Maitland Club > Works of Sir Thomas Urquhart

THE TRISSOTETRAS. 101
the sustentative leg of a perpendicular's exterior demission must alwayes be continued :
but, in this case, the outward falling of the perpendicular is onely from one extremity
of the given side ; for, if it be demitted likewise from the other end, it falleth then in-
wardly, and so produceth the third tenet of this mood, which is Dadisgatin, that is, if
the given angles be of a different quality, and that the perpendicular be internally de-
mitted, the double verticall is one of the proposed angles.
The nature of the perpendicular's demission in all the cases of this mood being thus
to the full explained, we may, without impediment, proceed to the performance of all
the orthogonosphericall operations, each in its own order thereto belonging.
To begin therefore at the first, whose quassitum, as I have told you already, is a
verticall angle, we must know, seeing the work is orthogonospherically to be perform-
ed, that the forementioned praenoscendum cannot be obtained without the helpe of one
of the sixteen datoquaeres ; and therefore, in my Trissotetras, considering the nature
of what is given, and asked in the cases of this mood, I have appointed Upalam to be
the subservient of its praenoscendum ; for by the resolver thereof, To— Tag— Nufir^
Mir, the subtendent and an angle being given ; for one of the given sides of every
loxogonosphericall, if the perpendicular be rightly demitted, becomes a subtendent,
and sometimes two given sides are subtendents both, we frame these three peculiar
problems for the three praenoscendas ; to wit, Utopat, for the double verticall, by the
raeanes of the great subtendent side, and the prime opposite angle ; secondly, Udo-
baud, for obtaining of the first verticall in the little rectangle, by vertue of the lesser
subtendent in the same rectangle, and the next opposite angle ; lastly, Uthophauth,
for the first co-verticall, by meanes of the first co-subtendent, and first co-opposite
angle, all which is at large set downe in the first partition of Alamebna in my table.
The first and chiefe operation being thus perfected, the verticall angles so found
out must concurre with each its correspondent opposite, for the obtaining of the
perpendicular necessary for the accomplishment of the second operation in every one
of the cases of the foresaid mood ; to which effect, Amaner is made the subservient,
by whose resolver, Say— Nag— Ta-fcfNyr, these three datoquaeres, Opatca, Obautca,
and Ophauthca come to light, and is manifestly shown how, by any paire of three
severall couples of different angles, the perpendicular is acquirable.
Now, though of this work, as it is a single one, no more then of the other succeed-
ing it in the same mood, nor of the last two in any of the disergeticks in their full
analogy, I doe not make any mention at all in my table ; but, after the couching of
the first operation for the praenoscendas, supply the roomes of the other two, with an
equivalent row of proportionals out of them specified, for attaining to the knowledge
of the maine quaesitum ; yet in this comment upon that table, for the more perspicui-
ties sake, and that the reader may as well know what way the rule is made, as how
thereby a demand is to be found out, I have thought fit to expatiat mv selfe for hU