Maitland Club > Works of Sir Thomas Urquhart

92 THE TRISSOTETRAS.
preseetion being observed, we shall thereby have the tangent or toucher of the ambient
side desired ; for it is, As the radius to the tangent of the angle given, so the sine
of the containing side proposed to the side required ; and because the tangent com-
plement and tangent are reciprocally proportionall, the sine likewise and secant com-
plement, for To — Tag — SeC^Tyr, we may say, keeping the same proportion, To —
Keg — Ma(£f°Myr, that is, As the radius to the secant complement of the given side,
so the tangent complement of the angle proposed to the tangent complement of the
side required. The truth of all these operations dependeth on Sbaprotca, the second
axiome of the sphericals, as is evidenced by 6. the fifth characteristick of its directory,
Pubkutethepsaler.
The fourth figure is Erollumane, which includeth all orthogonosphericall questions
wherein an ambient and an opposite oblique being given, the subtendent, the other
oblique, or the other ambient is demanded : It hath likewise, conforme to the three
former figures, three moods belonging to it ; the first whereof is Ezolum.
This Ezolum comprehendeth all those orthogonosphericall problems wherein one of
the legs with an opposite angle being given, the subtendent is required ; and by its
resolver, Sag— Sep — RadC^Sur, or by putting the radius in the first place, To —
Se — RegC^Sur, sheweth, that the abstracting of the radius from the sum of the sine
of the side and secant complement of the angle given, residuats the sine of the hypo-
tenusa required ; for it is, As the sine of the angle given to the sine of the opposite
side, so the radius to the sine of the subtendent ; or more refinedly, As the totall sine
to the sine of the side, so the secant complement of the angle given to the sine of the
subtendent side ; and because of the sine's and antisecant's, or secant complement's reci-
procall proportionality, To — Sag — Ret5°Ru, that is, As the radius to the sine of the
angle given, so the secant complement of the proposed side to the secant complement
of the subtendent required. The reason of all this is grounded on the third axiom,
Seproso, as is made manifest by the third syllable of its directory.
The second mood of this figure is Exoman, which comprehendeth all those problems
wherein a containing side and an opposite oblique being given, the adjacent oblique is
required ; and by its resolver, Ne — To — Nag(£f Sir, or more refinedly, To — Le —
NagG3=Sir, sheweth that the summe of the sine of the angle, together with the arith-
meticall complement of the antisine of the leg, which in the table I have so much re-
commended unto the reader, is set downe for a secant, the usuall preseetion being ob-
served, affordeth us the sine of the angle required ; and because of the reciprocall pro-
portion betwixt the sine complement and secant, and betwixt the sine and secant
complement, the theorem may be composed thus, To — Neg— LaC^Rir ; that is, As
the radius to the sine complement of the given side, so the secant of the angle proposed
to the secant complement of the angle demanded. The reason of this is likewise
grounded on Seproso, as you may perceive by the fourth characteristick of its
directory.