And, if we take the four first terms, making

we have the proportion of Metius, more exact than either of the preceding. These results are alternately greater and lefs than the truth.

5. Among continued fractions, thofe have been peculiarly diftinguished in which the denominators, after a certain number of changes, are continually repeated in the fame order. Such is the fraction

計六
2+ 3+, &c.

;

The amount of this fraction, though continued ad infinitum, may be easily found for, if x be the amount of all the terms but the first, which is an integer, (= 1), so that

I

2

+ 3+ 2+3+, &c.

Then, fince, after the 2d, all the fractions return in the fame order, their amount is alfox; therefore

x=

the denominators did not return in the fame order till after a greater interval, the value of the fraction would ftill be expreffed by the root of a quadratic equation; and converfely, the roots of all quadratic equations may be expreffed by periodical continued fractions, and by that means may often be very readily approximated in numbers, without the extraction of the fquare root.

6. The reduction of decimal, into the form of continued fractions, fometimes renders the law of their continuation evident.

Thus,

Thus, we know that

21.4121356+;

but, from the bare inspection of this decimal, we difcover no rule for its farther continuation. If, however, it be reduced

into a continued fraction, we find it

and we see in what manner it may be car

ried on to any degree of exactness.

APPENDIX.

No. II.

Indeterminate Problems.

HE indeterminate problems of the

TH

firft and fecond order, after all the unknown quantities, but two, have been exterminated, lead to equations of four different forms.

In all these formulas, a, b, c, d, denote given numbers. In the three first, x is to

be