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 truth 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 is 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 an inch, 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 mul * It is recorded, that in the time of Henry the First, the length of the king's arm was the standard yard: this gives an idea of the rudeness of the age. tiple of an inch, a foot, a yard, &c., any lengths of the other lines are. 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, &c. 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. This 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. D 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 any point D he stretches a string to A, and if turning it on D he finds the same length exactly reach to B, CD is at right angles to AB. A C B 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. If he take his square, and place one side on the line AC, the other will fall in a direction perpendicular to it, and he could run his chalk line along the edge. The teacher would also point out, that when the lines are perpendicular, the angles ACD DCB are equal; that if CD lean more towards A than towards B, the angle ACD will be less than DCB, &c. C Again, another very easy application of a simple proposition in the first book, to show that if AB is a straight line, Ca point without it, the perpendicular CD is the shortest line from C to D; any other line CF, CG, &c. would be greater than CD, as being opposite to the greater angle in the same triangle, and although every successive line CF, CG, keeps lessening as it gets nearer to D, yet at D it is least, and when it passes through that point, the length of a line from C to any point in AD goes on increasing as that point gets farther from D. It will easily be seen on what proposition in Euclid these remarks depend, and the young schoolmaster may profit by them, and apply other propositions in the same way. A A E K D G F B B Take this as a case where the eye may be made to help the mind: take a square thin piece of deal, say one foot on a side, and a circle of the same one foot in diameter; place the circle on the square so that it becomes inscribed on it, the figure will be this. They see clearly the difference between the area of the square and the inscribed circle is the sum of the four irregular corners AEKF, &c., contained between the sides of a triangle and the arc in each case. F H Find this difference, divide it by four, that will give any one corner AEKF: then inscribe a square in the circle: the difference between this and the first square will be the four triangles AEF, &c., and which will be found equal to the inscribed square FEHG. Dividing by four will give one of the triangles. Let the side of the square =a, which will also be the diameter of the circle: Then the area of the square will a2 (3.14159 being the area of a (circle, radius 1. 4 ; (78539) a2, .. the area of the four corners, FBEG, &c., =a2(-78539), and area of one of them = (051365) a2. AE2 + AF2 1 area of the inscribed square, and is one half of (a2), the circum scribing one; and any one of the triangles AEF will = = a2 as by the other method. These are given merely because of the pieces of wood making visible what is to be done. The following offers a practical application of the 47th and other propositions in the first book of Euclid. Imagine a line drawn from the eye of the spectator to the top of a tower, or any other object standing on a plain, and at right angles to it, and another from the same point parallel to the horizon, making an angle of 45°; then the height of the tower above the level of the eye is equal to the distance at which the observer is standing from the base; adding to this the height of the eye, would give the height of the object above the surface of the ground; if, when the observer takes his station, the angle is above 45°, he must recede from the tower—if less, he must advance-bringing to bear upon this the proposition "that the exterior angle of a triangle is greater than the interior and opposite." An approximation to accuracy in observing an angle of this kind may be made by making a sort of quadrant out of a piece of deal; holding one side horizontal, and looking along a line drawn from the centre through the middle of the arc. At all events, this is sufficient to make boys understand the theory of it, and the object of this is obvious to make them reason, that- If one angle of a triangle is a right angle, the other two taken together must be a right angle, because the angles of a triangle are 180°, or two right angles; if one of the two is 45°, or half a right angle, the remaining one must be the same -as the angles are equal the sides are equal.* The particular propositions bearing upon this the teacher will easily see. In teaching them land-measuring, they should be made to understand on what principle it is that they reduce any field complicated in shape to triangles, squares, and parallelograms; why they make their offsets at right angles to the line in which they are measuring; to be able to prove the propositions in Euclid as to the areas of these figures, &c.; that a triangle is half the parallelogram on the same base and altitude, &c., and not to do everything mechanically, without ever dreaming of the principles on which these measurements and calculations are made. Some time ago the observation was made to me, arising out of some boys having been seen to attempt carrying the above into practice: "Well, the worst thing I have heard of you lately is, that you are having trigonometry taught to the boys in the Somborne School." This odd sort of compliment has often come across me, not knowing exactly what it could mean. I suppose those who make such observations do not mean that there is anything positively wrong in teaching trigonometry; but that it is wrong to teach it to that class of boys usually attending our parish schools. Now, one of the leading features of the school here, and, in my opinion, one of the most important, is, that it unites in education the children of the employer with those of the employed; and that to many children of the former class the elements of this subject may be most usefully * A stick 5 feet 3 inches high is placed vertically at the equator, what is the figure traced out by its shadow during the 12 hours the sun is above the horizon? What is the length of the shadow, and of a line joining the top of the stick and the extremity of the shadow, when the sun's altitude is 45° and 60°? Work out the result in the latter case to four places of decimals. |