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**Extra info for Lectures on Ten British Mathematicians of the Nineteenth Century**

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GEORGE BOOLE (1815-1864) 36 Fig. 1. The square is a collection of points: it may serve to represent any collection of homogeneous units, whether finite or infinite in number, that is, the universe of the problem. Let x denote inside the left-hand circle, and y inside the right-hand circle. U xy will denote the points inside both circles (Fig. 1). In arithmetical value x may range from 1 to 0; so also y; while xy cannot be greater than x or y, or less than 0 or x + y − 1. This last is the principle of the syllogism.

Their great principle is the impossibility of taking the pants from a Highlander. Their only conception of the analytical processes of addition and subtraction is throwing into a heap and taking out of a heap. It does not occur to them that the processes of algebra are ideal, and not subject to gross material restrictions. If x + y denotes a quality without duplication, it will satisfy the condition (x + y)2 = x + y, x2 + 2xy + y 2 = x + y, but x2 = x, y 2 = y, ∴ 2xy = 0. CHAPTER 4. GEORGE BOOLE (1815-1864) 38 Similarly, if x − y denote a simple quality, then (x − y)2 = x − y, x2 + y 2 − 2xy = x − y, x2 = x, y 2 = y, therefore, y − 2xy = −y, ∴ y = xy.

He goes on to say, “I remark that the imaginary of ordinary algebra—for distinction call this θ—has no relation whatever to the quaternion symbols i, j, k; in fact, in the general point of view, all the quantities which present themselves, are, or may be, complex values a + θb, or in other words, say that a scalar quantity is in general of the form a + θb. ” To the pocketbook illustration it may be replied that a set of coordinates is an immense wall map, which you cannot carry about, even though you should roll it up, and therefore is useless for many important purposes.