ALL signs, between which there does not exist any familiarity in any of the modes above specified, are inconjunct and separated.
For instance, all signs are inconjunct which are neither commanding nor obeying, and not beholding each other nor of equal power, as well as all signs which contain between them the space of one sign only, or the space of five signs, and which do not at all share in any of the four prescribed configurations: viz. the opposition, the trine, the quartile, and the sextile. All parts which are distant from each other in the space of one sign only are considered inconjunct, because they are averted, as it were, from each other; and because, although the said space between them may extend into two signs, the whole only contains an angle equal to that of one sign: all parts distant from each other in the space of five signs are also considered inconjunct, because they divide the whole circle into unequal parts; whereas the spaces contained in the configurations above-mentioned, viz. the opposition, trine, quartile, and sextile, produce aliquot divisions. 1
27:1 It has never been very clearly shown how the followers of Ptolemy have reconciled the new aspects [called the semiquadrate, quintile, sesquiquadrate, biquintile, &c.] with the veto pronounced in this chapter. Kepler is said to have invented them, and they have been universally adopted; even Placidus, who has applied Ptolemy's doctrine to practice better than any other writer, has availed himself of them, *1 and, if the nativities published by him are to be credited, he has fully established their importance.
Salmon, in his "Horæ Mathematicæ," beforementioned, gives a long dissertation (from p. 403 to p. 414) on the old Ptolemaic aspects, illustrative of their foundation in nature and in mathematics; and, although his conclusions are not quite satisfactorily drawn, some of his arguments would seem appropriate, if he had but handled them more fully and expertly; particularly where he says that the aspects are derived "from the aliquot parts of a circle, wherein observe that, although the zodiac may have many more aliquot parts than these four (the sextile, quartile, trine, and opposition), yet those other aliquot parts of the circle, or 360 degrees, will not make an aliquot division of the signs also, which in this design was sought to answer, as well in the number 12, as in the number 360." The passage in which he endeavours to show that they are authorized by their projection, also deserves attention.
All Salmon's arguments, however, in support of the old Ptolemaic aspects, militate against the new Keplerian ones; and so does the following extract p. 28 from the "Astrological Discourse" of Sir Christopher Heydon: "For thus, amongst all ordinate planes that may be inscribed, there are two whose sides, joined together, have pre-eminence to take up a semicircle, but only the hexagon, quadrate, and equilateral triangle, answering to the sextile, quartile, and trine irradiated. The subtense, therefore, of a sextile aspect consisteth of two signs, which, joined to the subtense of a trine, composed of four, being regular and equilateral, take up six signs, which is a complete semicircle. In like manner, the sides of a quadrate inscribed, subtending three signs, twice reckoned, do occupy likewise the mediety of a circle. And what those figures are before said to perform" (that is, to take up a semicircle) "either doubled or joined together, may also be truly ascribed unto the opposite aspect by itself; for that the diametral line, which passeth from the place of conjunction to the opposite point, divideth a circle into two equal parts: the like whereof cannot be found in any other inscripts; for example, the side of a regular pentagon" (the quintile) "subtendeth 72 degrees, of an octagon" (the semiquadrate) "but 45; the remainders of which arcs, viz. 108 and 135 degrees, are not subtended by the sides of any ordinate figure."
27:*1 Except the semiquadrate, which he has not at all noticed.