Elements of Algebra

CHAP. II. ADDITIONS. 525 natural numbers, that are not squares, as far as 100. This Table being useful on various occasions, and particularly for the solution of indeterminate numbers of the second degree, as we shall afterwards find (Chap. 7), we shall here present it to our readers. It will be observed, that there are two series of integers answering to each radical number; the upper is that of the numbers $r^0, -r', P'', -P'''$, \ldots and the under that of the numbers, $\mu, \mu', \mu'', \mu'''$, \ldots ✓ $\sqrt{2}$ $1\ 1\ 1\ 1\ \ldots$ $1\ 2\ 2\ 2\ \ldots$ ✓ $\sqrt{3}$ $1\ 2\ 1\ 2\ 1\ 2\ 1\ \ldots$ $1\ 1\ 2\ 1\ 2\ 1\ 2\ \ldots$ ✓ $\sqrt{5}$ $1\ 1\ 1\ 1\ \ldots$ $2\ 4\ 4\ 4\ \ldots$ ✓ $\sqrt{6}$ $1\ 2\ 1\ 2\ 1\ 2\ 1\ \ldots$ $2\ 2\ 4\ 2\ 4\ 2\ 4\ \ldots$ ✓ $\sqrt{7}$ $1\ 3\ 2\ 3\ 1\ 3\ 2\ 3\ 1\ \ldots$ $2\ 1\ 1\ 1\ 4\ 1\ 1\ 1\ 4\ \ldots$ ✓ $\sqrt{8}$ $1\ 4\ 1\ 4\ 1\ 4\ 1\ \ldots$ $2\ 1\ 4\ 1\ 4\ 1\ 4\ \ldots$ ✓ $\sqrt{10}$ $1\ 1\ 1\ 1\ \ldots$ $3\ 6\ 6\ 6\ \ldots$ ✓ $\sqrt{11}$ $1\ 2\ 1\ 2\ 1\ 2\ 1\ \ldots$ $3\ 3\ 6\ 3\ 6\ 3\ 6\ \ldots$ ✓ $\sqrt{12}$ $1\ 3\ 1\ 3\ 1\ 3\ 1\ \ldots$ $3\ 2\ 6\ 2\ 6\ 2\ 6\ \ldots$ ✓ $\sqrt{13}$ $1\ 4\ 3\ 3\ 4\ 1\ 4\ 3\ 3\ 4\ 1\ \ldots$ $3\ 1\ 1\ 1\ 1\ 6\ 1\ 1\ 1\ 1\ 6\ \ldots$ ✓ $\sqrt{14}$ $1\ 5\ 2\ 5\ 1\ 5\ 2\ 5\ 1\ \ldots$ $3\ 1\ 2\ 1\ 6\ 1\ 2\ 1\ 6\ \ldots$ ✓ $\sqrt{15}$ $1\ 6\ 1\ 6\ 1\ 6\ 1\ \ldots$ $3\ 1\ 6\ 1\ 6\ 1\ 6\ \ldots$ ✓ $\sqrt{17}$ $1\ 1\ 1\ 1\ 1\ \ldots$ $4\ 8\ 8\ 8\ 8\ \ldots$ ✓ $\sqrt{18}$ $1\ 2\ 1\ 2\ 1\ 2\ 1\ 2\ 1\ \ldots$ $4\ 4\ 8\ 4\ 8\ 4\ 8\ 4\ 8\ \ldots$ ✓ $\sqrt{19}$ $1\ 3\ 5\ 2\ 5\ 3\ 1\ 3\ 5\ 2\ 5\ 3\ 1\ \ldots$ $4\ 2\ 1\ 3\ 1\ 2\ 8\ 2\ 1\ 3\ 1\ 2\ 8\ \ldots$ ✓ $\sqrt{20}$ $1\ 4\ 1\ 4\ 1\ 4\ 1\ 4\ 1\ \ldots$ $4\ 2\ 8\ 2\ 8\ 2\ 8\ 2\ 8\ \ldots$ ✓ $\sqrt{21}$ $1\ 5\ 4\ 3\ 4\ 5\ 1\ 5\ 4\ 3\ 4\ 5\ 1\ \ldots$ $4\ 1\ 1\ 2\ 1\ 1\ 8\ 1\ 1\ 2\ 1\ 1\ 8\ \ldots$