can be easily understood by learners whose mathematical knowledge does not go much beyond the simplest parts of Arithmetic, in the hope that it may prove useful to teachers and scholars in our elementary schools, and students in our training colleges, as well as to that interesting class of learners who are endeavouring to teach themselves from books, without the aid of a teacher.

H. L

CULHAM, December 1874.



LESSON I.-DEFINITIONS AND EXPLANATIONS. The word Mensuration really means “ measuring.”— If a farmer wishes to know the exact size of a field, or a builder that of a wall, or if a paperhanger wishes to calculate the cost of papering a room, or a painter that of painting the front of a house; if you had to estimate how many gallons of water a tub or cistern would hold, or how many cubic feet of air there are in a certain room-all these useful operations would have to be performed by mensuration, which is the art of calculating dimensions of length, area, and volume.

In studying this subject we shall have to make constant use of certain names, or terms, which some of you doubtless are already familiar with, having had to learn them when commencing the study of Euclid or Geometry. But for the sake of those who may not yet have begun this study we shall insert here a few of the simpler definitions or explanations, and introduce the others, as occasion requires them, farther on in the book.

It ought not to be necessary to remind those who will use this book that accurate definitions are very important, and that therefore, especially in such a study as this, it will be necessary to commit them to memory with great care.

1. A point is that which has no dimensions, i.e., neither length, breadth, nor thickness.

2. A line has length but no breadth.

3. A superficies or surface has length and breadth, as A.

4. A solid body has length, breadth, and thickness, as B.

5. A straight line (or right line) is one e which lies evenly between its extreme points. Sometimes it is said, and quite truly, that a straight line is the shortest distance between two points.

If you look at the water in a pond or cistern, where it is perfectly still, you will notice that its surface is level or horizontal, any straight line on the surface of the water, or parallel to it, is a horizontal line; any line which is perfectly upright on it is a vertical line; lines which are neither horizontal nor perpendicular are oblique.

6. An angle is the corner formed by the meeting of two straight lines, as BA and AC.

This would be called the angle BAC, three letters being used, andthe letter at the corner always placed in the middle; sometimes for the sake of shortness, and where there would be no ambiguity in such a case, it would

be spoken of simply as the angle

-C at A. Angles are said to be of three kinds, right, acute, and obtuse.


A right angle is formed by one straight line standing erect or perpendicular on another: thus, ABD, DBC, EFG, are three right angles.

Those of you who have begun to learn Euclid will have met with the following further and more perfect explanation of the term “ right angle, but it is, perhaps, a little too difficult for beginners:

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