By E. A. Maxwell

As Dr Maxwell writes in his preface to this e-book, his target has been to coach via leisure. 'The basic idea is improper concept may well usually be uncovered extra convincingly through following it to its absurd end than by way of only asserting the mistake and beginning back. hence a couple of by-ways look which, it really is was hoping, may perhaps amuse the pro, and aid to tempt again to the topic those that suggestion they have been wasting interest.' the traditional of data anticipated within reason straight forward. generally a simple assertion of the improper argument is through an publicity within which the mistake is traced to the main undemanding resource, and this technique usually ends up in an research that is usually of unforeseen intensity. Many scholars will notice simply how mathematically minded they're after they learn this publication; neither is that the single discovery they'll make. lecturers of arithmetic in colleges and technical colleges, faculties and universities may also be certain to locate whatever right here to delight them.

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These are the circles through the apses of the curve, that is, the points where the tangent is perpendicular to the radius. (The formula p = rdrjdp requires care at such points of the curve, since dr and dp are both zero. ) FALLACY 3. The explanation of this fallacy is comparatively simple, for it is a case of pure misconduct. The formula c b sinfJ sina requires D to be the middle point of BO; but D' is not the middle point of B'O', so the differentiation is not legitimate-a and fJ would have to vary too.

60 FALLACIES IN MATHEMATICS OX Then - = ou I (2u)-~ = X-I. Since ;: is independent of x, it follows that l/x is independent of x. II. FALLACY THE COMMENTARY l. The difficulty may best be explained by using the language of differentials. The relation r2 = 2ax-a 2+b 2 is perfectly correct, so that, taking differentials on both sides, rdr = adx. For a maximum or minimum value of r, dr must be zero, and this happens (a not being zero) precisely where dx = o. The turning values of r arrive at the turning values of x.

Cos 2 X = 1 - sin2 x, it follows that 1 + cosx = 1 + (l-sin2 x)l, or, squaring each side, = {I + (l-sin 2 x)I}2. (1 +COSX)2 In particular, when x = 1T, = {I + (1- O)lp, 0= (1 + 1)2 (1-1)2 or = 4. ~ 2. THE Since FALLACY THAT 1= +1 = -1. i = -1. The two fallacies which follow are somewhat similar at the critical point. It is perhaps worth while to include both for the sake of the geometry with which they are clothed. ([ 3. THE FALLACY THAT ALL LENGTHS ARE EQUAL. GIVEN: A, C, D, B are four points in order on a straight line.