A manual of information and suggestions for object lessons

Front matter

MANUAL

OF

INFORMATION AND SUGGESTIONS

Part I. First Lessons In Drawing

Some teachers begin their lessons in drawing with the formation of straight lines, vertical, horizontal, oblique, and

* We have represented the letter w in two forms : one in the word owl, and the other as seen in the alphabet. We prefer the latter. We have given on the Chart the " Spencerian" form of the letters. If the teacher should prefer any other forms, there is no objection to his using them, if he can make them verv ncatlv.

angular, and these they follow with exercises in forming curved lines. Chart No. XI. will furnish an abundant variety of such lessons, if teachers think best to follow this system.

Others very justly object to this mode of teaching drawing as unnatural and devoid of interest to the pupil, and prefer to begin simultaneously with both kinds of lines, as they are found combined in the outlines of the forms of real objects, as we have represented them in the upper part of Chart No. X. It is not often that we are required in drawing to make a continuous straight or curved line with one stroke of the pencil, and hence there is little reason for practicing upon such lines.

If the picture of a cap like that on Chart No. I. be given a child for a drawing lesson, the child, if it has had no previous instruction in drawing, will be very apt to begin with making heavy black marks for the outline, and these marks will be very certain to be wrong. They are then so heavy that they can not be corrected, and the child has to begin a new drawing. This will, perhaps, be equally faulty, and another drawing will be begun ; and so on until half a dozen or more attempts have been made, and the child becomes discouraged.

m The child should be taught to begin with making the faintest outline possible of some important part of the picture : it should then examine this outline carefully, and, without rubbing out any thing, correct it by a new light tracing. Let it continue thus until the outlines of the whole picture are satisfactory. The correct tracings may then be made a little firmer, like the outlines of the cap as given on this Chart. It will then be easy to fill up with lines of shading, as they are seen in the picture of the same cap on Chart No. I. These pencil lines of shading, however, should be made much lighter than they are in the engraved and printed picture. Children are inclined to go to the extreme of making a picture as black as possible. They should be encouraged to make all their drawings light, even when they are fully shaded ; and they should be shown that the most striking parts of a picture are usually those which are left almost or wholly xohite.

After the pupil has drawn an outline of the cap, as in Chart No. X., and shaded it, as in Chart No. I., he should be required to make a drawing, in the same order, of a real cap.

After this introduction of the subject, we would advise that pupils should be required to make drawings from Chart No. XIX. of the forms of leaves, stems, roots, and flowers represented there, being careful to sketch the outlines very lightly at first ; and that after each drawing — if it be of a leaf, for example — a similar real leaf should be placed before them as a copy, that they may thus be continually chawing from Nature.

They may next return to Chart No. X., and copy, in order, the outlines of objects there represented. After making one of these outlines, as directed with reference to that of the cap, and then shading it as shown elsewhere in the completed picture, the object itself, or some other familiar object, and a similar one if possible, should be placed before them to copy. In all cases the picture should be used merely to teach the children how to draw similar natural objects*

The complete picture of the pear, jug, and face will be found on Chart No. I. ; that of the fish in the Fifth Reader, page 227 ; but that on Chart No. I. may be drawn as well. The completed picture of the cat may be found on Chart No. VI. ; that of the boy reading on Chart No. V. ; domestic fowls on Chart No. XVII. ; and the horse, etc., on Chart No. XV. In drawing a picture of the human face,

* Drawing, as usually taught in our schools, is little more than copying pictures — a very profitless exercise, except as it does cultivate a taste for pictures or paintings generally ; hut it fails in cultivating a taste for Nature ; and its utility is the very doubtful one of copying a copy, instead of copying the originals. Drawing and painting lessons should have direct and constant reference to drawing and painting from Nature ; and if the system we have here so briefly sketched be tenaciously adhered to, the desired results will not be difficult of attainment.

the eye should be first drawn, as, if that should be wrong, all the rest would be spoiled.

The teacher will find on the Charts, and throughout the School and Family Readers, a great variety of pictures, from which he may select copies for his pupils; and he should be careful to select, in the early lessons, such as they are most familiar with, and of which they can find representatives in real objects in Nature to copy from. Let them copy from the upper part of Chart No. XX. the parts of flowers as there represented, and then find similar parts to copy from in the natural flowers. Let them copy, from Chart No. XXI. the forms of the roots of well-known plants, and let them bring in drawings of similar roots from Nature. Be careful to have them begin each picture by sketching the outlines very lightly ; and do not let them shade any of them too heavily. Accustom them to copy much from Nature. Let them make drawings of books, tables., chairs, and desks, and of old fences, the stumps of trees, rocks, etc. They should not attempt buildings until they have learned something of Perspective.

Part II. Geometrical Drawing

Children delight to exercise their constructive powers in making things : the girls, in cutting out patterns and making dresses ; the boys, in using the hammer and nails, the gimlet, the hatchet, the saw, the plane, the rule and compasses, in planning and constructing toys and machines, and in making diagrams of geometrical figures. What their elders and superiors do, children not only desire to know how to do, but they wish to do it also : labor — work — is with them an instinct of their very being, fraught with health and happiness ; and if, throughout childhood and youth, they could have suitable training, by way of recreation and amusement, in the principles and practice of construction, they would not make the inefficient men and women that we now class among the drones of society. The strong propensity, especially in boys, for using "tools," and " making" things, should not, therefore, be checked, but judiciously guided and trained as a part of the business of their education. It will thus lead, naturally, and without impediment to any thing else useful, to those attainments in knowledge, and that skill in art, that combine to make the scientific and practical farmer, the master mechanic, or the engineer. Our educators should look more to this hitherto almost wholly neglected part of education.*

Carrying out the above principles in part, under the head of Geometrical Drawing, on the Chart, we have introduced some interesting and useful problems in lines and forms, for practice in the school-room. While these exercises are calculated to train the eye and the hand to accuracy and neatness in drawing, they will also cultivate the inventive faculties, and give children the gratification of knowing how easily some apparently difficult things are done ; such things, also, as will be apt to come up for use in many ways in their future studies, and in practical life. It should not be deemed sufficient for children to tell how these figures are drawn. They should draw them ; and for this purpose they should be provided with a ruler, also with a pair of dividers or brass compasses, to one leg of which a short lead pencil, shaved flat on one side, may be firmly tied, for drawing the circles.

We would also further remind the teacher that the pupil should not be expected, at this early stage, to proceed to the demonstrations of these problems. After the pupil has solved the first problem, he should be left for a while to his own ingenuity in solving each subsequent one. After he has learned to bisect a given straight line, he has the key to the next three problems ; and if, unaided, he can accomplish their solution, the knowledge thus acquired will be wholly his own, and will be prized accordingly, and the great value of systematic contrivances will begin to be

* Manual Labor Schools have not been eminently successful in an educational point of view, because the labor required has not been planned with reference to educating the faculties, but only with reference to material profits in dollars and cents. Hence the labor has often been, in an educational aspect, an injury rather than a benefit.

appreciated. Let this process of self-instruction be carried out extensively, but not so far as to check the ardor and dampen the curiosity of the pupil. Moreover, as the pupil progresses from one step to another, let the teacher bring in all the illustrations possible, showing the relationship between these problems and constructive art, and especially in the construction of tools and machines with which the pupil is familiar.

Fig. 1, — To bisect a given straight line; that is, to divide it into tioo equal parts.

Let AB be the given straight line. From the centre A, with a radius* greater than the half of AB, describe two arcs* of a circle, as at C and D ; and from B, with the same radius, describe the two arcs of another similar circle, intersecting the first at C and D. The straight lines connecting the points of intersection will bisect AB in O. (The line COD will also be perpendicular to the line AB.)

Fig. 2. — From a given point in a straight line, to draw a perpendicular^ to this line.

Let EF be the given line, and O the given point in it. In the straight line EF take any point, E, and make OE equal to OF. From E, with a radius greater than EO, describe an arc above O ; and from F, with the same radius, describe another arc intersecting the first at G. Draw GO, and this line will be perpendicular to EF.

[Explain what is meant by a perpendicular, and also show that a perpendicular line may be in a horizontal, or in any other position. What lines in the school-room are perpendicular to certain other lines? The explanation of a perpendicular will show that each of the angles EOG and GOF, in Fig. 2, is a right angle. Give pupils a straight line, and ask them to form a right angle at a given point in it ; then two right angles on each side of the line. Describe the form of the wooden level and plummet which masons and carpenters often use for determining whether a wall or a beam is level or not. Ask them how they would make this level. Could an iron square, such as carpenters use, be made without knowing how to draw one line perpendicular to another ? How would they make a wooden square, to answer the purposes of the iron square ?]

* Describe what is meant by a radius, and what by an arc. See page 80, and Chart No. XI.

t Describe what is meant by perpendicular. See page 80, and Chart No. XI.

After having drawn Fig. 2, see if the pupil, unaided, can draw an equilateral triangle* It is done by describing the arcs which intersect at C, with a radius equal to EF, or FE, and then connecting the three points E, C, and F. See, also, if the pupil can describe, on a given line, an isosceles triangle, each of whose equal sides shall be double of the base. Next describe a similar triangle on the other side of the line.

Fig. 3. — From a given 'point without a straight line, to draw a perpendicular to this line.

Let HI be a straight line of unlimited length, and O the given point without it. From O, with a radius greater than the nearest distance to the straight line, describe an arc cutting the line in two points, H and I. Then from the points H and I, as centres, and with radiif greater than the half of IH, describe two arcs cutting each other in the point J. Draw OK in the exact direction of the point J, and OK will be perpendicular to HI. [HK and IK will also, evidently, be perpendicular to OK at the point K.]

Fig. 4. — To draw a square on a given straight line.

Let BA be the given straight line. First, from the point A, draw AG perpendicular to BA, and take AE equal to AB by measuring. Then, from the point E in the line EA, draw EH perpendicular to AE, and make EH equal to AB. Draw HB in a similar manner, and the square will be completed.

In order to draw AG perpendicular to BA, extend BA in the direction ofD; take any distance, such as AC, and make AC and AD equal. Then from the centres C and D, with equal radii greater than CA, draw the two arcs intersecting at G. Draw a line from G to A, and this line will be perj:>endicular to BA. The point G might have been taken below E. In a similar manner EH is drawn from the point E perpendicular to AE.

Fig. 5. — To find the centre of a given circle or arc.

Let JKLM be the given circle, or JKL the given arc

* A triangle having three equal sides, t Explain that radii is the plural of radius. C 2

Take any three points in the arc, or any three in the circle, as J, K, L ; join JK and KL ; bisect each by the perpendicular lines UT and RS, and their point of intersection, O, will be the centre of the given circle or the given arc. JK and KL are bisected by perpendiculars, similar to Fig. 1.

Now give the pupils this problem : To describe the circumference of a circle through any three given points.

Fig. 6. — To divide a given angle, or a given arc, into two equal parts. (That is, to bisect them.)

Let ACB be the given angle. From the centre C, with any radius, describe an arc AB. Draw the straight line (the chord*) AB. Bisect AB by the perpendicular CE, and the angle ACB will be divided into two equal parts.

If AB be the given arc, draw the straight line (or chord) AB, bisect it by the perpendicular EC, and this will bisect the arc.

Fig. 7. — To inscribe a circle in a given triangle.

Let GHF be the given triangle. Bisect H and F, any two angles of the triangle, by the lines HN" and FM. Their point of intersection at O will be the centre of the required circle. Then from the centre O, with a radius equal to the nearest distance of any one of the sides of the triangle, describe a circle, and its circumference will also touch the other sides of the triangle.

Figi 8. — To describe a circle about a given triangle.

Let ABC be the given triangle. Bisect AB and AC, any two of the sides of the triangle, by the perpendiculars GF and HI. Their point of intersection at O will be the centre of the required circle. From this centre, with a radius equal to the distance of any one of the angles of the triangle, describe a circle, and its circumference will also pass through the other angles.

Fig. 9. — Five problems.

1. To inscribe a square in a given circle.

If you do not know the centre of the circle, find it, as described under Fig. 5. Through the centre, O, draw the diameter KL. Through O draw MN perpendicular to KL. * For the definition of a chord, see page 80, and Chart No. XI.

Connect the points M, L, N, and K, and the square will be formed.

2. To describe a square about a given circle.

Draw, as just described, two diameters of the circle, KL and MN, at right angles to each other. Draw perpendiculars at the extremities of these diameters, and the square will be formed.

3. To inscribe a circle in a given square.

Take the outer square PRTS, connect the opposite angles, and from their point of intersection inscribe a circle with a radius equal to the nearest distance to any one side of the square ; or bisect any two adjacent sides of the square by perpendiculars, and their intersection will give the centre of the circle, and the length of the radius for inscribing the circle.

4. To describe a circle about a given square.

Let KMLN be the given square. Connect the angles of the square by the two diameters MN and KL, and from their intersection at O describe a circle with a radius equal to either of the semi-diameters, OM or OL, etc.

5. To inscribe a regular octagon in a circle.

In the circle inscribe the square KMLN ; bisect the arc cut off by each of its sides, which will give the points U, V, etc. Connect KIT, UN, NV, VL, etc., and a regular octagon will be inscribed in the circle.

A continued subdivision of the arcs thus obtained would give figures of 16 sides, 32 sides, 64 sides, etc.

Fig. 10. — To inscribe a regular hexagon in a given circle.

From the centre of the circle, A, draw the radius AB. Apply the radius AB six times to the circumference, and the hexagon will be formed, as shown in the figure.

Now ask the pupils how they would form a regular polygon of 12 sides.

Ask them how they would draw six equilateral triangles in a circle. If they were required to draw one equilateral triangle, having a side of it equal to a given straight line, how would they do it? (Describe a circle whose radius should be the given straight line; then describe another circle, with the same radius, having its circumference pass through the centre of the first circle, and it will be readily seen how the triangle can be formed.) A knowledge of the construction of the hexagon will enable pupils, after a very little study, to construct the diagram of the Chromatic Scale of Colors, Chart No. XIV.

Fig. 11. — To inscribe a regular pentagon in a given circle.

The centre of the circle being found, draw the radius HI, and from the point H draw HL at right angles to it. Bisect HL in J ; connect JI ; then make JK equal JI. Now apply HK to the circumference of the circle, beginning at L, and it will be found to exactly measure the circumference ten times. Connect the alternate points obtained by this measurement, as shown in the figure, and a regular pentagon will be formed.

How, then, could a regular decagon be formed ? A regular polygon of 20 sides ?

Now suggest to the pupils the following problem :

To construct an adjoining pentagon on each of the Jive sides of the pentagon. Fig. 11, and each equal to Fig. 11.

This may be easily done in the following manner : Take a side of the pentagon, say 5 7, and bisect it by a perpendicular, such as HK extended outwardly. Then, on this perpendicular as a diameter, describe a circle that shall cut the circle of Fig. 11 in the points 5 and 1. Then 5 7 will be one of the sides of the adjoining pentagon, and the other sides may be easily laid off around the circle. In a similar manner construct each of the adjoining five pentagons.

The knowledge of this construction is essential to the forming of the solid called the dodecahedron. See page 62.

Let them also exercise their ingenuity upon the following problem : Describe three equal circles touching one another; and also describe another circle which shall touch them all three.

Fig. 12. — To describe an ellipse.

As an ellipse is a figure that in form approaches a circle on the one hand, and is contracted to almost a straight line on the other, we have drawn three of them.

1. To draw the inner ellipse.

Suppose it be required to draw this ellipse on a smooth board.

Take any two points, A and D, and into each point drive a pin ; tie a string, AD, to these pins, one end to each pin, and leave the string sufficiently loose to reach to T ; then putting a pencil-point inside the string, and fully stretching out the string, move the pencil-point around against the string, and it will form on the board the inner ellipse.

The points A and D are called the foci of the ellipse.

2. If C be taken as one of the foci, while the other remains at A, the same length of string will give the form of the second ellipse.

3. If B be taken as one of the foci, while the other remains at A, the same length of string will give the form of the outer ellipse.

If the two foci should be brought much nearer to each other, with the same length of string, the ellipse would approach closely, in form, to a circle. The farther apart the foci are, the more will the ellipse be contracted in width.

How would the pupil draw a large circle in the garden — so large that he could not use a pair of compasses, or dividers ?

He might use a narrow strip of board, with a peg at each end, driving one of the pegs firmly into the ground, and using the other peg to mark with. Or he might use a rope, with a peg at each end.

How would he draw a large ellipse in the garden ?

Additional Exercises.

To construct the Jive regular polyhedrons.

There are five solids, in each of which the faces are all equal polygons, and the solid angles of which are equal. By knowing how to draw an equilateral triangle, the square, and the pentagon, as already described, each of these solids may easily be made of pasteboard.

Having drawn the figures as we have given them, on pasteboard, and cut them out accurately through their boundary lines, cut the other lines half through the pasteboard, turn up the parts and glue them together, and the forms of the solids will be obtained.

1. The Tetrahedron, or equilateral pyramid, is a solid bounded by four equal and equilateral triangles.

2. The Hexahedron, or cube, is a solid bounded by six equal squares.

3. An Octahedron is a solid bounded by eight similar triangular faces. The triangles may be either isosceles or equilateral. The latter give what is called a regular octahedron.

4. A regular Dodecahedron is a solid bounded by twelve equal pentagons.

5. A regular Icosahedron is a solid bounded by twenty equilateral triangles.

Let the pupil construct, out of pasteboard, a pentagonal prism • that is, a prism having a pentagon for each end, and equal parallelograms for the five sides.