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"It is not a question, how much a man knows, but what use he can make of what he knows; not a question of what he has acquired, and how much he has been trained, but of what he is and what he can do."

-J. G. Holland.



HIS everybody demands of the teacher; and he is scarcely in danger of being without fair pretensions in this branch. He should, however, know it by its principles, rather than by its rules and facts. He should so understand it, that if every arithmetic in the world should be burned, he could still make another, constructing its rules and explaining their principles. He should understand Arithmetic so well, that he could teach it thoroughly though all text-books should be excluded from his schoolroom. This is not demanding too much.

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Arithmetic is a certain science, and used every day of one's life, the teacher should be an entire master of it. And when he reflects that without Arithmetic the wonderful exchanges made through the net-work of modern business must be reduced to the simple barter of barbarous times; that without Arithmetic the manufacture and manipulation of delicate or highly effective machinery must cease; that the almost miraculous processes of the chemical and physical laboratory must be suspended; and that without the proportion of numbers, architecture, sculpture, painting, and even poetry and music must all lose their charms, then he can comprehend to what an extent Arithmetic lies at the foundation of mod. ern civilization and contributes to the physical, intellectual, and even spiritual welfare of mankind; then he can see why the teacher should be an entire master of it.




The custom of leaving all work in mensuration until pupils are about to leave the grammar grades has happily disappeared in most schools. To-day the pupil early becomes familiar with the simpler phases of the work in lines, surfaces, and solids. Pupils enjoy it when the presentation is concrete, and they quickly grasp the meaning of terms used.

With the new methods of teaching the subject, the old plan of working by rule for an answer has to some extent gone out of use here, as well as in the other parts of arithmetic. Then, too, since the child is to know and deal with lines, surfaces, and solids throughout his life, it is well that he becomes familiar with them as early as possible.

Do not attempt to teach mensuration without objects. Cut pieces of paper, cardboard, or wood to illustrate what you are teaching. Model solids from clay, if necessary. The piece of paper and pin used for a compass in "Measures for Little People" may be made to do good service here.

Mensuration is the measuring of lines, surfaces, and solids.


A line has only one dimension, length. The end of a line is a point. A point has position only.

A straight line is one that does not change its direction at any point. It is the shortest distance between two points.

-B The line AB is a straight line.

A curved line is one that changes its direction at every point.


MN is a curved line.


A broken line is one that changes its direction at some points but not at all,

AB is a broken line.


Parallel lines are lines which will never meet no matter how far extended.

BAB and CD are parallel lines. A horizontal line is one parallel to the horizon. A vertical line is one which makes an angle of 90° with the horizon.

An angle is the difference in direction between two lines which meet.


The angle ABC here is the difference in direction between the lines AB and BC, which meet at B. We read it as angle ABC, or angle CBA, placing the letter at the vertex in the middle.

The vertex of an angle is the point where the lines meet. B is at the vertex of the angle illustrated.

The sides of an angle are the lines that meet to form it.

Perpendicular lines are those which make an angle of 90°.


The line AB is perpendicular to the line CB, and the line CB is perpendicular to the line AB.

NOTE. Teach the pupil that in whatever direction the lines extend, so long as the angle formed is 90°, the lines are perpendicular to each other. Many pupils get the mis

taken idea that the lines must extend just as they do in the



Here the line AB is perpendicular to the line CD, and CD is perpendicular to AB.

A right angle is one of 90°, or one whose sides are perpendicular to each other.

The angles shown in the two preceding drawings are all right angles. An acute angle is one less than 90°.

Angle ABC is an acute angle.

An obtuse angle is one greater than a right angle.



Angle AED is an obtuse angle.


A surface has two dimensions, length and breadth. A plane surface is one that does not change its direction. A straight line laid upon it in any direction will touch at all points.

NOTE. Let the pupil apply the edge of a ruler to a table. Try a warped board. Ask the pupil to notice a carpenter test a planed board.

A quadrilateral is a plane figure having four sides.

A parallelogram is a quadrilateral whose opposite sides are parallel.

A rectangle is a parallelogram whose angles are right angles.



ADEG is a rectangle.

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