Maitland Club > Works of Sir Thomas Urquhart

104 THE TRISSOTETRAS.
Nop — Seudfr3°Nob*Kir, that is, As the sine of the double verticall to the antisine of
the prime cathetopposite, so the sine of the second verticaline, or verticall in the
lesser rectangle, to the antisine of the next cathetopposite, whose complement to a
semicircle is the angle required.
But when, the affection of the given angles being different, the perpendicular is
made to fall without, as in Dadisforeug, the second case of this mood, the resolver
thereof is particularised thus, Saud — Nob — SatC3=Nop*Ir, that is, As the sine of the
first verticaline, or verticall in the rectanglet, to the sine complement of the nearest
cathetopposite, so the sine of the double verticall to the sine complement of the prime
cathetopposite, which is the angle required.
And, lastly, if with the different qualities of the given and demanded angles the
perpendicular be let fall within, as in Dadisgatin, the third case of this mood, then
is the finall resolver to be determined thus, Sauth — Noph — SeuthC3=Nops*Ir, that is,
As the sine of the first co-verticall to the co-sine of the first co-opposite, so is the sine
of the second co-verticall to the co-sine of the second co-opposite, which is the angle
required.
The originall reason of all these operations is grounded on the axiome of Nabad-
prosver, as the first syllable of its directory, Alama, giveth us to understand, which
we may easily perceive by the analogy, that is onely amongst the angles without any
intermixture of sides in the termes of the proportion.
The second mood of the first angulary figure, that is to say, the first two termes
of whose datas are angles, is Allamebne, which comprehendeth all those disergetick
questions, wherein two angles being given and a side betweene, one of the other sides
is demanded ; which side, the perpendicular being let fall, is alwayes one of the second
subtendents, viz. in the first case a second subtendent of the lesser triangle, in the
second a second subtendent in the great rectangle, and in the last a second co-sub-
tendent.
To the knowledge of all these, that we may the more easily attaine, we must
consider the generall maxim of the cathetothesis of this mood, which is Cafyxe-
geq, that is to say, that in all the cases of Allamebne the perpendicular falleth from
the side required, and from that point thereof where it conterminats with the given
side upon the third side, continued if need be ; and according to the variety of the
second subtendent, which is the side demanded, there be these three especiall tenets
of this generall maxim, to wit, Dasimforauxy, Dadiscracforeug, and Dadiscramgatin.
Dasimforauxy, the first especiall tenet of the generall maxim of the cathetothesis
of this mood, sheweth, that when the proposed angles are of the same quality and ho-
mogeneall, the perpendicular falleth externally, and the first verticall is one of the
given angles, and annexed to the required side.
The second tenet, Dadiscracforeug, which pertaineth to the second case of this
mood, sheweth, that when the given angles are of a discrepant nature and heteroge-