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exceedingly useful and important. And in this respect, the art I have undertaken to elucidate is inferior to none, arithmetic orly excepted. Its use in most of the different branches of the Mathematics is so general and extensive, that it may justly be considered as the mother and mistress of all the rest, and the source from whence were derived the various properties and prin ciples to which they owe their existence.

As a testimony of this superior excellence, I need only mention a few of those who have studied and improved it; in which illustrious catalogue we have the names of Euclid, Archimedes, Thales, Anaxagoras, Pythagoras, Plato, Apollonius, Philo, and Ptolemy, amongst the ancients: and Huygens, Wallis, Gregory, Halley, the Bernouillies, Euler, Liebnitz and Newton, amongst the moderns; all of whom applied themselves to particular parts of it, and greatly enlarged and improved the subject. To the latter especially we are indebted for many valuable discoveries in the higher branches of the art; which have not only enhanced its dignity and importance, but rendered the practical application of it more genera and extensive.

The degree of estimation in which the art was held by these, and other eminent characters, will, in general, it is apprehended, be thought a sufficient encomium on its merits. But, for the sake of young people, and those of a confined education, it may not be amiss to give a few more instances of its advantage, and show that its importance in trade and business is not inferior to its dignity as a science. Artificers of almost all denominations are indebted to this invention for the establishment of their several occupations, and the perfection and value of their workmanship. Without its assistance, all the great and noble works of Art would have been imperfect and useless. By this means the architect lays down his plan, and erects his edifice: bridges are built over large rivers; ships are con

structed; and property of all kinds is accurately measured, and justly estimated. In short, most of the elegances and conveniences of life owe their existence to this art, and will be multiplied in proportion as it is well understood, and properly practised.

From this view of the subject, it is hardly to be accounted for, that, in a commercial nation, like our own an art of such general application should have been so greatly neglected. Mechanics of all kinds, it is well known, are but ill acquainted with its principles; and those who have been the best qualified to afford them any assistance, have thought it beneath their attention. Till within a few years past, there could not be found a regular treatise upon this subject in the English language. Some particular branches, it is true, had been greatly cultivated and improved; but these were only to be found in their miscellaneous state, interspersed through a number of large volumes, in the possession of but a few, and in a form and language totally unintelligible to those for whom they were more immediately necessary.

Dr. Hutton was the first person, in this country, who undertook to collect these scattered fragments, and to treat of the subject in a scientific, methodical manner. A small treatise by Hawney, and some others of little note, had indeed been long in the hands of the public; but these were extremely defective, both in matter and method; neither the principles nor practice of the art being properly or clearly explained. Before the publication of the treatise above mentioned, Mr. Robertson's may be considered as the only book, of any value, that could be consulted, either by the artisan or mathematician; and had he given the theory as well as the practice of the art, and divested his rules and examples of their algebraical form, there would have been 119 want of any other elementary treatise. To thcau two writers I am greatlv indebted for many

things in the following pages, and am ready to acknowledge, that I have used an unreserved freedom in se lecting from their works, wherever I found them to answer my purpose. To Dr. Hutton I am particularly obliged, and am so far from desiring to supersede the use of his performance by this publication, that I only wish it to be thought a useful introduction to it. His treatise is excellent in its kind; and had it been as well calculated for the use of the uninformed Artist as it is for the Mathematician, the following compendium had certainly never been published.

The method I have observed in composing this work, is that which was used in the Scholar's Guide to Arithmetic;" and, as my object has been to facilitate the acquirement of the same kind of useful knowledge, I am not without hopes of its being received with equal candor and approbation.

In school-books, and those designed for the use of learners, it has always appeared to me, that plain and concise rules, with proper exercises, are entirely sufficient for the purpose. In science, as well as in morals, example will ever enforce and illustrate precept; for this reason, an operation, wrought out at length, will be found of more service to beginners than all the tedious directions and observations that can possibly be given them. From constant experience I have been confirmed in this idea, and it is in pursuance of it that I have formed the plan of this publication. I have not been ambitious of adding much new matter to the subject: but only to arrange and methodize it in a manner more easy and rational than had been done before.

The text part of the work contains the rules in words at length, with examples to exercise them; and, in order that the learner may not be perplexed and interrupted in his progress, the remarks and demonstrations are confined to the notes, and may be consulted or not, as shall be thought necessary. To those who would

wish not to take things upon trust, but to be acquainted with the grounds and rationale of the operations they perform, they will be found extremely serviceable: and for this purpose I have endeavored to make them as easy as the nature of the subject would admit. But they can be consulted only by such as have made a previous acquaintance with several other branches of mathematical learning.

Some of the most difficult rules relating to the surfaces of solids, &c. could not be conveniently given, but by means of algebraical theorems; and as this was foreign to my purpose, I have not scrupled to omit them; being well persuaded that what is done upon that head will be fully sufficient to answer most practical purposes. In the Practical Geometry, likewise, which is prefixed to this treatise, such problems only are introduced as were known to be most intimately connected with the subject. And as this part of the work is a proper and necessary introduction to the rest, I have spared no pains in making it as clear and intelligible as possible.

Upon the whole, I have endeavored to consult the wants of the learner, more than those of the man of science. And if I have succeeded in this respect, my purpose is answered. I have not sought for reputation as a mathematician, but only to be useful as a tutor.

N. B. The favorable reception this work has met with, has induced me in this edition to make such alterations and additions as have since occurred to me, and which are such as I hope will render it still more acceptable to the public. Royal Academy, Woolwich,

July 14, 1807.


The favourable reception and great demand for BONNYCAS TLE'S MENSURATION and PRACTICAL GEOMETRY, since its first publication in this country, induced me to publish the present edition, which contains not only the whole of that valuable work, but all that is most useful in Hutton, Hawney, Ingram, and other modern works on the same subject.

To this edition is also added an article on MECHANICS and Dynamics, containing the principal problems in Brunton's Mechanics :—that is, Falling Bodies ; the Pendulum ; the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw, which are usually called the six Mechanica. Powers ; Velocity of Wheei's ; Steam Engine ; Water Wheels ; and Pumps.

JAMES RYAN New York, Oct. 1st, 1833.

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