to Newton by the falling of an apple from a tree in his garden or that the invention of the cotton jenny was suggested to Har greave by the circumstance of a common spinning-wheel continuing in its ordinary motion while in a state of falling to the ground-let him be well assured, that, had the minds of Newton and Hargreave not been previously stored with knowledge, these discoveries never would have been made by them. Apples and spinning wheels had fallen a thousand and a thousand times, but the knowledge necessary to turn these circumstances to good account was first concentrated in the minds of these two illustrious benefactors of mankind.

In Smith's Wealth of Nations it is related that the ingenious apparatus for opening and shutting the valves of the steam engine was introduced by the accident of an idle boy having fastened a brick as a counterweight to the handles which opened and shut the valves, and thus allowed him time to leave the machine and go to play. This simple trick of an idle boy, it is said, gave rise to the apparatus which superseded the constant attendance of a person while the engine was at work. This, however romantic, is not the fact the invention originated in necessity, no doubt, but it was begun and perfected by a thorough mechanic, Mr. H Brighton, about the year 1717.

While we are on this subject we cannot pass over another very common prejudice, which we conceive has a very hurtful tendency on the progress of the young mechanic. We allude to the pride that some men take in boasting that all their knowledge is original; or that they are self-taught. This is, in other words, stating, that no assistance has been taken either from teachers or books; and goes only to prove, that the knowledge of the individual so circumstanced must be very limited indeed. The unassisted exertions of one man must be very feeble, when compared with the collected exertions of the many who have gone before him in the career of discovery. That man must know little of geometry who has not availed himself of the use of Euclid's Elements, or some work of a similar nature; and the Elements of Euclid would have been meager and confined, had he not availed himself of the discoveries of his contemporaries and predecessors. A like remark may be made on the cultivation of ⚫ every department of knowledge; and to those whom we are now

addressing we say-learn from others all that you possibly can, and when you have done so, try to correct and improve what you have obtained. We know of no dishonourable means of acquiring knowledge, and therefore wherever we meet it we are disposed to respect it, even though it should not contain one particle of originality, if such be possible; for it is not easy to conceive how any man should be in possession of useful knowledge, and not make some new application of it; and a new application of an old principle is certainly one constituent of originality. With a knowledge of what others have done, that workman will be less likely to waste his time in enterprises which may ruin him by their failure, or in speculations which are unsupported by the principles of science.

In the museum of the mechanics' class of the university founded by the venerable Anderson of Glasgow, there is preserved the model of a machine to procure a perpetual motion. For the contrivance and execution of this beautiful specimen of workmanship, we are, we believe, indebted to an ingenious clock-maker of Dundee, who has proven himself a master in the use of his tools. But had he been acquainted with the first principles of mechanics, or with the nature and failure of the various attempts which had been made before his time for the same purpose, he would have seen the utter folly of his enterprise, and would have spent the seven years which he occupied in the construction of this truly beautiful model in some more useful employment. These seven years might have been devoted to the construction of timepieces which would have been of infinite service to the commerce and navigation of his country-in guiding the lonely mariner when far away on the billow-in determining the exact distance and direction of the part for which he is bound-whereas, the model of his perpetual motion is preserved in the museum as a lasting monument of this clock-maker's ignorance, perseverance, and handicraft.

It is another common error to suppose that genius alone can make a man a great mechanic, a great chemist, or a great any thing. Some one makes the remark, that every man is more than half humanity; and we do believe that the differences of the degrees of knowledge of different men arise more from thei difference of application than from original differences of capa

city. Let, therefore, the young workman earnestly try to learn and we do assure him that he will make advances which will be proportional to his application.

This book has been written with the view of assisting the young workman in obtaining a knowledge of the calculations connected with machinery. The first part is devoted to such parts of arithmetic as workmen generally require, and in which they are most commonly deficient. Nor is this deficiency to be wondered at, since the school books in our language contain, generally speaking, no explanation of the nature of the rules which they give, and are, moreover, embarrassed with so many divisions and subdivisions, that the mind of the scholar is perfectly perplexed, nor can it lay hold of the great leading principles which pervade the whole system. As this is the great instrument used throughout the book, we have endeavoured to make its use and management easily understood. The examples which we have given are indeed few and simple; but, if carefully considered, they will be found sufficient to establish the principle. The mere habit of calculation cannot be said to constitute a knowledge of arithmetic; it is easily obtained, but is of no avail without the principles. This is well illustrated by an occurrence of but recent date. To construct a set of mathematical tables requires, not only a knowledge of principles, but also immense calculation. M. De Pronney was desired by the government of France to construct a very large set of such tables; a task which would require the labour of a mathematician for many years. But Pronney fell upon an expedient which was every way worthy of a man of science. A change in the fashions of the Parisians had thrown about five hundered wig-makers idle, and Pronney contrived at once to give employment to these barbers, and at the same time to serve the purposes of science. He digested the principles of the calculation of these tables into short and simple rules, and printed forms of them, which he gave into the hands of these workmen, who, in a few months, produced a set of tables, the most correct and extensive that ever has been made. The peruke-makers may, so far as the construction of the tables was concerned, be regarded as mere machines, under the guidance of M. de Pronney. The same principle has been of late years carried to a far greater extent by our countryman, Professor Babbage, who has invented a machine by which logarithms and astrono

mical tables may be calculated and printed with the nos unerring certainty, thus obviating the necessity of employing either calcu lators or compositors. Let not these statements induce you, however, to neglect the practice of calculation; on the contrary, improve yourself in it wherever you can, but be also careful to learn the principle.

In that part devoted to geometry, we have given such information without demonstration as was necessary to the right understanding of the rest of the book; and the like may be said of the conic sections, mensuration, and useful curves. Thus far the book may be said to be a compend of certain branches of the mathematics. It is hoped that the reader, to whom such studies are new, will not be contented to stop here; but will be induced to investigate these subjects in theory; and for such as may be desirous of entering on such a course of study, where there is nothing to be met with but unsophisticated truths connected together by a chain of the most beautiful relations, we intend to offer a few words of well-meant advice as to the order and means of prosecuting such studies.*

In the first place, let the Elements of Euclid be studied so far as the end of the first book, in the course of which it should be borne in mind, that there is nothing really difficult to be met with. The greatest difficulty is, we believe, this, that, to a proposition which is so simple as to be almost self-evident, there is often

* In a very creditable work, recently published, "Stuart's History of the Steam Engine," it is stated that mathematics is not necessary to make a great mechanic, and Watt is cited as an instance. The instance chosen is most unfortunate for the author's assertion. Watt was descended from a family of mathematicians, and inherited in the highest degree the genius of his ancestors. One instance will sufficiently prove this. With a desire to determine what relation the boiling point bore to the pressure of the atmosphere on the surface of the water, he made several experiments with apothecaries' phials, and having found the relation between the pressure and temperature of ebullition, under different circumstances, he laid the temperatures down as abscissæ, and the pres sures as ordinates, and thus found a curve whose equation gave that wer known formula, the equation of the boiling point. No man but a mathe matician of high attainments would have thought of such a method of proceeding. To this we may add, that mechanics is a branch of mathe matics; for, as Sir Isaac Newton has defined it, "mechanics is the geometry of motion."

attached a long demonstration, which is apt to lead the reader to suppose that there is really something mysterious in it, which he does not understand. This proceeds from the fact, that it often requires a greater deal of circumlocution to show the connection of simple propositions with first principles, compared with propositions which are more complex; but we have no hesitation in saying, that if the steps of the propositions are carefully considered, one by one, they will be easily understood, and will lead at last to perfect conviction; for, as Lord Brougham has well observed, "Mathematical language is not only the simplest and most easily understood of any, but the shortest also ;" and Euclid has transmitted to posterity a specimen of the purest mathematical language. Of Euclid's Elements, there are various editions. Those of Simpson and Playfair are generally used in this country, and are deservedly popular. That of Dr. Thomson is a very valuable work, and very correct. But we beg to recommend to the workman the edition of Mr. Robert Wallace, of Glasgow, both for its execution and cheapness. The demonstrations are clear and short; many new propositions are added, and the connection of theory with practice is never omitted where it can be introduced.

When the first book of Euclid has been read, the study of algebra should be commenced, on which subject there are few good treatises to be found. That which we think best is the treatise of Euler, a book which has come from the hand of a master, and is therefore characterized by great simplicity. Another good book is the treatise of Saunderson. Let either of these works, or others if they cannot be had, be read carefully so far as to equations of the second degree. If any one part of this department can be said to be difficult, it is that of powers and roots, which is a subject of the greatest importance; and should, on that account, receive the most careful attention; and, if the treatise of Euler be used, we have no hesitation in saying, that little difficulty will be experienced. It may be necessary to observe, that attention should be paid all along to the intimate connection of arithmetic and algebra, which will tend to the better understanding of them both. Having advanced thus far, Euclid must again be returned to; and, after revising the first book, read on to the sixth inclusive. Occasional revision of the algebra is recommended, and an advancement as far as equations of the