 Membership
 MAA Press
 Meetings
 Competitions
 Community
 Programs
 Students
 High School Teachers
 Faculty and Departments
 Underrepresented Groups
 MAA Awards
 MAA Grants
 News
 About MAA
WALLIS
It is my opinion, which I submit for the consideration of the present illustrious company, that the ratio of a positive number to a negative number may be supposed greater than infinity.
LEIBNIZ
What?! Can such a monstrous conclusion emerge so soon to disturb our complacency? This is no mere instance of stray property or mistaken domain! I declare  were this proposition put forward by any less august personage and less wellrespected mathematician than Dr Wallis, I should dismiss it out of hand as nonsensical. But come, Sir, your reasons, if you will!
WALLIS^{5}
Certainly, Herr Leibniz! Observe that the reciprocals of the positive numbers taken in descending order: oneoverfive, oneoverfour, and so on, are increasing up to the last term oneovernothing which must be infinite;

therefore if we proceed further we should conclude that the reciprocals of negative numbers: oneoverminusone, oneoverminustwo, and so on, are greater than infinity.
LEIBNIZ
There must be something amiss here ... but I cannot at the present moment fault the argument.
EULER [to audience, in agitation ]
This is intolerable! He is in grave error. I cannot be silent: Heaven grant me intercourse with the past!...
[moves to join the group ]
Pardon! my name is Euler; I beg your indulgence. I am acquainted with your work and deeply honoured to be  ah  present. But, with regard to Dr Wallis' conclusions about mixed ratios of positive and negative numbers, allow me, I pray you, to demonstrate why I reject your argument, Sir!
The fundamental error lies in supposing that anything whatever can be concluded from such sequences of reciprocals. For if, instead of reciprocals as previously, we take numbers which are the reciprocals of squares, then we obtain the sequence:


Surely, from that, noone will venture to claim that these positive terms are greater than infinity? If anyone has lingering doubts, consider the sequence of reciprocals of square roots:


LEIBNIZ
Negative quantities are problematic, and imaginary quantities worse still. What intolerable contradictions may we let loose when these be combined with infinities! The road to madness is paved with such infinite sequences! And yet  take courage, friends! Here is a notorious example I am proud to have completely solved, although when first I wrestled with this Gordian knot I despaired of success:
The problem is to find the sum of the series: one minus one plus one minus one, and so forth.



EULER [laughing ]
No doubt, Herr Leibniz, you are expecting that I will shrink from this, as an insoluble contradiction, so that you can enlighten me. But I have a good argument which shows the sum to be one half, the happy medium between your two previous sums.
LEIBNIZ [deflated ]
Oh, indeed? Let us see it.
EULER ^{7}
I take the geometric series: one plus x plus x squared plus x cubed, etc., which is well known to sum to the reciprocal of one minus x.


LEIBNIZ
I concede that you have obtained the correct sum. But I have a marvellously subtle method by which I reach that conclusion with no resort to the banalities of geometric series.
Observe! If one takes the first term, then the first two terms, and then the first three, and so on, one obtains successively the partial sums 1,0,1,0,1, and so on ad infinitum . Therefore, as candidates for sum of the infinite series we started with, both 1 and 0 are equally probable, and so clearly the most probable value of all will be their arithmetic mean, namely ^{1}/_{2}, which is therefore the true sum.^{8}
EULER
There would appear to be a far greater deal of metaphysics than mathematics in that argument.
LEIBNIZ
Oh, I will not deny that. But the celebrated Bernoullis^{9} have accepted the argument as valid, and, if the truth be told, I really do believe that there is far more metaphysical truth in mathematics than is generally realised.^{10} And I am not by any means alone in holding this opinion. Our sagacious colleague Grandi, of the University of Pisa, was so moved by the discovery that this series, (whose true sum he found to be ^{1}/_{2} by a method similar to yours) could in another form sum to nothing, that he announced that he had a proof that the world could be created out of nothing.^{11} As he saw it, by merely looking at the series in a different way, the value nothing can be transmuted into something substantial. Fiat, ex nihilo , one half!
Gavin Hitchcock , "Alien Encounters  Act 1 (continued)," Convergence (May 2011)