cal course. I have condensed those which I have admitted, as much as was compatible with clearness and perspicuity, that the book might be small and consequently 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, problem, 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 a 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 be 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 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.

PREFACE TO THE SECOND EDITION.

THE speedy sale of the first edition of this work, has justified the belief of the author, that such a work was wanted. In placing the present edition before the public, it is only necessary to notice some slight changes, which have been made. The technical terms, problem, theorem, corollary, scholium, and axiom, which were not adopted in the first edition, are used in this. The form of reasoning is rendered rather more synthetic, by uniformly placing the propositions or definitions at the commencement of the sections. The proportions are placed in lines by themselves, that their connection may more readily be perceived by the eye. Some demonstrations have been enlarged for the sake of greater clearness. In the figures on the plates, some of the lines are made larger than the rest, for the purpose of rendering the order of construction more obvious. On the whole, it is believed that these changes will be found to be improvements. They have been suggested, not only by the experience of the author, in teaching from the book, but also by several friends, whose intelligence gives weight to their opinion.

Cambridge, Nov. 4th, 1829.

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 inundations 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 camé Anaxagoras. He was imprisoned on account of his opinions respecting astrono my, and during his confinement employed himself in attempting to find the quadrature of the circle, or the

ratio of the circumference to the diameter. It is remarkable that the first attempt to solve the most famous problem in Geometry, should have been a prison

amusement.

Pythagoras was born about 580 years before Christ. After having travelled into Egypt and India, he gave himself up to the study of geometry with wonderful ardour and success. It was he who discovered that the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. To express his joy and gratitude for this great discovery, we are told that he sacrificed one hundred oxen to the Muses. He also discovered that the circle is the greatest of all figures of the same perimeter.

The first man who digested the Elements of Geometry into a regular treatise, was Hippocrates, who lived soon after Pythagoras. This work has not come down to us; but history informs us, respecting Hippocrates, that he was originally a merchant; that he visited Athens on business, and was one day tempted by mere curiosity to visit the schools of philosophy; that he there heard of geometry for the first time, and was so charmed that he renounced all other pursuits and gave his whole mind to this. No wonder that with such fervent devotion to the study, he soon became one of the best geometers of his time.

We now come to the celebrated school of Plato, in which, during the life of its founder, geometry formed the basis of instruction. It is delightful to think of the enthusiasm which so great a man as Plato felt for this study. He placed an inscription over the door of his school, saying, "let no one who is ignorant of Geometry enter here." He also declared to his disciples his belief, that the mind of the Deity was constantly occupied with the truths of geometry. For some time the disciples of Plato shared the enthusiasm of their master, and accordingly from them geometry received immense accessions. Leon, a pupil of one of Plato's disciples, arranged, for the second time, the elements of Geometry into a regular treatise. And Eudoxus, an intimate friend of Plato, found out the solidity of a pyramid and It is also supposed that he was the inventor of the theory of geometrical proportion, as presented by Euclid, of whom we are next to speak.

cone.