Comments on Baron's Paper on "Power"

Article IX in The Mathematical Correspondent (1804, pp. 59-66), which is by George Baron (1769-1812), is entitled "A short Disquisition, concerning the Definition, of the word Power, in Arithmetic and Algebra." The definition that Baron criticizes is that of power:

the powers of any number, are the successive products, arising from the continual multiplication of that number, into itself. [p. 59]

In these and other quotations the original capitalization, font, punctuation, and spelling have been preserved. I say this because earlier today I read a review of Eats, Shoots & Leaves, The Zero Tolerance Approach to Punctuation, by Lynne Truss [The New York Times Book Review, April 25, 2004, p. 7]. Having read this review, I could not help but noting that Baron is in love with the comma.

Baron argues that from this definition "it must incontrovertibly follow, that, the first power of a number, is produced by multiplying that number, by itself". Thus we have

5¹ = 5 x 5 = 25,     5² = 25 x 5 = 125,    5³ = 125 x 5 = 625,  .  .  .

Sadly, he is correct, so Baron proposes the following definition:

The powers of any number, are the successive products, arising from unity, continually multiplied, by that number. [p. 61]

Thus, using this "universally correct" definition, Baron obtains

5¹ = 1 x 5 = 5,     5² = 1 x 5 x 5 = 25,    5³ = 1 x 5 x5 x x 5 = 125,  .  .  .

Admittedly, this is a point of pedantry, but Baron is correct. Actually, this is a rather clever way to state the definition. Today we define powers recursively:

a¹ = a

aⁿ⁺¹ = a¹ x a

[See, for example, Garrett Birkhoff and Saunders MacLane, A Survey of Modern Algebra, 1953 edition, p. 12.]

So far, so good. But next Baron considers "the nothingth powers of numbers". First Baron tries to explain the concept of nothing and then quotes "The learned Mr. Emerson". William Emerson (1701-1782) wrote books on Geometry, Algebra, Combinatorics, The Arithmetic of Infinites, and Fluxions. He is not to be confused with Frederick Emerson (1788-1857) who later locked horns with Charles Davies over copyright on an arithmetic text. We shall spare the reader this discussion, for it is quite unclear. But there is a footnote of interest

Mr. Emerson wrote this problem in the English language, but it has lately been translated into nonsense, by Jared Mansfield of Connecticut, and published in a wonderful book, which this translator calls Mathematical Essays. The indefinitely small nothings of Connecticut, are infinitely great absurdities, in the regions of science and common sense. [p. 63]

Now Baron is certainly aware that Mansfield succeeded him as professor of mathematics at the Military Academy, so it is strange that he refers to him as "of Connecticut." However, it is quite understandable that Baron is antagonistic to Mansfield for he succeeded him as professor of mathematics at the Military Academy.

Let us examine Baron's argument.

if  the multiplication by x, be abstracted from this first power of x, by means of division; the power will become nothing, but the unit will remain: for x¹/x = (1 * x)/x = 1, [for clarity, I have used * instead of x for multiplication] and hence it is plain that x⁰ = 1, when x represents any number whatsover. [p. 63]

Thus, in particular, it follows that  x⁰ = 1 and even  0⁰ = 1.

Now in these conclusions, Baron was ahead of his time. While the arguments he gives in support of his conclusions are certainly dubious, the results he obtains are correct [alas, many readers today will disagree, but they are wrong].

In the Benjamin Vaughn () papers at the American Philosophical Library in Philadelphia, there is a box of "Misc. Mathematical Notes" (box 6) that contains about an inch of loose notes. They are very incoherent, but some information can be gained from them. Most of them deal with Baron's paper on powers. There are numerous drafts, with many passages crossed out. It would take a good deal of close work to sort these out. Folder 7, dated July 19, 1805 of the Vaughn papers contains a clean copy of Vaughn's commentary, but I found the loose notes more interesting. From this clean copy one confirms that Baron is a native of England "who is said to have considerable skill in certain branches of mathematical investigation".

More items from the "Misc. Mathematical notes":

There is an envelope addressed to "Benjamin Vaughin Esq / Hallowell / Kennebock", confirming that Vaughan was once in Hallowell. Unfortunately, the document is not dated.