are distinct problems, as distinguished from those which are instruments of further investigation, do not require so much labour in the acquiring as the latter. The parts of the present volume which the student may look upon as instruments of investigation, without a thorough knowledge of which 弗 further advances in Mathematical Science are simply impossible, are these :-Fractional, Decimal, and Logarithmic Arithmetic. The Elementary processes of Algebra, the first part of Trigonometry (pages 292 to 329), and the six books of Euclid's Geometry, the earnest student ought not merely to understand, but to be thoroughly familiar with—as familiar as he probably is with the common operations of Arithmetic. He will find, in these parts of the subject, a variety of exercises, all of which we strongly recommend him to perform, and to test his knowledge of the text by writing out from memory—and that more than once-the substance of articles above indicated. This may seem rather stern counsel, but it is necessary. Self-examination by writing is the only way in which real knowledge of any kind can be tested; and it is specially needed in Mathematical Science, in which the least vagueness is not partial knowledge but ignorance. In regard to the other parts of the volume-that is, in the exposition of the mode of constructing tables of Logarithms-it is not so essential that the student should always have that ready at hand to refer to on a moment's notice. It is sufficient that he do not leave it till he thoroughly understands the method of investigation. It is also to be observed, that the student should always make a point of forming a distinct conception of every proposition he reads; otherwise he can never really understand his subject. Perhaps one of the most valuable results of Mathematical studies is, that they compel the student to form clear conceptions of things that can be clearly conceived; and from this circumstance they derive their name of Mathematical, that is, disciplinary studies. The value of the mental training derived from obtaining a thorough mastery of even as much as the first six books of Euclid, can hardly be over-rated. Compared with this, the practical applications of the science, which are themselves by no means unimportant, are of trifling value. Of course, to obtain this benefit the student's powers must be employed actively. He must think, as well as attend; and must exercise care and judgment in ascertaining whether he has really understood what he has read. There are comparatively few people ready to undergo this labour, and this circumstance at one time made us feel doubtful whether the mathemathical volumes of the "CIRCLE OF THE SCIENCES " would meet with so favourable a reception as some other departments of our work. The increased sale of the later numbers, however, has proved our fears to be unfounded, and seems to show that we have a large number of readers who are ready to accompany us through the more abstruse sciences. AMEN CORNER, October, 1854. THE EDITOR. |