Maitland Club > Works of Sir Thomas Urquhart

THE TRISSOTETRAS. 91
The third figure is Enarrulome, whose three moods are Etalum, Edamon, and
Ethaner.
This figure eomprehendeth all those orthogonosphericall questions wherein one of
the ambients with an adjacent angle is given, and the subtendent, an opposite angle,
or the other containing side is required.
Its first mood, Etalum, involveth all those orthogonosphericall problems wherein
a containing side with an insident angle thereon is proposed, and the hypotenusa de-
manded ; and by its resolver, Torp — Me — Nagft^Mur, or, by inverting the demand
upon the scheme, Tolp — Me — NagC^Mur sheweth, that the cutting off the first left
digit from the summe of the tangent complement of the ambient proposed and the sine
complement of the given angle, affords us the tangent complement of the subtendent
required ; for the theorem goes thus, As the totall sine to the tangent complement of
the given side, so the sine complement of the angle given to the tangent complement of
the hypotenusa required. And because the totall sine hath the same proportion to the
tangent complement which the sine hath to the sine complement, we may as well say,
To — Meg — Sa(j3°Nur, that is, As the radius to the tangent complement of the am-
bient side, so the sine of the angle given to the sine complement of the subtendent
required. The progresse of this mood dependeth on the axiom of Sbaprotca, as you
may perceive by the fourth consonant of its directorie, Pubkutethepsaler.
The second mood of the third figure is Edamon, which eomprehendeth all those
orthogonosphericall problems wherein an ambient and an adjacent angle being given,
the opposite oblique, viz. the angle under which the ambient is subtended, is requir-
ed ; and by its resolver, To — Neg — Saft3=Nir, sheweth, that the addition of the co-
sine of the ambient and of the sine of the angle proposed, affordeth us, if we omit
not the usuall presection, the cosine of the angle we seek for ; for it is, As the radius
to the cosine or sine complement of the given side, so the sine of the angle proposed to
the antisine or sine complement of the angle demanded ; now, the radius being alwayes
a meane proportionall betwixt the sine complement and the secant, we may for To
Neg — SaC^Nir say, To — Leg — RafcfLir, or To — Rag — Left^Lir ; that is, As
the totall sine to the secant, or cutter of the side given, or to the cosecant or secant
complement of the given angle, so is the secant complement of the angle, or secant of
the side, to the secant or cutter of the angle required. The reason of all this is
grounded on Seproso, because it runneth upon the proportion betwixt the sines of
the sides and the sines of their opposite angles, as is perspicuous to any by the second
syllable of the directory of that axiome.
The last mood of the third figure is Ethaner, which eomprehendeth all those or-
thogonosphericall problems wherein an ambient with an oblique annexed thereto is
given, and the other arch about the right angle is required ; and by its resolver, Torb
— Tag — SeCrf=Tyr, sheweth, that if we joyne the logarithms of the two middle pro-
portionals, which are the tangent of the given angle, and the sine of the side, the usual