Maitland Club > Works of Sir Thomas Urquhart

THE TRISSOTETRAS. 87
are alwayes to be understood, a digit being prescinded from the left, is equall to the
tangent complement of the angle required ; for the proposition goeth thus, As the
radius to the tangent of the angle given, so the sine complement of the subtendent side
to the tangent complement of the angle required ; and because tangents and tangent
complements are reciprocally proportionall, instead of To — Tag — NuC3=Mir, or, To
— Lu — MagOcfTir, which, for that the radius is a meane proportionall betwixt the L,
and N., the T. and M., is all one for inferring of the same fourth proportionall, or
foresaid quaesitum, we may say, Mag — Nu — Tot? Mir, that is, As the tangent com-
plement of the given angle to the cosine of the subtendent, so the totall sine to the
antitangent of the angle demanded ; for the totall sine being, as I have told you, a
meane proportionall betwixt the tangents and cotangents, the subtracting of the cotan-
gent or tangent complement from the summe of the radius and antisine, residuats a
logarithm equall to that of the remainder, by abstracting the radius from the sum of
the cosine of the subtendent and tangent of the angle given, either of which will fall
out to be the antitangent of the required angle.
Kotandum.
[Here alwayes is to be observed, that the subtracting of logarithms may be
avoyded, by substituting the arithmeticall complement thereof to be added to the loga-
rithms of the two middle proportionals ; which arithmetical complement, according to
Gellibrand, is nothing else but the difference between the logarithm to be subtracted
and another consisting of an unit, or binarie with the addition of cyphers, that is, the
single or double radius, for so the sum of the three logarithms, cutting off an unit or
binarie towards the left hand, will still be the logarithm of the fourth proportionall
required.
For the greater ease therefore in trigonometricall computations, such a logarithmicall
canon is to be wished for, wherein the radius is left out of all the secants, and all the
tangents of major arches, according to the method prescribed by Mr Speidel, who is
willing to take the paines to make such a new canon, better then any that ever hitherto
hath been made use of, so that the publike, whom it most concerneth, or some potent man
well minded towards the mathematicks, would be so generous as to releeve him of the
charge it must needs cost him ; which, considering his great affection to, and ability
in those sciences, will certainly be as small a summe as possibly he can bring it to.]
This parenthesis, though somewhat with the longest, will not, I hope, be displeas-
ing to the studious reader.
The second mood of the first figure is Ubamen, which comprehendeth all those pro-
blems wherein the subtendent and one oblique angle being given, the ambient adjoyn-
ing the angle given is required, and by its resolver, Nag — Mu— Torpft^Myr, shew-
eth that, if to the summe of the logarithms of the two middle proportionals we adde
the arithmeticall complement of the first, the cutting off the index from the aggregat