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picture would be distorted; and would not resemble the original.
The same is the case with the resemblance, produced in any other kind of drawings; but particularly in geometrical figures.
If the two triangles, ABC, ab c, are to be similar to each other, it is necessary that they should be constructed after the same manner, and that the side AC should be exactly as many times greater than the side ac, as the side BC is greater than b c, and the side AB, than a b. If (Fig. I. and II.) the side AB, for instance, be twice as great as the side ab; that is, if the side ab be half of the side AB, the side ac must also be half of the side AC, and the side bc half of the side BC; that is, the three sides, ab, a c, b c, of the triangle a b c, must be in proportion to the three sides, AB, AC, BC, of the triangle ABC. Again, if (Fig. I. and III.) the side AB be three times as great as the side ab; that is, if the side ab be one third of the side
AB, the side ac must also be one third of the side AC, and the side b c one third of the side BC; or the triangles ab c, ABC, would not be similar to each other. The same holds true of all other geometrical figures, composed of any number of sides. "If they are similar, their sides are proportional to each other.
There are different ways of denoting a geometrical proportion. Some mathematicians would express the proportionality of the sides, ab, ac, of the triangle abc (Fig. II.), to the sides AB, AC, of the triangle ABC (Fig. I.), in the following manner :
AB : ab: : AC :.ac;
AB : ab: :AC = ac, and also
AB : ab = AC : ac,* which is read thus :
AB is to ab, as AC is to a c. As a proportion is nothing else than the equality of two ratios, the third way of denoting a proportion, in which the sign of equality is put between the two ratios, seems to be the most natural. The reason why the sign of division (see Notation and Significations), is put
* The first manner of expressing a proportion is now in general use among the English and French mathematicians; the second is sometimes met with in old English writers, and the third way is adopted in Germany.
between the two terms, AB, ab, of a ratio, is obvious; for a ratio points out how many times one term (the side a b) is contained in another, (the side AB).
The first and fourth term of a proportion, together, are called extremes"; because one of them stands at the beginning, and the other at the end, of a proportion: the second and third terms, standing in the middle, are, together, called the means.
The following principles of geometrical proportions ought to be well understood and remembered.
1st. It is important to observe, that in every geometrical proportion the two ratios may be inverted; that is, instead of saying,
AB : ab=AC : ac, you may say, ab : AB = ac : AC. For the order of terms being changed in both ratios, they will still be equal to one another; but leaving one ratio unaltered, if you change the order of terms in the other, the proportion will be destroyed. You cannot say,
ab : AB = AC : ac; for the smaller side, ab, is contained twice in the greater side AB (Fig. I. and II.); but the greater side AC, is not contained once in the smaller side a c.
2d. Another remarkable property of geometrical proportions is, that you may change the order of the means, or extremes, without de
stroying the proportion. Thus the proportion AB : ab = AC:ac.
(I.) may be changed into AB : AC = ab:ac
(II.) or, by changing the extremes into ac: ab = AC: AB.
(III.) The reason, why you have a right to do this, is easily comprehended. If, in the first proportion, the side AB (Fig. I.), is exactly as many times greater than the side ab (Fig. II.), as the side AC is greater than the side a c, the ratios of AB to AC will be the same as that of a b to
In Fig. I. and II., for instance, we have ab equal to one half of AB; consequently a c is also equal to one half of AC; and therefore, let the ratio of the two lines, AB to AC, be whatever it may, their halves, ab and ac, must be in the same ratio. No one will deny, that the same ratio, which one dollar bears to one hundred dollars, exists between one half dollar and one hundred half dollars; or that one hundred dollars are just as many times greater than one dollar, as one hundred half dollars are greater than one half dollar. The second proportion would still be correct, if, as in Fig. I. and III., the sides, AB, AC, were three times as great as the sides, ab, ac; for then the thirds of AB and AC would still be in the same proportion as the whole lines AB and AC. Nothing can now be easier than to extend this mode
of reasoning, and show the generality of the principle here advanced. The correctness of the third proportion might be proved precisely in the same manner as that of the second ; for the third proportion differs from the second only in the order in which the two ratios are placed; and of two equal things, it does not matter which you put first. The correctness of the second proportion proves, therefore, that of the third proportion.
3d. If you have two geometrical proportions; which have one ratio common, the two remaining ratios will, again, make a proportion; for if two ratios be equal to the same ratio, they must be equal to each other. (See Axioms, Truth. 1.). If you have the two proportions :
AB : a b
AB : a b = BC : b c you will also have the proportion
AC : ac=
BC:bc. For an illustration of this principle, we may take the two triangles, ABC, abc (Fig. I. and II.): If the sides, AB and ab, are in proportion to the sides, AC and a c, and also in proportion to the sides, BC and b c, the three sides of the triangle ABC, will be in proportion to the three sides of the triangle abc; therefore, any two sides of the first triangle will be in proportion to the two corresponding sides of the other triangle.