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increasing the scope of the work, its size has been kept down by such an arrangement of the problems and theorems as secures the simplest demonstrations.

Part I. treats of Plane Geometry. The order of propositions here adopted, seems to accomplish, as fully as possible, the two ends of keeping practice always in advance of theory, and removing difficulties, as much as strict logic will permit, from the path of the beginner. These ends gained, the introduction to the science becomes interesting and suggestive. Analyses of the proof, showing at a glance the relations of the different parts of the demonstration, are given at the close of some of the propositions ; thus the student has the choice of two statements, which will assist each other in giving a correct understanding of the methods. At the end of each book is a collection of exercises for original investigation and practical application. They are so explained that the average student should be able to solve them unaided--not so difficult as to cause him to give up in despair, nor so easy as' to be of no interest or value. A collection of rules for the mensuration of plane surfaces, and a number of examples in them, complete Part I. This part is bound separately for the convenience of those who desire such a limited course.

Part II. contains three books, treating respectively of the geometry of planes, of solids, and of spherical geometry, with accompanying exercises. The rules for the measurement of geometrical solids are collected; a few additional rules in mensuration, not previously referred to, are proved, and numerous examples given.

PART III. contains an introduction to Modern Geometry, the name usually given to the discoveries in pure geometrical science, made since the advance in this direction was stayed, for a time, by the brilliant prospects opened by the Analytical Geometry of DESCARTES. For much of this part the author is indebted to the TRAITÉ DE GÉOMÉTRIE of ROUCHÉ ET DE COMBEROUSSE.

INTRODUCTION.

EXPLANATION OF TERMS.

The study of form is the basis of Geometry. A cubic foot of matter may be in the shape of a ball, or of a cubical block, or it may be irregular. When thus regarding only the amount of material, no attention is paid to the outline; Geometry, however, considers the outline to the exclusion of the amount of matter which it encloses.

A geometrical sphere is not a sphere of iron or wood, but sphere of empty space. It is therefore an imaginary solid, which cannot be perceived by the senses, and for which we must use some representative, as a ball or a diagram, in order to describe it. It is a type of one class of geometrical magnitudes-solids.

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These ideal portions of space are bounded by surfaces without thickness. Here, again, Geometry deals with the form of the surface. The surface may be flat, so that a straightedged ruler, in whatever direction it be laid, will touch along its length, or it may be curved. It may, if flat, be limited by straight lines or curved, by lines of equal or unequal length, etc.

If, now, we suppose the edge of the ruler to be without breadth, we obtain an idea of a new kind of magnitude-a geometrical line. As solids are bounded by surfaces, so surfaces are bounded by lines. Lines are also imaginary, having neither breadth nor thickness. We use marks to represent

them in our books. The form of the line is again important, whether it be straight or curved, long or short. These remarks

prepare

the
way

for the more concise definitions which follow.

one.

1. Geometry is that science which treats of the properties, relations and measurement of magnitudes.

Any solid may be considered as having three dimensionslength, breadth and thickness. A geometrical solid is the portion of space enclosed within the boundaries of a physical solid. These boundaries are surfaces; the boundaries of surfaces are lines, and lines are limited by points. The term magnitudes applies to solids, surfaces and lines. A solid has extension in three directions, a surface in two, and a line in

A point has no magnitude. 2. A proposition is a general statement. It may be a theorem, problem, axiom or postulate.

3. A theorem is a statement which it is required to prove. 4. A problem is a question which it is required to solve.

5. A demonstration is the course of reasoning by which the theorem is proved or problem solved.

A theorem may be proved either directly or by showing that an absurdity would result if it were supposed untrue.

6. An axiom is a self-evident proposition requiring no proof.

Such as: The whole is greater than a part.

7. A postulate is something to be done which is so simple that no one will hesitate to allow it.

Such as: Two points may be joined by a straight line.

8. A corollary is a consequence drawn from a preceding proposition.

9. A scholium is a remark made upon a preceding proposition, showing its limits or application.

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