Maitland Club > Works of Sir Thomas Urquhart

88 THE TRISSOTETRAS.
of the three, will residuat the tangent complement of the side required ; and therefore,
with the totall sine in the first place, it may be thus propounded, Torp — Mu — Lag
fr5*Myr ; for the first theorem being, As the sine complement of the angle given to
the tangent complement of the subtendent side, so the totall sine to the tangent comple-
ment of the side required : just so the second theorem, which is that refined, is, As the
totall sine to the tangent complement of the subtendent, so the secant of the given angle
to the tangent complement of the demanded side. Here you must consider, as I have
told you already, that of the whole secant I take but its excesse above the radius, as
I doe of all tangents above 45 degrees ; because the cutting off the first digit on the
left, supplieth the subtraction requisite for the finding out of the fourth proportionall ;
so that by addition onely the whole operation may be performed, of all wayes the
most succinct and ready. Otherwise, because of the totall sines meane proportionality
betwixt the sine complement and the secant, and betwixt the tangent gnd tangent
complement, it may be regulated thus, To — Tu — NagttrTyr, that is, As the radius
to the tangent of the subtendent, so the sine complement of the angle given to the tan-
gent of the side required. The reason of the resolution both of this and of the former
datoqurere, is grounded on the second axiom, and the proportion that, in severall rect-
angled spherieals which have the same acute angle at the base, is found betwixt the
sines of their perpendiculars and tangents of their base, as is shewne you by the two
first consonants of the directory of Sbaprotca.
The third and last mood of the first figure is Uphaner, which comprehendeth all
those problems wherein the hypotenusa and one of the obliques being given, the oppo-
site ambient is required; and by its resolver, Tol — Sag — Suf^ Syr, sheweth, that
if we adde the logarithms of the sine of the angle and sine of the subtendent, cutting
off the left supernumerarie digit from the summe, it gives us the logarithm of the sine
of the side demanded ; for it is, As the totall sine to the sine of the angle given, so the
sine of the subtendent side to, the sine of the side required ; and because by the
axiom of Rulerst it was proved, that when the sine of any arch is made radius, what
was then the totall sine becomes a secant, and therefore secant complement of that
arch; instead of Tol — Sag— SuCdrSyr, we may say, To — Ru — RagC5?"Ryr, that is,
As the totall sine is to the secant complement of the subtendent, so the secant com-
plement of the angle given to the secant complement of the side demanded. The re-
solution of this datoqusere by sines, is grounded on the first axiom of Spherieals, whicli
elucidates the proportion betwixt the sines of the hypotenusas and perpendiculars, as
it is declared to us by the first syllable of Suprosca's directory.
The second figure is Vemanore, which containeth all those orthogonosphericall
questions wherein the subtendent and an ambient being proposed, either of the obliques
or the other ambient is required, and hath three moods, viz. Ukelamb, Ugemon, and
Uchener.
The first mood, Ukelam, comprehendeth all those orthogonosphericall problems,