By George Boole
This Elibron Classics e-book is a facsimile reprint of a 1877 variation by way of Macmillan and Co., London.
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Additional info for A treatise on differential equations
An important property of a continuous function f which we can prove immediately is that if f( x) is equal to a for some value of x and to b for some other value, then it must take all values between a and b as x varies from one value to the other. This property can be used to prove that, if n is a positive non-zero integer, every positive number d has an nth root; because we can show that xn is a continuous function of x, and can be made to range from 0 to a value greater than d. Therefore for some value of x it must be equal to d.
12) could be taken as the axioms of a certain arithmetical system. If we were to replace the law (x + y)·z = x·z + y·z by the law x·(y + z) = x·y + x·z, we should get a slightly different but equally good set of axioms for the same system—the same because clearly either law can be deduced from the other, using the commutative law for multiplication. In order to know the simplest properties of whole numbers we must know what a whole number is, that is, we must consider not only axioms but also definitions.
The tribesmen can then distinguish between groups of 17 and of 18 by counting, but not between groups of 25 and of 75. By the comparison method, on the other hand, the 25 and the 75 are far more easily distinguished than are the 17 and the 18. Comparison does not, however, depend on merely looking at the two groups. The crude method can be refined. 2. Let us imagine that a master of ceremonies wants to know whether there are more men or more women at a dance, and that he uses the comparison method, not just by glancing at the dancers but by asking everyone to take a partner.
A treatise on differential equations by George Boole