Maitland Club > Works of Sir Thomas Urquhart

THE TRISSOTETRAS. 93
The last mood of this figure is Epsoner, which containeth all those orthogonosphe-
ricall problems wherein an ambient and an opposite oblique being given, the other
ambient is demanded ; and by its resolver, Tag — Tolb — TeC^Syr, or more elabou-
redly, Tolb — Mag — Tefr3~Syr, sheweth that the praescinding of the radius from the
summe of the tangent of the side and antitangent of the given angle, residuats the sine
of the side required ; for it is, As the tangent of the angle proposed to the totall sine,
so the tangent of the given side to the sine of the side demanded ; or, As the radius to
the tangent complement of the angle given, so the tangent of the given side to the
sine of the side required ; and because of the reciprocall analogy betwixt the tangents
and cotangents, and betwixt the sines and cosecants, we may with the same confidence
as formerly set it thus in the rule, To — Meg — Ta(j3*Ryr, and it will find out the
same quaesitum. The reason of the operations of this mood, because of the ingredi-
encie of tangents, dependeth on Sbaprotca, as is perceivable by the sixth determinater
of its directory Pubkutethepsaler.
The fifth figure of the orthogonosphericals is Achave, which containeth all those
problems wherein the angles being given, the subtendent or an ambient is desired,
and hath two moods, Alamun and Amaner.
Alamun comprehendeth all those problems wherein the angles being proposed, the
hypotenusa is required ; and by its resolver, Tag — Torb — MaC^Nur, or more com-
pendiously, Torb — Mag — Mafr3°Nur, sheweth that the summe of the cotangents not
exceeding the places of the radius, is the sine complement of the subtendent required ;
for it is, As the tangent of one of the angles to the radius, so the tangent complement
of the other angle to the sine complement of the hypotenusa demanded ; or, As the
totall sine to the tangent complement of one of the angles, so the tangent complement
of the other angle to the sine complement of the subtendent we seek for. The run-
ning of this mood upon tangents notifieth its dependance on Sbaprotca, as is evident
by the seventh determinater of the directory thereof.
The second mood of this figure is Amaner, which comprehendeth all those orthogo-
nosphericall problems wherein the angles being given, an ambient is demanded ; and
by its resolver, Say — Nag — Tuft^Nyr, or more perspicuously, Tw — Noy — Rayfcf
Nyr, sheweth that the summe of the logarithms of the antisine of the angle opposite
to the side required, and the arithmeticall complement of the sine of the angle adjoyn-
ing the said side, which we call its secant complement, with the usual presection, is
equall to the sine complement of the same side demanded ; for it is, As the sine of the
angle adjoyning the side required to the antisine of the other angle, so the totall sine
to the antisine of the side demanded ; or, As the radius to the antisine of the angle
opposite to the demanded side, so the antisecant of the angle conterminat with that
side to the antisine of the side required : and because of the analogy betwixt the anti-
sines and secants : and likewise betwixt the antisecants and sines, we may expresse it,
To — Say — LaC5"Lyr ; that is, As the radius to the sine of the angle insident on the