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and by increasing the polygon to BCDFA, the contour would be increased in the proportion, as side Bc or diagonal Bd to contour of smaller polygon so will BC or diagonal BD be to contour of larger polygon.
39. PROPORTIONALS OF AREAS OR SURFACES.
Introductory remarks. The previous Sections in this Division are applicable to ratios and proportions of lines, but the subsequent Sections apply to the still more important and interesting subject of proportionals as applied to areas or surfaces, the former term being more generally used.
Although on the first view of the subject it might appear that the consideration of proportionals in regard to areas must, from the varieties of forms in different figures, be attended with great complexity and difficulty, yet on investigation it will be found that the questions likely to arise may be answered by the application of a few general principles and theorems or formulæ.
This remark is not confined to figures which are similar, but extends to figures which materially differ in form, by converting them into equivalent triangles, an operation only requiring the use of a few parallel lines; and then ascertaining the side of an equivalent square, or more simply still by the use of a few more parallel lines, as subsequently explained (Sect. 42, &c.), and as shown in Fig. 29.
40. In parallelograms and also in triangles with equal altitudes ; the areas will be proportioned to the bases. (Fig. F 26.)
Note that unless otherwise expressed, altitude in geometry means the perpendicular altitude or height and not the slant
height, and in Fig. 26 it will be the line ax and not the line ab.
In the figure with the equal bases, ac, cf, and fh, the area of parallelogram acbc will equal abdc (C 17).
The areas of the parallelograms having the equal bases cf and fh will be also each equal to the area of axbc (C 17). The same also as between the triangles (C 20).
Taking the base af, the parallelogram or triangle with that base will contain twice the area of the base ac or cf, and with base ah three times the area, as will appear obvious on inspection of the figure.
41. In parallelograms and triangles with equal bases, the areas will be proportional to the altitudes. (Fig. F 27.)
In this figure the base will equal ab and the altitudes al, a2, and a3, and it represents a square divided into nine equal squares numbered from 1 to 9, but, as shown in Fig. 28, different ratios in length of altitude and base might be taken.
A mere inspection of either figure will show that with altitude al, the area will include the three squares 1, 2, and 3, or the half for the triangle, with altitude a2, the area would be the squares 1 to 6 or the half of those squares as to triangles, and with altitude a3 all the nine squares or the half as to triangles will be included.
42. In parallelograms and triangles generally, the areas will be according to the product of the bases by the altitudes.
The product of the bases by the altitudes will give the area in each case as a deduction from the last two Sections,
and the proportions will be ascertained by a comparison of the areas.
The proportions are ascertained geometrically in a very simple manner by converting one of the parallelograms into its equivalent having the same base as the other.
For example, in Fig. F 29, in which rectangle abcd represents the larger area and fghi the smaller, with different altitude and base, the length fl being equal to the larger base.
Substitute for the smaller area an equivalent rectangle having the base fl=ad.
This will be effected most easily by drawing from g to l and through point i the parallels 1 and 1', and through point k drawing km parallel to fl, and through I and parallel to fg drawing lm, which completes the rectangle flmk, having the required base and altitude = an and the areas bearing the proportion of lines an and ab.
The use of the parallels is shown in Div. C, Sect. 23b, and Fig. C, 206.
In practice the operation would be even more simple, and would be confined to marking with pencil the point in the base continued, cut by the first parallel, and the point k cut by the second parallel, and then from point k drawing kn parallel to ad.
43. The areas of similar triangles are according to the squares of their corresponding sides.
. This proposition is not confined to triangles, but extends to all similar figures, and is not only one of the most interesting propositions in geometry, but one which it is most desirable that a student should clearly understand.
Its demonstration as to triangles extends the demonstration in effect to all similar figures, or, at all events, directly to all right-lined figures which may be converted into similar triangles, and practically the application may be made to figures having curved sides.
To revert, however, to the case of similar triangles, and with a desire to give the demonstration in a more simple form than in Euclid, a demonstration in accordance with a French work (Puissant Cours de Mathématiques pour l'usage des Ecoles Militaires, p. 238) is given ; a different figure being, however, adopted in order to make the matter still more clear to students.
In Fig. F 30, let ABC and abC be similar triangles, and then (F Sect. 11);
(1) AB : ab :: AC : a'c'. (2) CF :c'f :: AC : a'c'. (3) Multiplying 1 by 2. AB x CF : ab x c'f':: AC? : ac?.
(4) For area, divide by 2. ABX CF:-2 : ab xc'f '= 2 :: AC: a'c'.
But the first and second terms in the last proportion are respectively the areas of the similar triangles, and the third and fourth terms are squares of their corresponding sides. Therefore, as area of triangle ABC is to area of triangle a'b'c', so is square of side AC to square of corresponding side a'c'.
The proportions as regards the sides BC and bc' may be shown by substituting those sides for AC and ac'.
44. The areas of similar polygons are according to the squares of their corresponding sides or diagonals.
As to parallelograms which may be divided into two
similar and equal triangles, it will be obvious that the demonstration in Sect. 43 will be also applicable.
As regards all other similar right-lined polygons, they may be divided into the same number of similar triangles (see Fig. F 21), to every one of which, taken separately, Sect. 43 would apply.
Thus in Fig. 21, with the similar figures divided into the similar triangles BCD and Bcd, BDF and Bdf, and BFA and Bfa (F, Sectns. 11 and 43), the proportion between the areas of the figures will be according to the squares of any similar sides, say BC and Bc, or similar diagonals, say BD and Bd.
The present Division only applies to proportional areas, and not the ascertainment of the actual areas, to which “ mensuration” will be applicable, but it may be as well to state that as to the areas of triangles, the area will be half the altitude multiplied by the entire base, or half the base multiplied by the entire altitude, and by substituting (Sect. 42) an equivalent triangle with a base of one inch, half the perpendicular height of the triangle will give the area in square inches and parts of an inch.
Figures with four or more sides may be converted into an equivalent triangle in a most rapid and simple manner. (See Div. C, Sect. 22.)
45. Proportionate areas in circles.
The estimates as well of the proportionate circumference of circles as also the proportionate areas are based on the sub