The following Treatise on Elementary Arithmetic has been designed as the first of a continuous series of Treatises on those sciences which are usually comprehended under the somewhat indistinct name MATHEMATICS.

During a long experience as preceptor in the higher departments of mathematics, the Author of this work has observed that almost all the difficulties which the student encounters are traceable to an imperfect acquaintance with arithmetic. It seems as if this subject were never regarded as having in it anything intellectual. Arithmetic is considered as a kind of legerdemain, a talsamic contrivance, by means of which results are to be obtained in some occult manner, into the nature of which the student is forbidden to inquire; hence many a one, as he advances in life, finds himself compelled to resume the study of the principles of arithmetic, and discovers that all along he has been working in the dark. Now, truly, there is hardly any branch of human knowledge which affords more scope for intellectual effort, or presents a more invigorating field for mental exercise than the science of number. It has therefore been the Author's aim to prepare a text-book which should call mind, not memory, into exercise, and from which all mere dogmatism should be scrupulously excluded.

In arranging it, he has endeavoured to explain the reasons of the various operations by an examination of the nature of the questions which give rise to them, and in this manner has sought to prepare the student for understanding their applications to the higher branches of science. The gradual formation of systematic numeration has been traced, and the operations of palpable arithmetic have been taught in order to reflect a clearer light on the true nature of our figurate processes. The few lines that are devoted to the explanation of the ancient Greek and Arabic Notations may not be unacceptable to those who study the history and phenomena of the human mind. We are accustomed to designate the ordinary numerals as Arabic. The Arabs themselves call them Rakam Hindi in contradistinction to those described at page 23, which they call Rakam Arabi.

It is hoped that, independently of the arrangement of its parts, this treatise contains enough of new and original matter to prevent it from being regarded as an uncalled-for addition to a class of books already sufficiently numerous. The method of computing from the left hand, which has been practised and taught by the author for thirty years, is now published for the first time. Besides the merit of novelty, this method has the higher merit of great usefulness. A very slight acquaintance with it augments one's power over numbers in an unexpected degree, and the continued practice of it renders computation a pastime. In the ordinary mode of computing we have never occasion to add more than nine to nine, or to take the product of more than nine times nine, and hence the limit of rapidity is soon reached. But when we begin to work from the left hand, every operation adds to our previous experience, and we soon become familiar with large numbers, so much so that the rapidity of mental calculation comes far to exceed the swiftness of the pen.

The multiplication of one large number by another by help of a movable slip of paper, though not to be recommended in actual business, is interesting, and the ability to perform it enables us to follow the operations for shortening the multiplication and division of long decimals.

The subject of prime and composite numbers has been introduced as preparatory to the theory of fractions; and the doctrine of proportion has been reached by means of the method of continued fractions invented by Lord Brouncker. The immense power of this method, the almost unlimited range of its applications, as well as the beautiful simplicity of the idea from which it arose, recommend it to the close attention of every calculator. The doctrine of continued fractions is here divested of its technicalities, and presented in such a form as to be intelligible to beginners; the formation of the successive approximating fractions being deduced without the aid of artificial symbols, perhaps more clearly than with that aid.

Among the minor improvements which have been introduced, the mode of obtaining at once the continued product of several numbers, the plan for shortening division by a number of two or three places, and the arrangement of the work for finding the greatest common divisor of two numbers, may be mentioned; Fourier's Division Ordonnée is also new to the English reader.

It was a matter of anxious deliberation whether the answers to the questions should be printed, and ultimately it was resolved to give these answers in a separate key. The Author, however, most earnestly entreats the student to trust to his own thorough comprehension of the matter, and to the care with which he works. Let him carry the feeling with him, that if his result differ from that given in the key, the key is likely to be wrong: above all things, cultivate self-reliance. In very many cases the manner in which the result is worked out is of more importance than the mere obtaining of it; and in some cases, for obvious reasons, the answers ought not to be given at all.

The Author is in hopes of speedily laying before the public, in continuation, a treatise on the higher arithmetic, in which the doctrines of Powers, Roots, and Logarithms, are completely investigated.

Edinburgh, June 1856.

In this Second Volume on Arithmetic an account is given of the doctrines of Powers, Roots, and Logarithms, so far as that can be well done without the aid of general symbols. The Treatise is intended not merely as a Text-Book on these subjects, but also as an introduction to Algebra: indeed, if we adopt the original meaning of the Arab words (ylim ul jibr, the science of powers), the present work forms the first, and not the least important chapter of that science.

To those who have only considered the subjects of direct, inverse, and fractional powers, and the cognate subject of Logarithms, in the light which the modern notation throws upon them, it may seem vain to attempt to explain these matters with no aid beyond that of our ordinary numeral notation; but an examination of the following pages may serve to show that the mind does not require the aid of artificial symbols to detect and appreciate even recondite properties of numbers; and the Author flatters himself that he has brought the leading properties of Logarithms completely within the bounds of arithmetic.

This has been accomplished by the help of a new method for extracting all roots, of which the previously well-known processes for extracting the square and cube roots are the two simplest cases. This method was given, by implication, in a small treatise "On the Solution of Algebraic Equations of all Orders, Edinburgh, 1829;" it is here simplified and adapted to ordinary arithmetic. By its means we obtain the root, and all the inferior powers of the root, with great rapidity; the simplicity of the arrangement being the better seen, the higher the order of the root which we extract.

In the actual construction of the first Decimal Logarithmic Tables, Briggs used the repeated extraction of the square root, until the results exceeded unit by fractions so small as to render the excesses sensibly proportional to the exponents. Had he known the method of extracting fifth roots, his labour would have been greatly lessened. The principle used by Briggs is, in essence, identic with that adopted by Dodson inl the construction of his Anti-Logarithmic Canon, and with that which is followed at page 119; the only difference is, that the ability to extract fifth roots has given us a much greater command of the subject than either Dodson or Briggs possessed.

The direct computation of the logarithm of a number, that is, in the language of modern algebra, the direct solution of the equation $a_{z} = n$, has not heretofore been obtained; for although the well-known formula

          $z=\Large\frac{(n-1)-\frac 1 2(n-1)^2 +\frac 1 3(n-1)^3-\frac 1 4(n-1)^4+ \text{etc.}}{(a-1)-\frac 1 2(a-1)^2 +\frac 1 3(a-1)^3-\frac 1 4(a-1)^4+ \text{etc.}}$

be a symbolical solution, it is only susceptible of direct application when $a$ and $n$ differ from unit by small fractions. In common logarithms $a - 1$ has the value 9, and $z$ has to be computed indirectly through the intervention of other numbers.

The student of the Higher Algebra will, therefore, be somewhat surprised to find an exceedingly simple and rapid solution, obtained by a train of reasoning which requires only a clear perception of the nature of powers, and which is altogether independent of notation.

This is another to be added to the rapidly accumulating testimonies of the usefulness of Lord Brouncker's continued fractions; for although the algorithm and definition of these fractions have not been employed, the essential idea has been freely used.

These two new processes, viz. the extraction of all roots, and the direct solution of the exponential equation, have enabled the Author to place the whole subject in a clear light, and to complete the Theory of Practical Arithmetic without calling in the dangerous aids of indefinite symbols and arbitrary notation.

In order to prepare the student for following the reasoning to be afterwards used in algebraic investigations, and also for the purpose of fortifying his knowledge of what has already been gone over, a short notice has been added of various Numeration scales. The study of this part of the work may serve to free the mind from those prejudices which are apt to attend the use of a single system, and may lead it to form just and comprehensive views of arithmetic in general.

Edinburgh, March 1857.