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PREFACE.

THE present volume is designed as a comprehensive introduction to one of the most important departments of practical mathematics. I have called it an Introduction, chiefly for the purpose of precluding criticism. "MENSURATION" is a term of somewhat ambiguous import. It might without palpable impropriety, be applied to every subject in which numerical measurements are concerned-Surveying, Engineering, Gunnery, Astronomy, &c.—and, accordingly, books bearing the same title as the present may be referred to, in which such matters have a place. But while so wide a field, and so great a variety of topics may not be absolutely forbidden by the label which they bear, yet there is a more circumscribed range, sufficiently defined by general usage and common consent, that is conceived to embrace the essentials of Mensuration, the omission of any of which would deprive a book on that subject of its just claim to the title it assumes.

It is within this range that I have here confined myself. I have endeavoured to give to every subject included in it a very full and complete discussion, and to conduct such discussions in a manner adapted to the capacity of every intelligent schoolboy. The several practical rules are accompanied with their demonstrations, and all these are given in the simplest form I could devise. There are only one or two problems in the more advanced part of the subject in which this plan has been interrupted. In these instances I was unable to contrive satisfactory demonstrations without the aid of the Integral Calculus-a subject which it would be preposterous to introduce into such a book as this. I have, therefore, been compelled, in these cases, to refer to other sources of information. The various demonstrations, though inserted in the text, are generally printed in a smaller type: all, or the more difficult of these, may be postponed or not, according to the degree of previous algebraical attainment on the part of the learner. It is desirable, however, that teachers should impart some amount of geometrical and algebraical knowledge to their pupils before setting them to Mensuration; a weary and mechanical drudgery would thus be converted into an agreeable as well as a profitable intellectual exercise; and a much firmer grasp of the rules would be secured, if they were always based upon the reasons, and not implicitly received upon the dictum of the teacher.

There is one feature which, strange to say, is almost peculiar to the present work. Every rule is illustrated by a practical example or two, exhibited at length, as indeed is generally done in books on this subject; but

the prevailing fault in such illustrations is, that no attention seems to be paid to arithmetical facilities. Every boy who has learnt arithmetic must have had his attention frequently directed to certain simplifications and abridgments which, in special cases, may often be applied with advantage in the way of contracting the work and economising numerical labour. But little or nothing of what he has thus had the pains of learning is ever brought into practical application in books of this kind: a few exceptions there may perhaps be; but in general the numerical operations alluded to involve a very unnecessary expenditure of figures. In the present work I have usually attended to this; all far-fetched "simplifications," as they are miscalled, I have uniformly avoided; but I have equally avoided spoiling what I hoped might prove a good Mensuration, by its bad Arithmetic.

Upon a subject on which so many treatises have been written, but little absolutely new can be expected. Departures, however, from the beaten track are not unfrequent in the present volume. These will be recognised in several of the demonstrations-in the rules for the areas of quadrilaterals, for the circumference of the circle, for the mensuration of the cylinder, &c. But I have been more intent upon perspicuity and scientific accuracy, than upon novelty either of matter or arrangement. Of the things not usually introduced into books on Mensuration, the short sections on Logarithms and Plane Trigonometry, and the Tables in connexion with these subjects, given at the end, will, I think, be especially acceptable to the learner. The Table of Natural Sines and Tangents to five places of Decimals will be found very serviceable. I do not know of any similar work containing such a table. Indeed, tables of Sines and Tangents to five places-beyond which extent they are not required in Mensuration—are not easily to be met with anywhere; those here inserted were prepared, not without some trouble, expressly for this work, from larger tables.

Although the volume is small, both in size and price, yet from the condensation observed in the printing, a considerable extent of matter has been introduced. No work on the subject, of the same bulk, that I am acquainted with, contains so full an exposition of the several topics of inquiry, nor so many examples for the exercise of the learner. It is very likely that in such a large amount of arithmetical computation, a few numerical oversights may have escaped detection; but I think I have nowhere fallen into an error of principle. Should any misprint occur in any of the answers to the examples, it will be corrected in the KEY, where the solutions will all be exhibited at length, accompanied with whatever additional explanations and comments may seem to be necessary for the information and guidance of the otherwise unassisted student.

LONDON, July 1, 1850.

J. R. YOUNG.

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