Part 13

In the example, fig. 94--a hand-wheel--the principal difficulty, even for the most advanced student in free-hand, will be in drawing the circles; a coin, if convenient, can be used to scribe about, in drawing these; the other parts can afterwards be filled in around the circle. Fig. 96 is introduced for practice in penciling and shading; the figure represents a water-wheel on a stone pier.

The familiar oil can is shown in fig. 95. These all are excellent objects for practice.

Geometrical Drawing.

Geometry is the science of measurement; it has been known for more than three thousand years; many lives have been devoted to its development, and it exists to-day as the foundation of all mathematics.

Geometrical drawing is the art of representing, to the eye, the problems “worked out” by geometricians, and the importance of a knowledge of geometrical drawing is paramount. The student will find that the figures delineated and explained in the next few pages constantly occur in mechanical drawing. Says Walter Smith, State Director of Art Education in Massachusetts, “I have never known a case where a student did not progress more satisfactorily in his studies after a course of practical geometry.”

The elementary conceptions of geometry are few:

1.--A point. 2.--A line. 3.--A surface. 4.--A solid, and 5.--An angle.

All of which elements are used in mechanical drawings.

From these, as data, a vast number of mathematical problems have been deduced; of which a few of the most elementary will be illustrated in this work; but these few will repay the attention of the student.

In “freehand” drawing the crayon and pencil are used; in geometrical drawings the dividers, as shown in illustration, fig. 97, together with a rule, are all that is necessary to accomplish the work.

A problem is something to be done, and geometry has been defined as the science of measurement; the relation between geometry and mechanical drawing is very close, hence the term “geometrical problem.”

Before proceeding with the examples, a few elementary statements belonging to the science of geometry are presented; these will be useful to the student, not only while “doing” the problems, but in many cases of every-day--future--experience.

Geometry is one of the oldest and simplest of sciences; it may be defined as the science of measurement; geometry is the root from which all regular mathematical calculations issue. It has claimed the best thought of practical men from the times of the Greeks and Romans two thousand years ago; they derived their knowledge of the science from the Egyptians, who in turn were indebted to the Chaldeans and Hindoos in times beyond any authentic history; hence it was under the operations of the laws explained in geometry, that the pyramids of Egypt and the temples of Greece were constructed, as well as the engines of war and appliances of peace of ancient times.

A point is mere position, and has no magnitude.

A line is that which has extension in length only. The extremities of lines are points.

A surface is that which has extension in length and breadth only.

A solid is that which has extension in length, breadth and thickness.

An angle is the difference in the direction of two lines proceeding from the same point.

Lines, Surfaces, Angles and Solids constitute the different kinds of quantity called geometrical magnitudes.

Parallel lines are lines which have the same direction; hence parallel lines can never meet, however far they may be produced; for two lines taking the same direction cannot approach or recede from each other.

An Axiom is a self-evident truth, not only too simple to require, but too simple to admit of demonstration.

A Proposition is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

A Problem is something proposed to be done.

A Theorem is something proposed to be demonstrated.

A Hypothesis is a supposition made with a view to draw from it some consequence which establishes the truth or falsehood of a proposition, or solves a problem.

A Lemma is something which is premised, or demonstrated, in order to render what follows more easy.

A Corollary is a consequent truth derived immediately from some preceding truth or demonstration.

A Scholium is a remark or observation made upon something going before it.