. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 46 GEOMETRICAL RECREATIONS [CH. Ill OG = OE. Similarly, since HO bisects GB and DA and is per- pendicular to them, we have OD = OA. Also, by construction, DC = AE. Therefore the three sides of the triangle ODG are equal respectively to the three sides of the triangle OA E. Hence, by Euc. I. 8, the triangles are equal; and therefore the angle ODG is equal to the angle OAE. Again, since HO bisects DA and is perpendicular to it, we have the angle OD A equal to the
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 46 GEOMETRICAL RECREATIONS [CH. Ill OG = OE. Similarly, since HO bisects GB and DA and is per- pendicular to them, we have OD = OA. Also, by construction, DC = AE. Therefore the three sides of the triangle ODG are equal respectively to the three sides of the triangle OA E. Hence, by Euc. I. 8, the triangles are equal; and therefore the angle ODG is equal to the angle OAE. Again, since HO bisects DA and is perpendicular to it, we have the angle OD A equal to the angle OAD. Hence the angle ADG (which is the difference of ODG and ODA) is equal to the angle DAE (which is the difference of OAE and OAD). But ADG is a right angle, and DAE is necessarily greater than a right angle. Thus the result is impossible. Second Fallacy*- To prove that a part of a line is equal to the whole line. Let ABG be a triangle; and, to fix our ideas, let us suppose that the triangle is scalene, that the angle B is. acute, and that the angle A is greater than the angle G. From A draw AD making the angle BAD equal to the angle C, and cutting BG in D. From A draw AE perpendicular to BG. The triangles ABG, ABD are equiangular; hence, by Euc. vi. 19, A ABG : A ABD =AG' : AD\ Also the triangles ABG, ABD are of equal altitude; hence, by Euc. vi. 1, A ABG : AABD=BG : BD, .-. AC>:AD*=BC : BD. AG* AD' " BG ~ BD' * See a note by M. Cocooz in VIllustration, Paris, Jan. 12, 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
