Elements of Algebra

486 ADDITIONS. CHAP. I. the fraction $ \frac{V}{V'} $ itself; therefore, this fraction must also terminate one of the above two series, but the other series will go on to infinity. In fact, suppose that $ \delta $ is the last denominator of the continued fraction; then $ \frac{D}{D'} $ will be the last of the principal fractions, and the series of fractions greater than $ a $ will be terminated by this same fraction $ \frac{D}{D'} $. Now, the other series of fractions less than $ a $, will naturally stop at the fraction $ \frac{C}{C'} $, which precedes $ \frac{D}{D'} $; but to continue it, we have only to consider that the denominator $ \varepsilon $, which must follow the last denominator $ \delta $, will be $ = \infty $ (Art. 3); so that the fraction $ \frac{E}{E'} $, which would follow $ \frac{D}{D'} $ in the series of principal fractions, would be $ \frac{\infty D+C}{\infty D'+C'} = \frac{D}{D'} $*; now, by the law of intermediate fractions, it is evident that, since $ \varepsilon = \infty $, we might insert between the fractions $ \frac{C}{C'} $ and $ \frac{E}{E'} $, an infinite number of intermediate fractions, which would be $ \frac{D+C}{D'+C'} $, $ \frac{2D+C}{2D'+C'} $, $ \frac{3D+C}{3D'+C'} $ &c. So that in this case, after the fraction $ \frac{C}{C'} $, in the first series of fractions, we may also place the intermediate fractions we speak of, and continue them to infinity. 19. *Problem*. A fraction expressed by a great number of figures being given, to find all the fractions, in less terms, which approach so near the truth, that it is impossible to approach nearer without employing greater ones. * Because an infinite quantity cannot be increased by addition; and therefore $ \infty D+C = \infty D $, and $ \infty D'+C' = \infty D' $; consequently, $ \frac{\infty D+C}{\infty D'+C'} = \frac{\infty D}{\infty D'} = \frac{D}{D'} $. B.