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PARALLELOGRAMS AND TRIANGLES.
the product—or, as it is the same to divide either factor before multiplying as to divide the product after, we may multiply the base by half the altitude, or the altitude by half the base, and the product in either case will be the area of the triangle.
4. But, in the case of a product, we multiply the product if we multiply either factor, and divide the product if we divide either factor. Therefore, triangles or parallelograms which have equal bases, are to each other in the ratio of their altitudes; and triangles or parallelograms which have equal altitudes, are to each other in the ratio of their bases. Therefore, again
Triangles or parallelograms which have different bases, are to each other in a ratio compounded of the ratios of their bases and altitudes.
ance, if the base of the triangle parallelogram a is 12, and its altitude 6; and the base of the triangular parallelogram B is 9, and its altitude 4—the numbers meaning equal measures in all those cases; then
A : B = 12 x 6 : 9 X 4 = 72 : 36; that is, the area of the triangle or parallelogram a is double that of the triangle or parallelogram B; and any others whose dimensions are known and expressed in the same measure, can be compared in the same manner; and those products which are compared are, after all, nothing more than the simple expressions for the areas of the figures as already explained.
5. If two triangles or two parallelograms are equal in area, but have their bases and altitudes different, then the bases and altitudes are inversely or reciprocally proportioned. Let a be the base and b the altitude of the one ; and c the base and d the altitude of the other; then, if the areas are equal, that is, if
a b = c d, then
for multiplying the extremes and means, we have a b = c d, as before.
6. A rectangle which has two sides containing one of its angles equal, has all its four sides equal, and is called a SQUARE; and the expression for the area of a square is the product of the side multiplied by itself; or if the side be called a, then the area of the square is expressed by a'.
If a square and rectangle have equal areas, the side of the square is a mean proportional between the base and altitude, or which is the same, between the length and breadth of the rectangle. For since they are both rectangles, and equal in area, the sides or factors whose products express their areas, are reciprocally proportional ; that is, if b is the length and c the breadth of the rectangle, and a the de of the square, then
b: a= a:c; and, multiplying the extremes and means, we have b c = a?, or the areas are equal, which was the proposition.
Before proceeding further with the general principles, we shall endeavour to show how the last mentioned figures may be constructed. To construct a square on a given line. At one extremity of the line draw a perpendicular toward that side of the line upon which the square is to be situated, and make this perpendicular equal to the given line. Then from the two extreniities of those lines as centres, and with a radius equal to either of them (for they are the same), describe arcs intersecting each other; the point of section of those ares will be the fourth angle of the square; and if lines are drawn from this to the second and third angles, the square will be constructed.
To construct rectangle under two given straight lines. Take in any straight line a portion equal to one of them, and at either extremity erect a perpendicular equal to the other ; then, on the opposite extremity of each with the other one as a
radius, describe arches intersecting each other, and the point of section will be the fourth angle of the rectangle, to which the two remaining sides may be drawn from the extremities of the others. An oblique parallelogram may be drawn exactly in the same manner, only an angle of it must be given as well as the sides; and the two sides must be placed so as to make an angle equal to this angle, and then the remainder of the construction is exactly the same as that of a rectangle.
It need hardly be mentioned, because it is apparent without any description, that the same data which suffice for constructing one square, rectangle, or parallelogram, must suffice also for constructing another in any other place or at any other time; and that, if the data, that is to say, the one side in the case of the square, the two sides in the case of the rectangle, and the two sides and the included angle in the case of the parallelogram, are exactly equal in any two instances, then the proof of the equality in every respect of the figures themselves, is as clearly established as it can possibly be, even though the one of them were applied to the other and observed to coincide with it. And we may repeat, that this is the useful proof in practice; and that the coincidence, how pleasant soever it may
be for closet mathematicians, is of little avail in real life. It is desirable, for instance, that an acre of land should be the same quantity or extent of surface in Cumberland as in Cornwall; and that a mile of the road between London and Bath should be exactly the same length as a mile of that between Edinburgh and Glasgow; but it would puzzle all the geometers that ever lived to bring the two acres of land in the different counties, or the two miles in length of the different roads, to any comparison except through the medium of some measure, or other means of connexion, which had nothing to do with the applying of the
one of them to the other, and observing whether they coincided or not.
The next step of our progress, in examining the relations of figures, and of the boundaries and other parts of figures, is naturally that which involves some medium of comparison ; and for this purpose we must have recourse to the doctrine of ratios, as explained in a former section; because equality of ratios is very often the only equality of which we can avail ourselves, in such comparisons. But before we can enter upon this with the proper advantage, it is necessary that we should understand a little more about the relations of the different parts of the same figure, both to each other and to the whole.
All the figures with which we are concerned in plane geometry, may be reduced to these four classes :- First, circles; secondly, squares; thirdly, triangles; and fourthly, rectilineal figures having more than three sides, of which last squares are a particular division ; but they are so simple as compared with the others, that they require to be separated.
The general condition which figures must have, in order that we may apply the doctrine of ratios to them, and reason from the one to the other, or from parts of one to other parts of the whole of the same, is the property of their being
1. Figures are said to be similar, when they are exactly of the same shape, but different from each other in size. Equal sized figures of the same shape are of course similar, as well as unequal sized ones; but they are equal as well as similar; and any ratios which they, or corresponding parts of them, may have to each other, are ratios of equality, not merely in the relation itself upon which the ratio is founded, but in the
terms of the ratio ; and it is evident, that from the comparison of two ratios of this kind, no useful conclusion can be drawn. Thus, if a ratio were stated
12:6=12, and the fourth term sought by an operation in the common rule of three, the fourth term would be the same as the second, or 6 again, and we should learn nothing by the operation. This is what is called an identical proposition: and it very
often happens that those who have not disciplined their minds in the practice of reasoning in a close and logical manner, make use of such propositions in succession, or reason in a circle," as it is called, that is, go round like a mill horse, and end just where they began,—or rather their saying, for it is not argument, like the circumference of a circle, has neither beginning nor end ; is founded on nothing and leads to nothing.
Thus, the species of similarity which is useful to us, is similarity without sameness. And there are three conditions requisite, in order to establish the similarity of any two plane figures. First, they must have the same number of sides in each, and, by necessary consequence, the same number of angles. Secondly, all the angles of the one must be each to each equal to all the angles of the other, when taken in the same order of succession ; and thirdly, the sides about the equal angles, and which are situated in the same order with regard to the angles, must be proportionals. This third condition is, in some figures, so closely connected with the second one, that the one of them cannot exist without the other; and therefore it might, perhaps, be possible to deduce the one from the other, at least in the case of one or two figures ; but still it is much better to take both parts into the definition, as then it is rendered perfectly general,
2. Circles have only one side each, and they have no angles ;