CERTAIN FALLACIES 397

REMARKS ON CERTAIN FALLACIES.

Similar to the common proof that 2==1 is the following proof that3===1; we may build up others, as interesting exercises in Algebra:

Let .z?==a (1)then ^==a2 (2)and x^a^x (3)

also Xs�a^c^x�a3 (4)==a2 {x�a)

d?2+.2?a+a2=a2 (5)substituting o for x,

Sa^a2 (6)or 3==1 (7)

It may well be said that the fallacy consists in assuming that(5) is true; while in fact we have divided equals by x�a, or 0, whichis meaningless. Is it not helpful to note that in (2) we have introducedthe root x == �a, and in (3) the root a? = o, then to get (5) from (4)we throw out the original root x=a, but to get (6) we use again: a?=a?If in (5) we had substituted either x=�a or a?===o, the only remainingvalues of a?, we should have had a2^^2, or 1=1.

II.

A pail half full==a pail half empty (1). �. A pail full==a pail empty -(2). Multiplying equals by 2.The fallacy consists in assuming that equations compare materials,

whereas they compare numbers. Of course the number of cubic inchesin one half the volume of a pail equals the number of cubic inchesin the other half. In this sense a pailful equals a pail empty.

III.Some writers, in deriving the equation of a conic, define an ellipse

as a curve such that the sum of the distances of any of its points Pfrom two fixed points F1 and F equals a constant, 2a; thus PF^PF==2a (1) ; for the hyperbola the difference, PF1�PF==2a (2) ; but insquaring, transposing rational terms, and squaring again they get from(1) the same equation as from (2), an equation equivalent to the fourequations –PF1–PF==–2?, but not equivalent to (1) or (2). So thedifference between the two definitions is entirely lost in the non-revers-ible process of squaring. Have not the blind been leading the -blind?Goshen College. D. A. LEHMAN.

TEACHERS OF HIOH SCHOOL MATHEMATICS, PLEASEREAD.

The Committee of the Central Association of Science and MathematicsTeachers on Unifying Secondary Mathematics, want the names of allteachers who are attempting to introduce a course in mathematics in whicharithmetic, algebra and geometry are taught as. much as possible as closelyrelated parts of one subject, mathematics. They urge any teacher who isengaged in working out such a .course, or who knows some one who is, tocommunicate the fact to the committee. Address Walter W. Hart, Short-ridge H. S., Indianapolis, Ind.