Maitland Club > Works of Sir Thomas Urquhart

100 THE TRISSOTETRAS.
Here must I admonish the reader, that he startle not at the mentioning of foure
especiall cathetothetick tenets, and foure severall cases belonging to each disergetick
mood, seeing, to the most observant eye, there be but three of either perceptible in
my Trissotetras ; for, the fourth, both tenet and case, being the same by way of ex-
pression in all the moods, and being fully resolved by the third case of every mood, it
shall suffice to speak thereof here once for all ; the tenet of this common case is Sim-
momatin, that is to say, when all the three angles in any of those disergeticks are of
the same affection, either all acute, or all oblique, the perpendicular falleth inwardly,
whether the double verticall be an angle given, an angle demanded, or neither. Yet
here it is to be considered, that, seeing triangles may be either calculated by their reall
and naturall, or by their circular parts, or by both together, and that for the more
facility, we oftentimes, instead of the proposed triangle, resolve its opposite ; it is not the
reall and given triangle that, in this case, we so much take notice of, as of its resolvable
and equivalent, the opposite triangle ; as, for example, if a sphericall triangle, with two
obtuse angles, and one acute, be given you to resolve, it will fall within the eompasse
of Simomatin, because its opposite sphericall is simply acute angled ; and also, if you
be desired to calculate a sphericall triangle, with two acute angles, and one obtuse, it
will likewise fall within the reach of the same case, because its opposite sphericall is
simply obtusangled. The reason of both the premises is from the equality of the op-
posite angles of concurring quadrants, which, that they are equal], no man needs to
doubt, that will take the paines to let fall a perpendicular from the middle of the one
quadrant upon the other ; for so there will be two triangles made equilaterall ; and
seeing it is an universally received truth, that equall sides sustaine equall angles, the
identitie of the perpendicular in both the foresaid triangles must needs manifest the
equality of the two opposite angles.
I have been the ampler in the elucidating of this case, that, it over-running all the
moods of the disergetick loxogonosphericals, the reader, in what mood or datoqusere
soever he please to resolve this foresaid ease, may, for that purpose, to this place have
recourse ; to the which, without any further intended reiteration of this tenet, I doe
heartily remit him.
The first especiall tenet of the generall maxim of the cathetothesis of this mood is
Dasimforaug, that is, when the given angles are of the same nature, but different from
the required, the perpendicular falleth outwardly, and the first verticall is a given
angle ; the second tenet belonging to the second case of this mood is Dadisforeug, that is,
when the proposed angles are of different affections, the perpendicular is externally de-
mitted, and one of the given angles is a second verticall ; yet this discrepance is to be
observed betweene the externall prosilieneie of the perpendicular arch in this case, and
that other of the former ; that in the former, it is no matter from which of the ends
of the proposed side the perpendicular be let fall upon one of the comprehending legs
of the angle required, which leg must be increased, for it is a generall notandum that