NO. II
ALMAGEST; BOOK II. EXTRACT FROM CHAP. IX
Of Circumstances regulated by Ascensions
IN any climate whatever, the magnitude of a given day or night is to be computed by the number of ascensional times proper to that particular climate. For example, the magnitude of the day will be ascertained by numbering the times between the Sun's zodiacal degree and the degree diametrically opposite, in the succession of the signs; and that of the night, by numbering the times, from the degree diametrically opposite to the Sun, onwards, in the order of the signs, to be degree actually occupied by the Sun: because, by dividing the respective amounts of these times so obtained, by fifteen, the number of equatorial hours belonging to each space will be exhibited; and if the division be made by twelve, instead of fifteen, the result will show the numbers of degrees equivalent to one temporal hour of either of the said spaces respectively.
The magnitude of any temporal hour may be, however, more easily found by referring to the annexed Table of Ascensions, and taking the difference between the respective aggregate numbers, inserted therein under the heads of the equinoctial parallel or right sphere, and of any particular climate for which the magnitude of the temporal hour is required; and, if the said hour be a diurnal hour, the aggregate times as stated against the zodiacal degree occupied by the Sun; but, if nocturnal, those stated against the degree diametrically opposite, are to be compared; and the sixth part of the difference between them is to be added, if the said degree be in the northern signs, to the fifteen times of an equatorial hour; but subtracted therefrom, if in the southern signs. The amount thus obtained will be the required number of degrees of the temporal hour in question.
And if it be required to reduce the temporal hours of any given day or night, in a certain climate, into equatorial hours, they must be multiplied by their proper horary times, whether diurnal or nocturnal, as the case may be; the product is then to be divided by fifteen, and the quotient will necessarily be the number of equatorial hours in the climate in question, on the given day or night. On the other hand, equatorial hours are also to be reduced into temporal hours by being multiplied by fifteen, the product of which is to be divided by the horary times proper to the given day or night in the said climate.
The degree ascending in the ecliptic, at any given temporal hour, may also be ascertained by multiplying the number of temporal hours since sunrise, if the given hour be diurnal, but if nocturnal, since sunset, by their proper horary times; and the product is to be added, in the succession of the signs, to the aggregate number (as shown by the ascensions proper to the climate) of the Sun's degree, if the given hour be diurnal, but, if nocturnal, to that of the degree diametrically opposite, and that particular degree of the ecliptic which shall correspond with the total number thus found in the ascensions of the climate will be the degree then ascending.
But, in order to ascertain the degree on the meridian above the earth, the number of temporal hours since the preceding noon are also to be multiplied by their proper horary times, and the product is to be added to the aggregate number of the Sun's right ascension; and that degree of the ecliptic, with which the total number as found in the aggregate times of right ascension shall correspond, will then be on the meridian. 1 The degree on the oriental horizon will, however, also show what degrees occupies the meridian; for, by subtracting 90 times (the amount of the quadrant) from the aggregate number ascribed to the said ascending degree in the Table proper to the climate, the number so reduced will be found, in the aggregate times of the Table of Right Ascension, to correspond with the degree on the meridian. And again, on the other hand, by adding 90 to the aggregate times ascribed by right of ascension to the degree on the meridian above the earth, the degree ascending may be obtained, for it will be that degree which corresponds to that total number, as stated in the Table proper to the climate. 2
The Sun always preserves an equal distance in equatorial hours from all parts of the same meridian; but his distance in equatorial hours from different meridians varies according to the degrees of distance between meridian and meridian.
The foregoing extracts have been made to show the entire agreement between the astronomy of the Tetrabiblos and that of the Almagest. The Tables herein given from the latter work are, of course, now, in some degree, superseded by others of modern calculation, infinitely more complete.