Halve Quotes
"Thank God for the French army." When we read about Germany, when we watch with surprise and distress the tumultuous insurgence of ferocity and war spirit, the pitiless ill-treatment of minorities, the denial of the normal protections of civilised society to large numbers of individuals solely on the ground of race-when we see that occurring in one of the most gifted, learned, scientific and formidable nations in the world, one cannot help feeling glad that the fierce passions that are raging in Germany have not found, as yet, any other outlet but upon themselves. It seems to me that, at a moment like this, to ask France to halve her army while Germany doubles hers... to ask France to halve her air force while the German air force remains whatever it is... such a proposal, it seems to me, is likely to be considered by the French Government at present, at any rate, as somewhat unseasonable.
Winston Churchill
Take a unit, halve it, halve the result, and so on continually. This gives-1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.;Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c;... We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added.
...We say that-1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c.; &c.;is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. ...it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please.
Augustus De Morgan