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IN THE STATEMENT, THE CALCULATION, THE DISTRIBUTION,

AND THE ARRANGEMENT

op

QUANTITIES,

LINEAR, SUPERFICIAL, AND SOLID.

BY THOMAS SMITH.

Not harsh, and crabbed, as dull fools suppose ;
But musical as is Apollo's lute.

MILTON.

LONDON:

LONGMAN, REES, ORME, BROWN, GREEN, AND LONGMAN,

PATERNOSTER ROW.

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LIVERPOOL : PRINTED BY WILLMER AND SMITH. PREFACE,

IN A CONVERSATION BETWEEN A FRIEND AND THE AUTHOR.

FRIEND.-And who, do you think, wishes to know anything about Evolution and Involution ; how many persons are there, think you, to whom such matters can ever be of practical use ?

AUTHOR. — But few, comparatively, I am aware. But all men wish to learn to think, to reason; and I have selected this subject, have availed myself of the rich vein of materials which it affords, for the purpose of winning the mind into a delightful and profitable exercise of its reasoning powers.

FRIEND. — You deem this subject, then, a favourable one to the accomplishment of such a purpose ?

AUTHOR. - I am not singular, as you are aware, in deeming the study of Mathematical learning favourable to the improvement of the mind. This study having long been recommended, by some of the most esteemed authorities in affairs of this nature.

FRIEND. - This appears strange from you! You, who, in your “ Lessons on Arithmetic," so earnestly caution your Pupil against the pursuit of this very study of Mathematics.

AUTHOR. — Appear strange, it may, if you do not attend to the distinction, which, in the passage to which you allude, I have, as I think, pretty clearly, drawn, between the useful, and the useless ; between what you must allow me to term, the substantial, and the merely abstract. Against the pursuit of abstract mathematics, and, indeed, against abstractions of all sorts, I would caution the student. It is not philosophy, but dreaming. I have, more than once in my life, in my desire to improve myself by the study of this science, entered upon the confines of abstraction, but have withdrawn, on finding that, like indulgence in mere reverie, the mind was sensibly sinking under the attempt.

FRIEND. —Then you do not indulge in abstractions, in this work?

AUTHOR.—In this, nor in any. Philosophy consists, or it merits not the name ; philosophy, like common sense, consists in a love of knowledge that is useful. It is a knowledge of substantial things; of Nature, in her various regions; and of the Laws by which She is governed. It is true, that learning, once deemed abstract, speculations once deemed useless, have not infrequently been found to have useful applications. So that we must not be very rigid in our abstinence from the dreams of science. However, I have here followed no abstractions. I have traced the operations of numbers to the verge, I have stopped at the boundary which stands between numbers as they relate, or as they refer, to matter, and the abstract quantities of Mathematicians. In short, I have offered to the consideration of the reader, figures of real and tangible quantities, as the subject of every operation and inquiry.

FRIEND. — Your chief purpose, you say, in the work, is to win the mind to a pleasing, and profitable exercise of its powers ?

AUTHOR.And thereby to strengthen and improve it. It is not for me to say how far I think that I have succeeded in my purpose, nor to speak of any other quality of the work. I found this branch of the science of numbers replete with matter, itself interesting and useful, but far more valuable, as a medium for the exhibition and illustration of effective methods of analysis, and of reasoning; valuable as a means of training the mind, without the irksomeness of dry and repulsive rules, into a vigorous and healthful exercise, and into a taste for the Principles of science.

Besides these, however, I presume to hope that the work will be useful in other ways. I have intended to make the Book complete, by treating of those few rules of Arithmetic to which, on quitting their schools, gentlemen, and professional men, can be supposed to have occasion to refer. And then, I presume, also, that the Book will prove to be a short and easy introduction to Mensuration, Surveying, and Gauging.

EVOLUTION, &c.

INTRODUCTION.

rs.

FAMILIARLY acquainted with the first four rules of arithmetic, the very rudiments of the most ordinary education, the enquiring student will proceed without difficulty into the study of this more engaging department of the science of numbers. And if, in those four rules we include the addition, subtraction, multiplication, and division of compound numbers, a knowledge of these, joined to the few rules developed in the following pages, will most probably comprehend all the arithmetic that a gentleman or man of liberal pursuits may require.

The Terms, and the Signs or marks employed throughout the work, are defined or explained as they severally arise. But, for obvious reasons, a list of them in a tabular form is desirable. Such a list will be found at the end of the work. And to make it more useful, some few other terms and marks, immediately connected with the object of the work, and used by other authors, will be comprehended in the Table. And this will be followed by Tables of English weights and measures.

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