Maitland Club > Works of Sir Thomas Urquhart

94 THE TRISSOTETRAS.
required side, so the secant of the other given angle to the secant of the side that is
demanded. Here the angulary intermixture of proportions giveth us to understand
that this mood dependeth on Seproso, as is manifested by the last characteristick of
Uchedezexam the directory of this axiom.
The sixth and last figure is Escheva, which comprehendeth all those problems
wherein the two containing sides being given, either the subtendent or an angle is
demanded ; it hath two' moods, Enerul and Erelam.
The first mood thereof Enerul, containeth all such problems as having the ambients
given, require the subtendent ; and by its resolver, Ton — Neg — Net3°Nur, sheweth
that the summe of the Logarithms of the cosines of the two legs unradiated, is the
logarithm of the cosine of the subtendent ; for it is, As the totall sine to the cosine of
one of the ambients, so the cosine of the other including leg given to the cosine of
the required subtendent ; and because of the cosinal and secantine proportion, we may
safely say, To — Leg — Le{?3"Lur. That is, As the radius to the secant of one shanke
or leg, so the secant of the other shanke or leg to the secant of the hypotenusa de-
manded. The coursing thus upon sines and their proportionals evidenceth that this
mood dependeth on Suprosca, the first of the sphericall Axioms, which is pointed at
by the third and last characteristick of Uphugen the directorie thereof.
The second mood of the last figure, and consequently the last mood of all the ortho-
gonosphericals, is Erelam, which comprehendeth all those orthogonosphericall pro-
blems wherin the two containing sides being proposed, an angle is demanded ; and by
its resolver, Sei — Teg — Torb&j-Tir, or by primifying the radius, Torb — Tepi— Rexi
(fc?Tir, giveth us to understand that the cutting off the radius from the summe of the
tangent of the side opposite to the angle demanded, and the cosecant of the side con-
terminat with the said angle, residuats the touch-line of the angle in question ; for it
is, As the sine of the side adjoyning the angle required to the tangent of the other
given side, so the radius to the tangent of the angle demanded ; or, As the totall sine
to the tangent of the ambient opposite to the angle sought, so the antisecant of the
leg adjacent to the said asked angle to the tangent or toucher thereof; and because
sines have the same proportion to cosecants which tangents have to cotangents, we
may say, To — Sei — meC3=mir, that is, As the radius to the sine of the side conter-
minat with the angle required, so the cotangent of the other leg to the cotangent of
the angle searched after ; or yet more profoundly by an alternat proportion changing
the relation of the fourth proportional!, although the same formerly required angle,
thus, To — Ilei — meG5 = mor, that is, As the radius to the antisecant of the side adja-
cent to the angle sought for, so the antitangent of the other side to the antitangent
of that side's opposit angle, which is the angle demanded. The reason hereof is
grounded on Sbaprotca ; for the tangentine proportion of the terms of this mood spe-
cifieth its dependance on the second axiom, which is showen unto us by the eight and
last characteristick of its directorie Pubkutethepsaler.