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536. The circle. This is stated in Sect. 52 to have for its area the square of the diameter x 785398, or as a very close approximation, eleven fourteenths of the square of the diameter.
In Fig. F 41, the remarks in the present Section are applied to a circle having the radius of one inch, and consequently the diameter of two inches.
The figure shows that the square of the diameter contains the four squares of the radius marked A, B, C, and D, the two equivalent squares E and F being drawn, and showing he square of diameter, also contains as area the squares A, B, E, and F.
According to the practical rule given in Sect. 52, the circle will contain eleven fourteenths of A, B, E, F.
To apply the rule. From c at any angle to the line cr draw the indefinite line cx, and on it set off fourteen equal parts. From the fourteenth part draw line 14r, and from the eleventh part a parallel 11p, cutting the line cr at s, and which will give eleven fourteenth parts of that line, and line st will give the like proportion of the four squares (F 11), the part beyond the third square being (Fig. 37) fourteen hundredth parts, and giving square of radius x 3.14, the only difference from the more exact area being only .0015927 of a foot.
54. General diagram for the drawing similar figures, with areas in any required proportions ; with suggestions for preparation of enlarged diagram for offices or workshops.
In Fig. F 16, and Sect. 27, a diagram applicable to lines was given and explained.
In the present diagram, Fig. F 36 is applicable to areas. It is in this work limited to three inches in diameter and
to twelfth parts, but, as afterwards explained, may be extended without inconvenience so as to give the proportions in fiftieth parts.
The diagram is thus prepared :-On the given diameter (ac) draw the semicircle abc, and divide the diameter into the desired number of equal parts, say twelve.
From the points of division draw the perpendicular to the semicircle and draw the chords la, lc, 2a, 2c, &c., as in the figure, and mark the numbers as therein given.
Then in drawing any similar figures, with the correspond ing sides equal to la and lc the figures would have the ratio in area of 1 and 11, and with la and ac as 1 and 12.
With 2a and 2c the ratio would be 2 and 10, with 3a and 3c, 3 to 9, and so on according to the numbers in the figure
The sides may be enlarged or contracted, as under Fig. 33 and Sect. 50.
For an office or workshop the diameter might be so extended as to be made applicable to areas of much larger extent, and might be divided into fiftieth parts if desired, even with a diameter of 123 inches.
For finding the proportionate area of different figures see Sect. 49, &c., ante.
55. Having the base (ab) of a triangle and angles or direction of the sides (az and bz) given (Fig. 42), to find geometrically and also arithmetically,
1. The point y in the base, of the perpendicular to the
(1.) To find point y. (Base of perpendicular to apex.) (Geometrically.)
(a) With any assumed length (say 12 parts) draw perpendicular fg (points ag and z covering), and with same length draw perpendicular hi (points bi and 2 covering), or on the ground only points f and g and h and i will be marked.
(b) From a, at any angle, draw indefinite line ax ; on it set off any convenient multiple (say 3) containing part of base af, extending to point j, which mark, and from j set off jk = three times hb.
(c) From k draw kb, and parallel to it jy, cutting line ab at y, the point required (F 11), ay:af:: by:bh. (Arithmetically.) Measurements given in figure entire base = 100.
(16 + 9) :.100:: 16 : 64 ::9:36. (2.) Length of perpendicular yz.
(Geometrically.) In similar triangles afg and ayz, af:fg :: ay: yz, .. yz =(fg xay) - af; or, bh: hi :: by : Yz, .. yz= (hi x by)=-bh. [To test accuracy in large operations.]
(Arithmetically.). 16:12:: 64:48; or, 9:12:: 36 : 48. (3.) Length of side az. af : ag :: ay: az, .: az=(ag xay) - af; or, 16:20 :: 64:
(4.) Length of side bz. bh : bi :: by: b2, .. bz=(bi x by)--bh; or, 9:15:: 36: 60.
5. Area of triangle azb (half perpendicular x base or converse), ab x tyz; or, kab.xyz; 100+ 24 ; or, 50 x 48= 2400=area.
The solutions here given may be made of great practical use in measuring distances or in surveying work.
1. As this Division is of a thoroughly practical character and depends on but few propositions, and as I took great pains in the preparation of my work on Practical Geometry to make the Division applicable to partitions as compre. hensive and as interesting as possible, I feel that my best course is to insert a considerable part of that Division in the present work with some slight alterations.
The Division is not only useful and interesting in its direct geometrical bearing, but is of the greatest value in the training it gives to the mind in the exercise of the reasoning powers for any purpose and in the most effectual way, namely, the habit of separating difficulties, and overcoming them successively.
2. The subject of this Division, although scarcely noticed in English works on geometry, is one to which French works are very extensively directed.
The great subdivision of land in France, necessitates the division or partition of fields and parcels of land of various forms amongst several persons, and not only calls for the application of geometry to the partition of the land into equal or proportional parts, but in order to secure to each owner the use of a pond, a well, a pit, a road, or a river, also calls for the division from or to particular points, so as to
include, as part of every share, the use of such pond, well, &c.
3. The most interesting work which has come to my notice as regards the partition of land is one by Col. Guy, a French officer of artillery, entitled, “Guide pratique du Géomètre arpenteur.” On the subject of mensuration generally that work will be found of very great value, though, like most French mathematical works, it is published at very moderate price (3fr. 50c.).
Several of the figures in the present Division are modifications of figures in M. Guy's work.
4. In almost all cases the partitions are effected solely by the application of the proposition that “triangles on equal bases and between. equidistant parallels” (that is, being of the same perpendicular height) “are equivalent,” or contain the same area. (See Div. C ante, Sect. 17.) In Euclid this forms several propositions in book 1, and the rule is even then left incomplete, as shown in Div. C ante, Sect. 17; also in the English editions the triangles are loosely and inaccurately stated to be equal instead of equivalent.
The same mistake is probably in the original, as in the Latin edition of the work by Tacquet, which gives propositions 37 aud 38 of Fuclid (the same and equal bases) in a combined form ; the words are “Triangula super basi eadem vel æquali, inter ejusdem parallelas constituta, sunt æqualia.”
This inaccuracy has been before referred to, but it has so important a bearing on the present Division that attention is again directed to it.