In conformity with the foregoing remarks, when it may happen that the prorogatory and preceding place may be actually on the oriental horizon, it will be proper to reckon, at once, the ascensional times which may intervene until the meeting of the degrees; because, after the same number of equatorial times, the anæreta will arrive at the prorogatory place; that is to say, at the oriental horizon. Should the prorogatory place be found on the meridian, the whole number of degrees by right ascension, in which the whole intercepted arc will transit the meridian, must then be taken. And if the prorogatory place be on the occidental horizon, the number of descensions, in which every degree of the distance will be carried down (or, in other words, the number of ascensions, in which their opposite degrees will ascend), is in that case to be reckoned.[160]
When, however, a prorogatory and preceding place may not be situated on any one of the three aforesaid points, but in some intermediate station, it must be observed that other times[161] will then bring the succeeding place to the preceding one; and not the times of ascension or descension, or transit of the mid-heaven, as above spoken of. For any places whatever, which have one particular position, on the same degree, in regard to the horizon and meridian, are alike and identical. This is the case, for instance, with all places lying on any one of those semicircles which are drawn through the arcs of the meridian and horizon; and each of these semicircles (all of which have position at the same equal distance from each other) marks one temporal hour[162]; and, as the time occupied in proceeding through the places[163] above described, and arriving at the same position of the horizon and meridian, is rendered unequal to and different from the time of transits in the zodiac; so, also, the transits of other spaces are made, agreeably to their position, in time again distinct from this.
There is, however, a method by which the proportion of time, occupied in the progress of a succeeding place to a prorogatory and preceding place, in whatever position, whether oriental, meridianal, or occidental, or any other, may be easily calculated. It is as follows:—
When it has been ascertained what degree of the zodiac is on the mid-heaven, as also which are the preceding and succeeding degrees, the period of whose meeting is to be calculated, the position of the preceding degree, and its distance in temporal hours from the meridian, are next to be noted; because any part of the zodiac, on becoming distant from the meridian in the same temporal hours, must fall on the same individual semicircle.[164] For ascertaining this distance, the number of ascensions, in a right sphere, found in the intermediate space between the said preceding degree and the mid-heaven, either above or under the earth, is to be divided by the number of the diurnal or nocturnal horary times of the said preceding degree: for instance, if that degree be above the earth, by its diurnal horary times; and, by its nocturnal, if it be under the earth. It is then to be discovered in what number of equatorial times the succeeding degree will be distant from the same meridian, by as many similar temporal hours as those by which the preceding degree is distant from it. And, to effect this, the hours in question must be noted, and it must first be observed, by the right ascensions again, how many equatorial times the succeeding degree, at its original position, is distant from the degree on the mid-heaven; and then it must be seen how many equatorial times it will be distant, on coming to the preceding degree’s distance in temporal hours from the said mid-heaven: this will be found, by multiplying those hours by the succeeding degree’s horary times; diurnal, if the future position be above the earth, and nocturnal if under; and the difference in amount, of these two distances, in equatorial times, will present the number of years inquired for.
CHAPTER XV
EXEMPLIFICATION
In order to exemplify the foregoing instructions, let the first point of Aries be supposed as the preceding place, and the first point in Gemini the succeeding; and let the latitude of the country, to which the operation relates, be such as will cause the longest day to consist of fourteen hours[165]; and where the horary magnitude of the beginning of Gemini will be about seventeen equatorial times.[166]
Let the first point of Aries be first placed on the ascendant, so that the beginning of Capricorn may be on the mid-heaven above the earth, and the first point of Gemini be distant from the said mid-heaven 140 equatorial times.[167] Now, since the first point of Aries is distant six temporal hours from the mid-heaven above the earth, the times of that distance will be found, by multiplying the said six hours by the seventeen equatorial times of the horary magnitude of the first point of Gemini, to be 102.[168] The whole sum of the distance to the mid-heaven above the earth, is 148 times; and as 148 times exceed 102 by 46, the succeeding place will consequently devolve into the preceding place after 46 times (being the amount of the times of ascension of Aries and Taurus[169]); since, in this instance, the prorogatory place is established in the ascendant.
In like manner, let the first point of Aries be next placed on the mid-heaven, culminating; so that the first point of Gemini, in its first position, may be distant from the said mid-heaven 58 equatorial times.[170] Now, as it is required to bring the first point of Gemini, in its second position, to the mid-heaven, the whole distance is to be reckoned, viz. 58 times, in which Aries and Taurus pass the mid-heaven; because, again, the prorogatory place was culminating.[171]
In the same way, let the first point of Aries be descending[172]; so that the beginning of Cancer may occupy the mid-heaven, and the first point of Gemini precede the mid-heaven at the distance of 32 equatorial times.[173] Therefore, as the first point of Aries is on the west, and again distant six temporal hours from the meridian, let these six hours be multiplied by the seventeen times; which will produce 102, making the sum of the distance[174] of the first point of Gemini, at its future descension, from the meridian.[175] But, as the first point of Gemini, at its first position, was already distant from the meridian 32 times; which number 102 exceed by 70; it will consequently arrive at its descension after 70 times, the amount of the excess; in which space Aries and Taurus will have descended, and their opposite signs Libra and Scorpio arisen.[176]
Again, let the first point of Aries have another position, not in any angle,[177] but, for instance, at the distance of three temporal hours past the meridian; so that the 18th degree of Taurus may be on the mid-heaven, and the first point of Gemini be approaching the mid-heaven, at the distance of thirteen equatorial times. The seventeen times must, therefore, be again multiplied by the three hours, and the first point of Gemini, at its second position, will be found to be past the meridian, at the distance of 51 times.[178] The distance of 13 times of the first position and 51 times of the second position are then both to be taken; and they will produce 64 times. In the former instances the prorogatory place performed in the same succession; viz. occupying 46 times in coming to the ascendant, 58 in coming to the mid-heaven, and 70 in coming to the west; so that the present number of times, depending on the intermediate position between the mid-heaven and the west, and being 64, also differs from each of the other numbers, in proportion to the three hours’ difference of position. For, in the other cases which proceeded by quadrants,[179] according to the angles, the times progressively differed by twelve, but, in the present case of a minor distance of three hours, they differ by six.[180]