Popular Science Monthly/Volume 34/January 1889/Inventional Geometry




INVENTIONAL geometry is the name given to a series of carefully graded problems, thought out and arranged by that able mathematical teacher, William George Spencer, the father of the distinguished philosopher.

The little book was published in this country in 1876.[1] An appreciative review in the New York "Evening Post" led the writer to procure a copy, and then to set to work to solve the

Fig. 1.

simple problems in the order given—the purpose being to form an impartial judgment of the value of the book for school use. A hundred or more problems were wrought, with increasing interest; and then, to make a further test, the book was given separately to a number of pupils, both girls and boys, each being

Fig. 2.

asked to work as many problems as he could. In each instance the pupil became interested in the work, and wished to continue. The remark of one girl was especially significant. "It's so different from ordinary study," she said; "there's something about it that leads me on."

A class was formed after these tests, and a few months' work proved the incalculable value of inventional geometry in a school course of study; and eleven years' experience in many classes and in different schools confirms that judgment.

Fig. 3. Fig. 4.

In all these classes the pleasure experienced in the study has made the work delightful both to teacher and taught; and there has always been a continuous interest from the beginning to the end of the term. This pleasure and interest came, not from any

Fig. 5.

environment, not from the peculiar individuality of the class, but because the problems are so graded and stated that the pupil's progress becomes one of self-development—a realization of the highest law in education.

The accompanying illustrations of the work of one class,[2] which began the study in September of the past school year and continued it till February, afford some suggestion of Fig. 6. the scope of inventional geometry. Each pupil is equipped with a ruler, which he uses merely as a straight-edge, a pair of drawing-compasses, and a right-line pen. For a short period I drawings are made with lead-pencil, till the pupil has acquired a little manual skill; then the change is made to pen, and all problems must be drawn in ink. Neatness and accuracy are insisted upon, and secured.

Fig. 7.

The class, having worked at their desks the problems assigned for a lesson, come up for recitation, and are directed to put their solutions upon the blackboard. The problems admit of such graphical representation on the board, and there are always so many different forms of solution, that pupils delight in this bold

Fig. 8.

drawing of what they have wrought out at their desks. Explanation follows the drawing, in which the pupil accounts for each step and every point he has used in constructing. At the start

Fig. 9.

he does not see what points he may take, nor what are given. For instance, the simple problem, "Can you place two circles to touch each other at a particular point?" will likely be drawn as shown in Fig. 1. The question, "Where was the particular point?" or, the point being marked, where he must make the circles touch, brings the correct solution (Fig. 2).

Many mistakes of like nature occur in the first lessons. In every such case the pupil must be led, by questioning, to see what is incorrect. He should not be told or shown, but thrown back upon himself; for, in inventional geometry, the knowledge is to be gained by growth and experience, through the powers he possesses and the method of acquirement peculiar to his mind. Occasionally the pupil is not a little baffled, and the skill of the

Fig. 10.

teacher is put to its best test to gain the solution without showing or telling him. Telling or showing is the method of the instructor—not of the teacher. The following problem (Fig. 3), "Given a circle, and a tangent to that circle, it is required to find the point in the circumference to which it is a tangent," is one of these difficult ones. Not many of these occur; the author, however, has a purpose in these few. For the most part the pupil is able by the grading to go on without questioning, as will readily be seen by examining the problem of which Fig. 4 is the solution, and the questions based upon it:

"Place three circles so that the circumference of each may rest upon the centers of the other two, and find the center of the curvilinear figure which is common to all the three circles."

"That point in an equilateral triangle which is equally distant from each side of the triangle, and equally distant from each of the angular points of the triangle, is called the center of the triangle."

"Can you make an equilateral triangle whose sides shall be two inches, and find the center of it?"

"Can you place a circle in an equilateral triangle?"

Fig. 10

"Can you divide an equilateral triangle into six parts that shall be equal and similar?"

"Can you divide an equilateral triangle into three equal and similar parts?"

To exercise to the utmost the pupil's power to invent, problems are given with certain restrictions: "Can you divide an angle into four equal angles without using more than four circles?" (Fig. 5).

"Can you construct a square on a line without using any other radius than the length of the line?" (Figs. 6, 7, 8, and 9).

Such problems as that solved in Fig. 10, "Can you place four octagons to meet in one point and to overlap each other to an equal extent?" delight the eye by beauty of form, and teach the pupil the basis of geometrical design.

Figs. 11 and 12, solutions to "Can you fit a square inside a circle, and another outside, in such positions with regard to each other as shall show the ratio the inner one has to the outer?"

Fig. 12.

and "Place a hexagon inside a circle, and another outside, in such positions with regard to each other as to show the ratio the inner one has to the outer," illustrate one way in which comparative area of figures is treated.

We have spoken of the pleasure a class experiences in putting their solutions upon the blackboard, and in examining the drawings of each other and following the explanations. There continually come up at these times incidents of this sort: Fig. 13 is given as a solution of "Can you raise a perpendicular to a line, and from the end of it?"

In his explanation the pupil points out the given line, the end from which he is to erect the perpendicular, the point from which as a center he sweeps each circle, why he may take that point, and

Fig. 13.

why he sweeps the circle. Another pupil discovers, before the explanation is finished, that the problem can be solved with one less circle, and there is the keenest interest while he draws and explains his way (Fig. 14).

The original and independent power acquired is shown in Figs.

Fig. 14.

15, 16, 17, and 18, solutions of the problem, "Can you make an octagon with one side given?"

Spencer's "Inventional Geometry" is one of the most original

Fig. 15.

and scientific contributions to school text-book literature ever made. What this little book teaches simply and naturally was

Fig. 16.

taught by going over demonstrative geometry, then taking up mechanical drawing, and adding to these personal experience. The author has secured all this and much more. He appeals first to the inventive faculty, seeks expression through the hand, and brings before the eye accurate and beautiful forms of the pupil's own constructing. The eye is trained to accuracy and similarity of forms, invention is quickened, comparison and judgment are constantly exercised, and inductive growth of mind is directly promoted. Besides the manual skill gained in constructing figures, and the power acquired to deal with original questions

Fig. 17.

through the constant appeal to invention, the pupil gains by his own efforts proofs of theorems really conclusive in themselves, though not the syllogistic form of proof belonging to the deductive science.

In but one of a great number of schools visited has the writer found inventional geometry used, and in that school quite out of the design of the author. It should precede demonstrative geometry, so as to give the pupil many concepts to draw upon when he takes up syllogistic demonstration. Demonstrative geometry then becomes an easier subject, and he is surer of what he is doing, because he has more general notions as a basis. In the school alluded to, the pupils were constructing figures and then demonstrating the questions, making the study simply supplementary to ordinary geometry. There was little invention. Nearly all the constructions were noticeable adaptations of what had been drawn for Fig. 18. demonstration in the deductive study.

Nor have the advocates of industrial training, with but one exception, so far as the writer has been able to learn, availed themselves of this study, which is not tentative, but directly in the line of what they urge.

Inventional geometry should be given a place in every school; and, if it becomes a question of time between that and demonstrative geometry, assign the time, in nearly every instance, to the former, because it is of far greater practical value, and many times more educative.

  1. New York, D. Appleton & Co., 1876.
  2. The average age was about fourteen years. The illustrations are reproductions of drawings made by the class, though in the usual class-work the lines would be finer.