Lectures on Ten British Mathematicians of the Nineteenth by Alexander Macfarlane

By Alexander Macfarlane

Initially released in 1916. This quantity from the Cornell collage Library's print collections was once scanned on an APT BookScan and switched over to JPG 2000 structure by means of Kirtas applied sciences. All titles scanned hide to hide and pages may perhaps contain marks notations and different marginalia found in the unique quantity.

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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.

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