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means of conveying useful knowledge, but of mental exercise and cultivation.
“There were seven different liquid measures, graduated according to the standard measures of the kingdom. The teacher took one in his hand, held it up before the class, and displayed it in all its dimensions. Sometimes he would allow it to be passed along by the members of the class, that each one night have an opportunity to handle it, and to form an idea of its capacity. Then he would take another, and either tell the class how many measures of one kind would be equivalent to one measure of the other, or, if he thought them prepared for the question, he would obtain their judgment upon the relative capacity of the respective measures. In this way he would go through with the whole series, referring from one to another, until all had been examined, and their relative capacities understood. Then followed arithmetical questions, founded upon the facts they had learned, - such as, if one measure full of anything costs so much, what would another measure full (designating the measure) cost, or seven other measures full? The same thing was then done with the weights.
“ In the public schools of Holland, too, large sheets or cards were hung upon the walls of the room, containing fac-similes of the inscription and relief --- face and reverse - of all the current coins of the kingdom. The representatives of gold coins were yellow, of the silver white, and of the copper, copper colour."- Mann's “Educational Tour," with Preface by W. B. Hodgson, LL.D.
A knowledge of some of the more simple parts of geometry is quite necessary for any schoolmaster who wishes to be thought competent to his work, or to stand in what
may be looked upon as the first class of teachers in our elementary schools. For this purpose, it is highly desirable that they should at least know so much of the subject as would enable them to teach the first three books
of Euclid, with a few propositions out of the other books. Many of the propositions in the first three books are of easy application to the mechanic arts; particularly to the carpenter's shop, to the principles of land measuring, etc., and an edition of these, pointing out such propositions and their application, with a few practical deductions, would be of great use in our elementary schools.
There are many of the appliances of the carpenter with his tools, and of other mechanic trades, so strictly geometrical and so easy of proof, as to be easily learned, and the workman who knows them instead of being a machine, becomes an intelligent being, and has sources of enjoyment opened out to him, which many of them would turn to a good purpose.
Even a knowledge of the axioms of Euclid, such as “ things which are equal to the same thing, are equal to one another.”
“If equals be added to equals the wholes are equal."
“If equals be added to unequals, the wholes are unequal," etc., suggest modes of reasoning, which are extremely useful; and a thorough knowledge of the kind of reasoning in the propositions of the three books, gives a man a habit and a power of drawing proper conclusions from given data, which he would scarcely be able to acquire with so little trouble, in any other way.
Children may easily be made to understand what is meant by the terms perpendicular, horizontal, right angle, and lines parallel to each other, by referring to the things in the room.
Thus the walls are perpendicular, or at right angles to the floor— the boards are horizontal and parallel to each other — the courses of bricks are parallel – the door-posts perpendicular to the floor, etc.; the beams, rafters, etc., of the roof, all might be referred to as illustrating things of this kind.
The way in which the circle is divided ought to be understood; the number of degrees in a quadrant, etc. ; that the three angles of a triangle are equal to two right angles; and therefore if a triangle is right-angled, or has one right angle, the remaining two must be equal to a right angle.
The proposition that if two sides of a triangle are equal, the angles opposite are equal, and the converse.
To bisect a given rectilineal angle.
The following is a very interesting and useful application of this proposition in showing how a meridian line may be laid down by it:
Tell the boys to stick in the ground, and in the direction of the plumb-line, a straight rod, to observe and mark out the direction and length of its shadow on a sunny morning before twelve o'clock, say at eleven: to observe in the afternoon when the shadow has exactly the same length; join to the extremities of the shadows, and on the line which joins them, which is the basis of an isosceles triangle, describe an equilateral triangle on the contrary side of the line to that of the stick; a line drawn from the point where the staff goes into the ground to the vertex of this triangle will be the true meridian, or by simply drawing a line from the stick to the middle of the line joining the extremities of the shadows.
Place the compass on the line, and let them observe how much the two meridians differ: that the length of the shadow, at equal intervals from noon, will be the same both in the morning and in the afternoon, etc.
To draw a perpendicular from a given point in a line, or let one fall on a line from a point without it.
The one, that either of two exterior angles is greater than the interior and opposite angle -- showing from this, how the angle under which an object is
diminishes you recede from, and increases as you advance towards it.
The proposition about the areas of triangles and parallelograms, as applying to the superficial measurement of rectilineal figures.
The 47th in the first book, that the square of the side opposite the right angle is equal to the sum of the squares on the other two sides. All these from the first book are particularly of practical application.
It will be found very useful for fixing on their minds any particular geometrical truths likely to be of use to them afterwards, if the teacher tests it by application to actual measurement, and not to rest satisfied with proving it
merely as an abstract truth ; for instance, in this schoolroom there is a black line, marked on two adjoining walls, about a foot from the floor; as the walls are at right angles to each other, of course these lines are also; they are divided into feet and divisions of a foot, numbered from the corner or right angle, then taking any point in each of these lines, and joining them by a string, this forms a right-angled triangle. The boys have learned that the sum of the squares of the two sides containing the right angle is equal to the square on the third side, the teacher will tell them, for instance, to draw a line between the point marked six feet on the one and eight feet on the other; square each number, add them together, and extract the square root, which they find to be 10; then they apply the foot rule measure the string, and find it exactly ten feet by measurement.
Again, draw the line between the point marked five feet on one and seven on the other: work it out, and they get a result 8.6 feet; the teacher would ask, is .6 half an inch or more?- More by a tenth.— They then measure the piece of string which reached between the extreme points, and find it perfectly correct.
The teacher would then point out that this would always be the case, when the walls stand at right angles to each other. The bricklayer knows this, and, laying out his foundation walls, measures eight feet along one line, and six along the other, from the same corner ; he then places a ten foot rod between the extreme points, and if it exactly reaches, he is satisfied his walls are square.
Through the middle of the line on the end wall a vertical line is drawn, and divided in the same way, and higher up on the wall are marked three parallel lines a foot and a yard in length; these are very convenient to refer to as a sort of standard of measure, and to show what multiple of an inch, a foot, a yard, etc, any lengths of the other lines are.
It is recorded, then, that at the time of Henry the First, the length of the king's arın was the standard yard: this gives an idea of the rudeness of the age.
A teacher with a little knowledge of geometry will see
numberless ways in which these lines may be made useful. I feel a difficulty in entering further into this without having recourse to diagrams, which in the printing of this book I did not contemplate.
The following occur to me as simple :- - Tell a boy to measure the width of the door and its height; now what length of string will it take to reach between opposite corners? work it out: then to take a piece of string and measure,—they correspond; the same for his book, slate, a table, etc. Measure the two sides of the room- find the line which would reach from corner to corner.
Again, let one of the boys hold the string against a fixed point in the upright wall, say four feet high, and another extend it to any point towards the middle of the floor they see this forms a right-angled triangle; another boy takes the rule, measures from the point where the string touches the floor to the base of the black line, taking this as one side, the height four feet as the other, they work it out, and then measure as before. The testing of theory by practice, gives them a great interest in what they are doing.
As an example of the carpenter applying a proposition in Euclid, take this :
Not having his square at hand, he wishes to draw one chalk line at right angles to another, from a given point in it.
From C in the straight line AB he marks off with his compasses on each side of it, CA and CB, equal to each other, he then places his rule in the direction CD, as nearly perpendicular as he can guess, and draws a line,
CD, along it; from the point D he stretches a string to A, and if turning it on D he finds the same length exactly to reach to B, CD is at right angles to AB.
If he wanted to fix a piece of wood CD in AB, and at right angles to it, he would of course measure in the same way; if AD were longer than DB, he would lean it towards A until they were equal, if shorter he would have to move it in the contrary direction.