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use of means; then this result is nothing more than a copy of the mental perception which was previously had of the use of the means, and the result that, as a matter of certainty, followed the use of them; and if this is done with sufficient knowledge and judgment, the person doing it can repeat it again and again with a feeling of perfect certainty that the result must be perfectly equal in all cases.
Though this maxim is perfectly general, yet we must understand it within the proper limits before we can depend upon it for that absolute certainty which belongs to the mathematical sciences, and to them only; though the nearer that we can approximate this certainty in other matters which are of a mixed nature, the better. This, by the way, is the grand practical use of mathematics, and of incalculably more value to mankind than all the technical applications, which are absolutely necessary upon comparatively few occasions in the common business of life.
Let us apply this general maxim to the simplest of all cases, namely, the ascertaining of what data are necessary for constructing, and for ascertaining the perfect equality, in every respect, of
1. If the three sides are given, we have already shown how the triangle may be constructed, with the limitation of this single condition—that any two of the three sides must be greater than the third one. We have to place one of the sides on a straight line, and on the extremities of this side, as centres, describe circles, having the radii equal to the other two sides, one at the one extremity and the other at the other; and if the intersection of those circles is joined by two lines drawn to the extremities which were made the centres of the circles, a trian
gle is constructed, the three sides of which are equal to the sides which were given.
It is of no consequence in what order the three given sides are taken, because the triangle must have exactly the same size and shape, in all cases of the same three sides ; but it may have several positions, which do not, however, in the least affect its form or its value.
If the same side is placed on the line, in order that its extremities may be the centres of the circles, there are four positions of the triangle in the case of its being scalene ; that is, of having all its sides unequal. That is to say, the triangle may be constructed upon either side of the line, and the longer or the shorter of the two remaining sides may be placed at one extremity, that is, made the radius of the circle there. Thus the triangle may be turned side for side, and also end for end, in both cases, which gives four positions ; but as the sides are all the same in each case, and subtend arcs of equal circles, it follows that the angles opposite the equal sides are equal in each of the four positions. But
any of the three sides may be placed on the line, in order to give the centres of the circles whose radii are made respectively equal to the other two sides; and this again gives three varieties of position in each of the former four, or twelve in all, though there is nothing in either which can produce the slightest difference in the size and shape of the whole triangle, or of any of the six parts—the three sides and three angles of which it is made up.
In all these twelve positions of the triangle, we have assumed that the position of the original line to which the side first used is applied to remain the same; but this line may have any direction whatever, without in the least affecting the value of the triangle, or of any part of it; and, therefore, the same
POSITIONS OF A TRIANGLE.
triangle may have an indefinite number of positions. This, without
any reference to the fact that position is not a datum in the construction of the triangle, would suffice to show that position has nothing whatever to do with form or the value; but that the triangle, of which the three sides are known, must be the same, in what part of the world soever it is constructed, and whether it is ever constructed or not, provided that the lengths of the sides are expressed in the same measure and properly understood. Thus, if the three sides of a triangle are respectively, 5 inches, 4 inches, and 3 inches, which have the condition necessary for forming a triangle—as any two of them are together greater than the third, then this triangle has a definite shape and size from which it cannot deviate ; and any one who remembers the numbers, and is possessed of a rule or scale divided into inches, and a pair of compasses for describing circles, can construct this triangle whenever he pleases; and if he should describe one in London, and another at any distant place-say Calcutta in the East Indies—he would have no more doubt of the equality of those triangles in every respect, than if they were cut out in two pieces of flat paper, so that the one could be applied to the other, and be seen to coincide with it, or fill the same space in the whole and in every part.
These numbers, 5, 4, and 3, expressing the three sides of a triangle in the same measure, are worthy of being remembered; for, as we shall see afterwards, the angle opposite to the side 5 must be a right angle; and therefore we can always get a right angle by constructing a triangle whose sides are 5, 4, and 3; and as they are the simplest lengths of sides which have this property, they are worth bearing in mind.
2. If two sides, and the angle included between them, are given, there are sufficient data for constructing the triangle ;
CONSTRUCTION OF TRIANGLES.
and if two triangles have two sides and the included angle of the one equal to two sides and the included angle of the other -it being understood that it is not the sum of the sides which is equal, but that each of the two sides of the one triangle, taken singly, is equal to one side of the other—then the triangles are every way equal.
This does not require a formal demonstration, or the application of the one triangle to the other, any more than the former case ; for, if the lengths of the sides and the measure of the angle between them are known, the triangle can be constructed at any time or in any place; and if two triangles, with equal including sides, and an equal included angle, are constructedor even imagined to be constructed-how far soever they may be asunder, we can no more doubt their equality than we can doubt that an inch is an inch, or any angle itself and not another.
The given sides determine their own lengths, and the given angle determines their position with regard to each other. By this means the position of those extremities of them which are most distant from the angle are also determined, and these determine the length of the third side which joins them. They do this not only in respect of the length of this third side, but in respect of its position with regard to each of the two given ones, at the points where it meets them; and this, of course, determines the remaining angles of the triangles; and if there are two such triangles, it necessarily follows that the third sides of both are equal, and also that the angles opposite the equal sides are equal.
3. If one side and two angles of a triangle are given, there are data sufficient for constructing the triangle ; and if two triangles have a side and two angles of the one equal to a side and two angles of the other, they are equal in every respect.
ANGLES OF TRIANGLES.
If two angles are given, then the third is also given, or, which is the same thing, there are data sufficient for finding it; for the three angles are equal to two right angles, or the angular space on one side of a straight line at a point is the same. Therefore, to find the third angle, we have only to draw a straight line, and take any point of it; then, at this point we make an angle equal to one of the given angles; and afterwards apply to this angle, at the same point, another angle equal to the second given angle; and the angular space between this last angle and the opposite end of the straight line, which is the supplement of the sum of the two given angles, must be the third angle of the triangle. When this is done, the given side and the three angles are quite sufficient for enabling us to construct the triangle, whether the two given angles are the ones adjacent to the given side—that is, the one at the one extremity of it, and the other at the other—or whether one of the given angles is opposite to the given side.
In order to construct the triangle, it must be stated in what order the given angles are to be arranged with regard to the given side. If they are both adjacent to it, the form and magnitude of the triangle are determined, but not the position, because that would be reversed, if the given angles were unequal and made to change places. Also, if only one of the given angles is to be adjacent to the given side, we must know to which extremity of it that angle is to be adjacent, otherwise we may reverse the position of the triangle. In every case, howver, the magnitude of the triangle is determinate ; and in comparing two triangles, which have the three angles and a side in the one equal, each to each, to the three angles and a side of the other, the sides opposité equal angles, and also the angles opposite equal sides, are equal to each other, and consequently the triangles are equal in every respect.