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Lett a

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a'c

a" therefore

and hence a' <a; but a" bd

T'à 4īd

ac > a, therefore

bd The proof is similar when any other of the numbers is interminate.

2. When any two of the numbers, as a and d, are interminate. a

a"

a'c
ū ū

Then
dd bd

(1st case), and a'c ac

a' c <üd

therefore bd' to ac or a" <a; but a'

d' ac

bở The proof is similar when any other two are interminate. The third case, when three are interminate, may be similarly proved by referring to the second case; and the fourth case, when they are all interminate, is similarly proved by referring to the third case.

X. The second property of numbers stated in the scholium after the definitions of the additional fifth book, may be proved thus. Let be any vulgar fraction. Reduce 1

to a decimal, and it is obvious, that, since the remainders cannot exceed n -1, after the first n -1 terms of the quotient are found, the remainder, if the decimal is not finite, must be the same as some of the preceding remainders, or else 1; and hence the decimal will be a circulate, the numbers of figures in which will not exceed n- -1. Then the product of this circulate by m, will be a circulate consisting of the same number of figures, and its value =

m

n

n

m

n

The tenth proposition of the book on the quadrature of the circle, may be easily proved by means of two of the propositions in the additional fifth book:

:

a 6

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:

=

=

Since AB : M=(A:M, B: M) (VI. 23), and (A:M, B:M)=(a:1,6:1) (Ad. V.7) = î i (Ad. V. 26)= M)

Ī ab; therefore A·B:M= ab, or A·B = abM2. In this demonstration a and 6

may

be either terminate or interminate.

The cause for introducing the new terms, terminate and interminate, is that the terms, rational and irrational, are not sufficiently general; for the term irrational, applied to numbers, properly comprehends only that class of interminate numbers, that results from extracting the roots of numbers that are not complete powers.

THE END.

Edinburgh : Printed by W. and R. Chambers, 19, Waterlon Place.

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