This post will discuss a curious approximation with a curious history.

## Approximation

Let *x* be a number near 1, written as a fraction

*x* = *p* / *q*.

Then define *s* and *d* as the sum and difference of the numerator and denominator.

*s* = *p* + *q*

*d* = *p* − *q*

Since we are assuming *x* is near 1, *s* is larger relative to *d*.

We have the following approximation for the *n*th root of *x*.

^{n}√*x* ≈ (*ns* + *d*) / (*ns* − *d*).

This comes from a paper written in 1897 [1]. At the time there was great interest in approximations that are easy to carry out by hand, and this formula would have been very convenient.

The approximation assumes *x* is near 1. If not, you can multiply by a number of known square root to make *x* near 1. There will be an example below.

## Examples

### Positive *d*

Let’s find the cube root of *x* = 112/97. We have *n* = 3, *p* = 112, *q* = 97, *s* = 209, and *d* = 15. The approximation tells says

^{3}√*x* ≈ 642/612 = 107/102 = 1.049019…

while the exact value is 1.049096… .

### Negative *d*

The value of *d* might be negative, as when *x* = 31/32. If we want to find the fifth root, *n* = 5, *p* = 31, *q* = 32, *s* = 63, and *d* = −1.

^{5}√*x* ≈ 312/314= 156/157 = 0.9936708…

while the exact value is 0.9936703… .

*x* not near 1

If *x* is not near 1, you can make it near 1. For example, suppose you wanted to compute the square root of 3. Since 17² = 289, 300/289 is near 1. You could find the square root of 300/289, then multiply the result by 17/10 to get an approximation to √3.

## History

The author refers to this approximation as Mercator’s formula, presumable Gerardus Mercator (1512–1594) [2] of map projection fame. A brief search did not find this formula because Mercator’s projection drowns out Mercator’s formula in search results.

The author says a proof is given in Hutton’s *Tracts on Mathematics*, Vol 1. I tracked down this reference, and the full title in all its 19th century charm is

TRACTS

ON

MATHEMATICAL

AND

PHILOSOPHICAL SUBJECTS,

COMPRISING,

AMONG NUMEROUS IMPORTANT ARTICLES,

THE THEORY OF BRIDGES,

WITH SEVERAL PLANS OF IMPROVEMENT,

ALSO,

THE RESULTS OF NUMEROUS EXPERIMENTS ON

THE FORCE OF GUNPOWER,

WITH APPLICATIONS TO

THE MODERN PRACTICE OF ARTILLERY.

IN THREE VOLUMES

BY CHARLES HUTTON, LL.D. AND F.R.S. &c.

Late Professor of Mathematics in the Royal Military Academy, Woolwich.

Hutton’s book looks interesting. You can find it on Archive.org. Besides bridges and gunpowder, the book has a lot to say about what we’d now call numerical analysis, such as ways to accelerate the convergence of series. Hutton’s version of the formula above does not require that *x* be near 1.

## Related posts

- Mentally approximating factorials
- Mentally computing common functions
- Connecting powers of two and decibels

[1] Ansel N. Kellogg. Empirical formulæ; for Approximate Computation. The American Mathematical Monthly. February 1897, Vol. 4 No. 2, pp. 39–49.

[2] Mercator’s projection is so familiar that we may not appreciate what a clever man he was. We can derive his projection now using calculus and logarithms, but Mercator developed it before Napier developed logarithms or Newton developed calculus. More on that here.

It looks to be due to the mathematician Nicholas Mercator. Hutton starts discussing Mercator’s Logarithmotechnia on p405 of vol 1 at https://archive.org/details/tractsonmathemat01hutt/page/404/mode/2up?q=Mercator . He shows the approximation starting on p411, at https://archive.org/details/tractsonmathemat01hutt/page/410/mode/2up?q=%22approximate+multiplication%22 and points out that x in that approximation must “not differ greatly from unity” on p412.

Hutton says this is Mercator’s “prop. 7”, and it matches Mercator’s Propoſito VII in the archive.org copy of Logarithmotechnia at https://archive.org/details/ita-bnc-mag-00000857-001/page/n28/mode/2up .

For those curious about the formula, you can reach it by writing x as (s+d)/(s-d) and then doing a 1st order Taylor approximation over d for both numerator and denominator of (s+-d)^t with t = 1/n. The validity of the approximation hinges on d being close to 0, which is equivalent as x being close to 1.