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In preparing the following work, two objects have been kept constantly in view. First, I have endeavoured to bring the essential principles of Geometry within a small compass; and, secondly, to make their connexion easy to be understood. That such a book is wanted, I am convinced from personal experience. The works of Euclid and Legendre, the two most generally studied in New-England, though each is nearly perfect in its kind, are, for that very reason, suited only to the highest seminaries of learning. They cost too much and they require too much time, to be generally studied in academies and schools. More over they are too abstruse and difficult for the comprehension of very young pupils. All this is a necessary consequence of their fulness and perfection, as treatises on this branch of Mathematics. They necessarily contain many propositions, which are not requisite for the understanding of subsequent branches, such as Trigonometry and Conic Sections; and which are not made use of in the more important practical applications, such as Mensuration, Surveying and Navigation. To study them would be an excellent discipline for the mind, if there were time; but this detains the pupil too long from the subsequent higher branches, which afford an equally salutary discipline for the mind, and, in addition to this, are absolutely essential to a complete practical education.

Under these impressions, I have omitted all such propositions as are not absolutely necessary for the understanding of the subsequent parts of a mathematical course. I have

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condensed those which I have admitted, as much as was compatible with clearness and perspicuity, that the book might be small and consequen. cheap. I have placed the problems immediately after the theorems upon which they depend, that this dependence might always be readily perceived. I have avoided the general use of the technical terms, problent, theorem, corollary, scholium and axiom, from a conviction that they confuse rather than assist young minds; and have used instead of them, the general term proposition. With regard to definitions, I have, for the most part, deferred giving them, until the magnitudes or figures defined were to be immediately considered, believing that in this way they would be more readily understood and remembered. Whenever I have ventured to depart from the definitions in common use, as in the case of à straight line and of parallel lines, it has been done, not for the sake of being original, but solely with a view to greater simplicity; remembering that the work was for youth and not for adepts. The same remark applies to those demonstrations which are believed to he original, such as the equality of the angles formed by parallel lines meeting a straight line ; and the approximate ratio of the circumference of a circle to its diameter: also several of the properties of a triangle by inscribing it in a circle.

The division of the work into three sections, is founded in the nature of the subject. Extension, or the space which matter occupies, has three dimensions, length, breadth, and thickness. These may be considered separately or in connexion. When we consider length alone, its representative is a line. Hence the first section treats of lines and their relations. When we consider length and breadth together, or length in two ways, their representative is a surface. . Hence the second section treats of surfaces. Lastly when we consider length, breadth, and thickness together, or length in three ways, their representative is a solid. Hence the third section treats of solids. The appendix is not designed to give a complete view of the applications of geometry to practical purposes, for this would require a separate volume ; but only to give the pupil a general notion of the uses of geometry, by some of the most important particular cases. Questions are placed at the end of the whole, because it is believed they will assist young pupils in reviewing. Those propositions and definitions

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which are thought proper to be committed to memory by the pupil, are printed in Italics and separated from the context by a dash at the beginning and end.

It is proper here to observe that the circle is uniformly treated in the following work, as a regular polygon of an infinite number of sides. This has done more than all other expedients, to reduce the dimensions of the work, without diminishing the number of results. If this principle had not been introduced, and the properties of the circle and figures depending upon it, had been demonstrated by the usual method of a reductio ad absurdum, at least thirty pages more would have been necessary to obtain the same results as are here obtained. This appeared to be a sufficient reason for introducing it.

Under the impression that every student, who is at all inquisitive or curious, must desire to know something of the history of geometry, its origin and progress are briefly traced in the Introduction. If the student should read this before studying the body of the work, it is recommended that he read it again, after he has finished the course of demonstration,

I shall make but one observation more. This work is prepared for young pupils, and does not profess to be a complete treatise on all the elements of Geometry. If, therefore it be honoured with criticism, it is but just that these things should be kept in mind. Its pretensions are humble; and that it has many faults, no one can be more sensible than

THE AUTHOR. Round Hill, Northampton, Feb. 2, 1829.

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INTRODUCTION,

CONTAINING

A brief history of Geometry.

GEOMETRY takes its name from two Greek words signifying the measuring of land, this being the first purpose to which it was applied. It is generally supposed to have originated in Egypt, and to have owed its invention to the necessity of determining anew every year, the land-marks which designated the share of land belonging to each proprietor, when the annual iņundations of the Nile had obliterated or removed them. This however is conjecture. But it is known with certainty, that the Egyptians had some little knowledge of the first principles of Geometry.

The scanty knowledge of the Egyptians was brought into Greece by Thales the philosopher, about 640 years before Christ; and there, geometry grew up, from a few scattered elements, into that exact and beautiful science which it now is. While in Egypt, it is said that Thales learned enough of Geometry to enable him to measure the heights of the pyramids by means of their shadows, and to ascertain the distance of vessels remote from the shore. Upon his return to Greece, he not only encouraged the study among his countrymen, but made some important discoveries himself. He first found out that all the angles inscribed in a semicircle are right angles, and was so delighted with the discovery that he made a sacrifice to the Muses.

Soon after Thales came Anaxagoras. He was imprisoned on account of his opinions respecting astronomy, and during his confinement employed himself in attempting to

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