Chapter XIV. Number of the’ Modes of Prorogation（续 2）

. the Future position be above the earth, and nocturnal if unders and the difference in amount, of these two distances, 4n equatorial = will present the number of years inquir-

In order t6 exemplify. the foregoing instructions, let-the first

point of Aries be supposed ag the preceding place, and the —

first point of 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* ; and where the horary magnitude of the beginning of Gemini will be about seventeen equatorial timest. ,

Let the first point of Aries be first placed on the ascendant, so that the beginning of Capricorn may be on the midheaven above the earth, and the first point of Gemini’ be distant from the said mid-heaven 148 equatorial timesf. 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§. 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 said excess, and also, as near as possible, the amount of the times of ascension of Aries and Taurus||), since, in this instance, the prorogatory place is established in the ascendant.

* This, in the Northern Hemisphere, would be the latitude of Alexandria, (where Ptolemy flourished) or, in his own words, that of the 3d Climate, passing through Lower Egypt, numbered 30° 22’— Vide extracts from the Tables of the Almagest, inserted in the Appendix.

t This is the magnitude of the diurnal temporal hour of the first pant of Gemini in the latitude prescribed.

} By right ascension, as shewn by the Extract, inserted in the Appendix, from the Table of Ascensions in the Almagest. The exact distance, however, according to that Table, is 147° 44/.

§ Or rather, according to the Table, 102° 39.

|| That is to say, of their oblique ascension, which is here required to be reckoned; because the prorogatory and preceding place is in the ascendant. Vide p. 139, and Note + in p. 188,—And the first point of Gemini, on arriving at the ascendant, will be distant from the mid-heaven 102° 39’ by right ascension; the 13th degree of Aquarius being then in culmination in the prescribed latitude. The

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*. Now, as it 1s 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 wae culminatingt.

In the same way, let the first point of Aries be descending} ; so that the beginning of Cancer may occupy the midheaven, and the first point of Gemini precede the midheaven at the distance of 32 equatorial times§. There- -

fore, 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]| of the first point of Gemini, at its future descension from the meridian]. ° 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. descensjon 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**. Oblique ascensions in the latrtude 30% 22’ N. are also shewn in the extract referred to in the preceding note: and it thereby pet, that. Aries and Taurus ascend in 45° 5’, instead of 46°.

* Or rather, 57° 44’ — by right ascension.—Vide extract abaya referred to.

’ § Or, rather, $2° 16-—by right ascension again.—Vide extract as before.

| By right ascension.—The amount according to the Table is, however, 102° 39’, as before'stated.

@{ On which the 10th degree of Virgo will then be posited. = - . ** By oblique descension and ascension: vide P. seoeee Table — shews the amount to neice 23’,

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a4 PTOLEMY's TETRaBIIes. j|§§§ [Beok FIT.

Again, let the first point ef Anes have another posuen, pot im any angie*, but, for imstzmce, at the distance of three temporal honors past the meridian; so that the 1th degree of Tasrus may be on the mid-beaven, and the frst poant of Gemini be approaching the mid-heaven, at the distance of thirteen equatorial times. The seventeen trams mast, therefore, be again multiplied by the three hours, and the first point of Gemini, at its second position, will i found to be past the meridian, at the distance of 51 timest. The distances of 13 times of the first position and 51 tines of the second position are then both to be taken; and they will produce 64 times. In the former instances the proregatory place performed in the same succession ; viz. occn— pying 46 times in coming to the ascendant, 58 in comme to the mid-heaven, and 70 in coming to the west; so thut the present number of times, depending on the imtenmediate “position between the mid-heaven and the west, and being 64, also differs from each of the other numbers, in propertion tothe three hours difference of position. For, in the © other cases which proceeded by quadrants}, 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§.

There is, however, another method which may be used, and which is still more simple ; for instance, should the preceding degree be on the aseendant, the following inter-

* In reference to p. 139, and note + in the same page.

+ The 18th degree of Cancer being then in eulmination.

} Or semi-diurnal arcs, each equal to six temporal hours. . & The amount of the progressive difference of the times of promo gation, as here mentioned, is of course only applicable to the perallel of declination of the first point of Gemini, in the latitude befere quoted. It must necessgrily vary in all other parallels of declinatien, and also in all other latitudes.

mediate times of ascension*, between it and the succeeding degree, may be reckoned; shuld it be on the mid-heaven, the times of ascension must be reckoned on.a right sphere ; and, if it be on the west, descending, the intermediate times of descensiont are to be reckoned. But, should the preceding degree be between any two of these angles, as, for "Instance, at the distance of Aries, just spoken of, the proper times for each angle must first be considered. And, since the first point of Aries was assigned a position between the two angles of the mid-heaven and the west, the proper times of the distances from these angles to the first point of Geminit would be found to be 58 from the mid-heaven, and 70 from the west. The distances, in temporal hours, of the preceding degree from each of these angles, are then to be ascertained; and whatever proportion these same temporal hours, contained in such distances between the said preceding degree and each angle, may bear to the temporal hours of the whole quadrant, the same proportion, out of the excess of the times of distance of one angle over those of the other, is either to be added to, or deducted from, the actual number of times of the respective angles. For instance, in the example before set forth, 70 times exceed:58 times by 12; and the preceding place was distant from the angles three eaual temporal hours, which are the half of six, the number belonging to the whole quadrant. Now, three being the half of six, and 12 being the amount of the excess, the half of 12 is therefore to be taken, giving 6 to be either added to the 58 times, or subtracted from the 70: