Defining powers is often carelessly done. Almost thirty years before Libri's first paper, George Baron published "A short Disquisition, concerning the Definition, of the word Power, in Arithmetic and Algebra" in *The Mathematical Correspondent* (1804). In this paper [1], Baron begins the discussion with the following definition:

*The powers of any number, are the successive products, arising from unity, continually multiplied, by that number*.

As an example, he writes that 1 × 5 = 5, which is the first power of 5, and 1 × 5 × 5 = 25, which is the second power of 5, etc. The first, second, etc., powers are then conveniently expressed as 5^{1}, 5^{2}, etc. In the same manner, the powers of any number *x* might be represented as *x*^{1}, *x*^{2}, etc., in which *x*^{1} = 1 × *x*, *x*^{2} = *x*^{1} × *x*, etc. After stating a few corollaries, Baron writes:

Let us, therefore, next inquire, whether the same definition, will not lead us to a clear and intelligible solution, of the mysterious paradoxes, resulting from the common definition, when applied, to what is denominated, the *nothingth* power of numbers.

Baron then addresses the rules for dividing powers (look back to the argument from the high school text), but he develops a different conclusion:

If the multiplication by *x*, be abstracted from the first power of *x*, by means of division; the power will become nothing but the unit will remain: for \(\frac{x^1}{x} = \frac{1\times x}{x} =1,\) and hence it is plain that *x*^{0} = 1, when *x* represents any number whatever. But since the number *x*, is here unlimited with regard to greatness, it follows, that, the nothingth power of an infinite number is equal to a unit.

Baron gives credit to both William Emerson (1780) [3] and Jared Mansfield (1802) [9] who wrote on the subject of "nothing." Baron takes their arguments one step further and postulates that the number *x* can be any number, great or small:

To pursue the application of our definition, to quantity in the ultimate extremity of smallness, let us suppose *x* to represent any fractional quantity; or in other words, let *x* denote any magnitude, expressed in numbers, by means of some part of its measuring unit: then by the definition *x*^{1} = 1 × *x*. Let now this multiplication by *x*, be abstracted; and for the reasons heretofore advanced, we have *x*^{0} = 1. Now since *x* here represents a fractional quantity, independent of any limitation, in respect to smallness; we may therefore suppose *x*, by means of continual diminution, or decrease, to pass from its present value, through every degree of smallness, until it become *nothing*; then it will be evident, that, during this diminution or decrease of *x*, *x*^{0} will continue equal to an invariable unit; and that *precisely at the instant*, when *x* becomes *nothing*, *x*^{0}, or 0^{0} = 1.

Baron never mentions the term *indeterminate form*, and he in fact ends his treatise with the following:

Also, since *x*^{0} = 1, whatever be the value of *x*; of consequence; in every system of logarithms, the logarithm of 1 = 0.

*Editor's note:* *This article was published in March of 2008.*