APPENDIX. No. I. Continued Fractions. HE manner of reducing fractions i. TH to their least terms has fuggefted another form, in which it is often convenient to have them expreffed. The nature of this form, or of what is called a continued fraction, will be easily understood by an example. 2. By the fame means, may any fraction be refolved into the form, and it is then called a continued fraction. This formula may be expreffed in words, by faying, that the reciprocal of d is to be added to c, the reciprocal of that fum to b, and the reciprocal of this last sum to a, in order to have the fraction required. 3. The inverse of the preceding operation, or that by which a continued fraction is reduced into the form of an ordinary one, is easily derived from this explanation. Thus, if the fraction be then the reciprocal of d added to c, or c+=+1, the reciprocal of which, I d d =cd+1' being added to b, gives b+ cd+1 bcd+b+d; and the reciprocal of this last, cd+1 viz. cd+1 viz. bcd+b+d being added to a, gives 4. One of the chief advantages derived from this manner of expreffing fractions is, that it enables us to find a feries of other fractions that approach in value to any given one, and each of them expreffed in the leaft numbers poffible. Thus, in the given example 314159, which is known to exprefs nearly the proportion of the circumference of a circle to its diameter, and which we may call, if we take only the two firft terms of the continued fraction into which it was refolved, viz. 3+/1/1, we have = و 22 7 nearly; and this is the theorem of Archimedes. If we take the three firft terms, we have which is nearer to the truth than the former. And |