# Who replaced the solid Halmos QED symbol with the open one?

Who replaced the Halmos solid QED symbol $\blacksquare$ with the open square $\square$? In his 1955 "general topology" book, page vi, John Kelley attributed the original solid square to Paul Halmos.

The end of each proof is signalized by $\blacksquare$. This notation is also due to Halmos.

To be precise, Kelley used a solid rectangle, about 7 points height above the line, 2 points depth below the line, and a width of 3.2 points. But I can't find such a solid rectangle in this TeX symbol manual for use in Mathjax: http://www.math.harvard.edu/texman/node21.html.

In 1958, Howard Eves in "Foundations and fundamental concepts of mathematics", page 149, wrote:

The modern symbol $\square$, suggested by Paul R. Halmos, or some variant of it, is frequently used to signal the end of a proof.

Even though Eves used the open square, he didn't attribute the change from solid to open to any particular author. The other books where I have seen any historical comments at all on this subject all simply refer to Halmos. No one gives any credit for who used the open square first.

Q1. Does anyone have any references which use the open square earlier than 1958?

Q2. Does anyone know of a definitive attribution for the open square?

Q3. Is it possible that the open square came first, before the solid square?

As you would expect, there are several mentions in wikipedia, but none of them definitive. The closest to an explanation is this apparently self-contradictory page: https://en.wikipedia.org/wiki/Tombstone_(typography). I don't have full confidence in the accuracy of the Halmos quote on that page. There's also Q.E.D., Mathematical proof and List of mathematical symbols.

• Is this really about history of science and math ? – Alexandre Eremenko Oct 8 '15 at 19:43
• Handwriting a filled square is no fun, I write open squares on the blackboard (and filled ones by LaTeX' AMS theorem environments). – vonbrand Oct 8 '15 at 21:53
• @Alexandre Eremenko: Yes indeed. It is a question about the history of mathematical notation. In mathematics more than in any other subject, it is generally recognized that the choice of notation has an enormous effect on the ability to express ideas and to think clearly about ideas. Many times in the history of mathematics, the invention of a new notation had substantial benefits. Conversely, the English insistence on using the Newtonian dot-notation for derivatives is generally believed to have held back English mathematics for a hundred years. I can give references for that if you like. – Alan U. Kennington Oct 9 '15 at 5:49
• @vonbrand: That makes me wonder if it's something like the development of the blackboard bold fonts as a way of indicating bold on a blackboard. Now BB is part of regular typography. My personal preference for the open square in printed mathematics is because printers sometime have difficulties printing solid black. It shows up the imperfections on the cylinder of a laser printer. But possibly it was the blackboard which incubated the open square. The quote from Halmos was 1985, and no doubt by then, the open square had become more usual. – Alan U. Kennington Oct 9 '15 at 5:56