. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. Ill] GEOMETRICAL RECREATIONS 49 OE perpendicular to AG produced. Draw OF perpendicular to AB produced. Join OB, OG. Following the same argument as before, from the equality of the triangles AOF and AOE, we obtain AF= AE; and, from the equality of the triangles BOF and GOE, we obtain FB = EG. Therefore AF-FB = AE- EG, that is, AB - A Thus in all cases, whether or not DO and AO meet, and whether they meet inside or outside the triangle, we have AB=AG: and
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. CH. Ill] GEOMETRICAL RECREATIONS 49 OE perpendicular to AG produced. Draw OF perpendicular to AB produced. Join OB, OG. Following the same argument as before, from the equality of the triangles AOF and AOE, we obtain AF= AE; and, from the equality of the triangles BOF and GOE, we obtain FB = EG. Therefore AF-FB = AE- EG, that is, AB - A Thus in all cases, whether or not DO and AO meet, and whether they meet inside or outside the triangle, we have AB=AG: and therefore every triangle is isosceles, a result which is impossible. Fifth Fallacy*. To prove that 7r/4 is equal to v/3. On the hypothenuse, BG, of an isosceles right-angled triangle, DBG, describe an equilateral triangle ABG, the vertex A being on the same side of the base as D is. On GA take a point H so that CH = CD. Bisect BD in K. Join HK and let it cut GB (produced) in L. Join DL. Bisect DL at M, and through M draw MO perpendicular to DL. Bisect HL at N, and through N draw NO perpendicular to HL. Since DL and HL intersect, therefore MO and NO will also intersect; moreover, since BDG is a right angle, MO and NO both slope away from DG and therefore they will meet on the side of DL remote from A. Join OG, OD, OH, OL. The triangles OMD and OML are equal, hence OD = OL. Similarly the triangles ONL and ONH are equal, hence OL = OH. Therefore OD = OH. Now in the triangles OGD and OGH, we have OD = OH, CD = GH (by construction), and * This ingenious fallacy is due to Captain Turton: it appeared for the first time in the third edition of this work. B. R, i. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly resemble the original Ball, W. W. Rouse (Walter William Rouse), 1850-1925. London, Macmillan
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Keywords: ., bookcentury1900, bookdecade1920, booksubjectgeometry, bookyear192
