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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, etc.
Again, another very easy application of a simple proposition in the first-book, to show that if AB is a straight line,
from C a point without it, the perpendicular CD is the shortest line from C to D: any other line CF, CG, etc. would be greater B 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 further 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. Take this as a case where the eye may be made to help
the mind; take a square thin A 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 F clearly that the difference be
tween the area of the square and the inscribed circle is the sum of the four irregular corners AEKF, etc., contained between the sides of a triangle and the
arc in each side. 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, etc., 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 = a?,
a? area of the circle = (3:14159) —; (3:14159 being the area)
I of a circle, radius 1. )
= (-78539) a”, ... the area of the four corners, FBEG, etc.,
= a’ (1–78539),
= al .21461); and area of one of them = ('051365) aż.
a? a? a? Again, EF2 = AE? + AF2 = - +- =-, which is the
4 4 2 area of the inscribed square, and is one half of (a”), thecircum
scribing one; and any one of the triangles AEF will =“.
AE.AF a? Again the area of the triangle AFE =- =
-, and this divided by two, or —, as by the other method.
8 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 of 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.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—and the angles being equal the sides are equal.
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.
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 triangle, 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, etc.; that a triangle is half the parallelogram on the same base and altitude, etc., and not to do everything mechani. cally, 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 any thing 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 taught, as applying to practical purposes connected with their after-life.
However, I have myself no objection that this or any other ometry, a knowledge of which may be likely to advance the interest and the civilization of mankind, half as much as trigonometry does, should be taught to promising boys in our parish schools, whose parents have been able to keep them there to a sufficient age, and who have acquirements enabling them to learn it-these will be exceptions, and not the rule.
But why among the words, supposed to be of suspicious termination, attack the ometries? they, of all others, are the most harmless — dealing in weight and measure of an exact quantitative kind; so demonstratively true, that there is no chance of getting wrong—no possibility of their being antianything whatever. There may be something of wrongness in some of the ologies, as they leave room for the wanderings of fancy, and do not deal in measured quantity as the ometries do. Here scientific men may, and perhaps sometimes do, become bold and speculative to a degree which may startle those of a more sober-minded temperament, and who have not paid attention to the subject on which they treat; still, I think we may rest satisfied that where theories are advanced not based on truth, they will be but short-lived, and not do much mischief in the end.
An instance of the force of meaning in a word when it once gets good hold on the public mind — happening to go to a book sale in my own neighbourhood, where there was a copy of an early edition of the Encyclopedia Britannica, when it was placed on the table for sale, a man
employed by one of the booksellers in London, rather drily, perhaps cunningly, observed, — “Why, you won't find the word railroad in it.” Not another word was said ; but after that I observed there was not a bidder besides himself.
This is a subject not mentioned in former editions of this work; but as it is most desirable it should be taught in our elementary schools, the following observations will, I trust, be useful to those for whom the book is intended:
Hitherto drawing has been a branch of instruction mainly confined to schools for the upper classes, in which it has too often been loosely and inefficiently taught as an accomplishment merely — but there is no reason why it should not form a recognized and most useful part of the routine of instruction of every school. By the aid of the appliances and opportunities of instruction now offered by the Department of Science and Art, any teacher may make himself master of as much of the theory and practice of drawing as will enable him to impart an extent of knowledge in this direction, better digested and really greater in amount than has been hitherto, in pine cases out of ten, given in schools in which it has formed an expensive extra. Efficient examples and illustrative manuals, following each other in proper sequence, are now supplied, on well-devised terms, by the new department of the Board of Trade; and by the aid of these the study of drawing may, in almost every case, be introduced to a certain extent.
By the term drawing, however, we must, in the outset, clearly understand that the end of the study, as introduced into elementary schools, is not necessarily “fine art” in the production of pictorial manifestations. Drawing, strictly speaking, should be looked upon as a mechanical exercise, analogous in fact to writing, and regarded, if I may so express it, as graphic language-that as ideas are expressed by words or in writing, so they may be embodied by drawing.