Elements of Algebra

CHAP. I. ADDITIONS. 489 manner: $4, \quad 7, \quad 1, \quad 3, \quad 1, \quad 16, \quad 1, \quad 1, \quad 15.$ $\frac{4}{1}, \quad \frac{29}{7}, \quad \frac{33}{8}, \quad \frac{128}{31}, \quad \frac{161}{39}, \quad \frac{2704}{655}, \quad \frac{2865}{694}, \quad \frac{5569}{1349}, \quad \frac{86400}{20929}.$ the last fraction being the same as the one proposed. In order to facilitate the formation of these fractions, we first write, as is here done, the series of quotients $4, 7, 1, \ldots$ and place under these coefficients the fractions $\frac{4}{1}, \frac{29}{7}, \frac{33}{8}, \ldots$ which result from them. The first fraction will have for its numerator the number which is above it, and for its denominator unity. The second will have for its numerator the product of the number which is above it by the numerator of the first, plus unity, and for its denominator the number itself which is above it. The third will have for its numerator the product of the number which is above it by the numerator of the second, plus that of the first; and, in the same manner, for its denominator, the product of the number which is above it by the denominator of the second, plus that of the first. And, in general, each fraction will have for its numerator the product of the number which is above it by the numerator of the preceding fraction, plus that of the second preceding one; and for its denominator the product of the same number by the denominator of the preceding fraction, plus that of the second preceding one. So that $29 = 7 \times 4 + 1$, $7 = 7$; $33 = 1 \times 29 + 4$, $8 = 1 \times 7 + 1$; $128 = 3 \times 33 + 29$, $31 = 3 \times 8 + 7$, and so on; which agrees with the formulæ of Art. 10. Now, we see from the fractions $\frac{4}{1}, \frac{29}{7}, \frac{33}{8}, \ldots$ that the simplest intercalation is that of one day in four common years, which is the foundation of the Julian Calendar; but that we should approximate with more exactness by intercalating only $7$ days in the space of $29$ common years, or eight in the space of $33$ years, and so on. It appears farther, that as the fractions $\frac{4}{1}, \frac{29}{7}, \frac{33}{8}, \ldots$ are alternately less and greater than the fraction $\frac{86400}{20929}$, or $\frac{24^h}{5^h.48'.49''}$, the intercalation of one day in four years would be too much, that of seven days in twenty-nine years too little, that of eight days in thirty-three years too much, and so on; but each of these intercalations will be the most exact that it is possible to make in the same space of time. Now, if we arrange in two separate series the fractions