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examination papers, but most of them are original, and have been expressly constructed with reference to the most important points, and to the usual difficulties of beginners. The miscellaneous collection of Examples at the end of the book is arranged in sets of ten in each set.

Although great care has been taken to ensure accuracy, it can hardly be hoped that a book involving so large an amount of numerical calculation will be free from error. Any corrections or remarks relating to the book will be most thankfully received.



April, 1869.



MENSURATION gives rules for estimating lengths, areas, and volumes.

We shall assume that the beginner in Mensuration is familiar with the elements of Arithmetic, including the process for the extraction of the square root of a number.

We shall also assume that he is familiar with the use of certain convenient symbols, namely that + denotes addition, denotes subtraction, x denotes multiplication, ; denotes division, and denotes the square root.

Some knowledge of Geometry is also necessary; and accordingly the first three Chapters of the book treat of that subject. The beginner should at once read carefully the first Chapter, in which various terms are defined, which we shall have to employ hereafter; he will however probably find that he is already familiar with the meaning of many of these terms from the common use of them. He can then proceed to the fourth and the following Chapters, referring to the second and the third as occasion may require.


T. M.




1. The words point and line are too well known to require any definition ; but a caution must be given with respect to the strict sense in which these words are used in Geometry.

A point is represented in a printed book by a spot of ink, which may be very small, but still has some size; we must not however suppose that a point in Geometry has

any size.

Lines may be straight or curved. A line is represented in a printed book by a band of ink, which may be very narrow, but still has some breadth; we must not however suppose that a line in Geometry has


breadth. 2. The word surface is also in common use. Surfaces may be flat or curved. A flat surface is usually called a plane surface in Geometry. We must not suppose that a surface in Geometry has any thickness.

3. Thus we may say that a point has neither length, breadth, nor thickness ; a line has only length ; a surface has length and breadth. A solid body has length, breadth, and thickness. We shall not consider solid bodies until we arrive at the Fourth Section of the book; in the first Three Sections we shall only consider lines and figures on a plane surface.

4. An angle is the inclination of two straight lines to one another which meet together, but are not in the same straight line.

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