<s xml:space="preserve">
Given two sides and the angle opposite one of those sides, the angle opposite the other is known.
<lb/>
Or,
<lb/>
Given two angles, and the side opposite one of the given angles, the side opposite the other is </s>
</quote>
<lb/>
<s xml:space="preserve">
Immediately after the statement of the proposition, Viète gave the following statement,
under the heading Syntomon.</s>
<lb/>
<quote xml:lang="lat">
<s xml:space="preserve">
Quæ per factionem sub sinibus peripheriarum & adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt.
<lb/>
Cum duæ peripheriæ angulum acutum componunt, est
<lb/>
Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia,
minus sinu complementi composita.</s>
</quote>
<lb/>
<quote>
<s xml:space="preserve">
What appears from a combination of the sine of the arcs, dividing the sine of the total,
is also shown by the operations of addition and subtraction.
<lb/>
When two arcs contain acute angles, then as the whole sine is to twice the sine of the first,
so is the sine of the second to the sum of
the sine of the complement of the difference minus the sine of the complement of the </s>
</quote>
<lb/>
<s xml:space="preserve">
In modern notation this statement may be written as:
<math>
<mstyle>
<mn>1</mn>
<mo>:</mo>
<mrow>
<mn>2</mn>
<mspace width="0.167em"/>
</mrow>
<mrow>
<mi>sin</mi>
<mo maxsize="1">(</mo>
</mrow>
<mi>a</mi>
<mo maxsize="1">)</mo>
<mo>=</mo>
<mrow>
<mi>sin</mi>
<mo maxsize="1">(</mo>
</mrow>
<mi>b</mi>
<mo maxsize="1">)</mo>
<mo>:</mo>
<mrow>
<mi>cos</mi>
<mo maxsize="1">(</mo>
</mrow>
<mi>a</mi>
<mo>-</mo>
<mi>b</mi>
<mo maxsize="1">)</mo>
<mo>+</mo>
<mrow>
<mi>cos</mi>
<mo maxsize="1">(</mo>
</mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo maxsize="1">)</mo>
</mstyle>
</math>
. This is the ratio Harriot has written next to diagram 1, where both angles are acute. The other diagrams are for cases where one or both the angles are obtuse. </s>
<s xml:space="preserve">]</s>
</p>
</div>
<head xml:space="preserve" xml:lang="lat">
Vieta lib. 8. resp.
pag. 39.
<lb/>
<foreign xml:lang="gre">Syntomon</foreign>
<lb/>
[
<emph style="bf">Translation: </emph>
Viète, Responsorum liber VIII.
<lb/>
]</head>
<p xml:lang="lat">
<s xml:space="preserve">
[???] in alia charta
<lb/>
[
<emph style="bf">Translation: </emph>
[???] in another sheet ]</s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
Quæ per factionem sub sinibus peripherieriarum et adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt.
<lb/>
[
<emph style="bf">Translation: </emph>
What appears from a combination of the sine of the arcs, dividing the sine of the total,
is also shown by the operations of addition and subtraction.</s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
, una peripheria
<lb/>
<math>
<mstyle>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
, altera peripheria
<lb/>
<math>
<mstyle>
<mi>d</mi>
<mi>c</mi>
</mstyle>
</math>
, differentia
<lb/>
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
, aggregatum &
<lb/>
[
<emph style="bf">Translation: </emph>
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
is one arc,
<math>
<mstyle>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
the other.
<lb/>
<math>
<mstyle>
<mi>d</mi>
<mi>c</mi>
</mstyle>
</math>
is the difference,
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
the sum. </s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
Hæc quarta analogia est re eadem
<lb/>
cum secunda.
<lb/>
porro, prima et tertia analogiæ
<lb/>
reducuntur ad unam si quartus
<lb/>
terminus ita
<lb/>
[
<emph style="bf">Translation: </emph>
This fourth ratio is the same thing as the second.
<lb/>
Further, the first and third ratios are reduced to one if the fourth term is written ]
<lb/>
[
<emph style="bf">Commentary: </emph>
Here the symbols that looks like an equals sign is to be read as a minus sign,
where the smaller quantity is always understood to be subtracted from the larger.</s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
Nota
<lb/>
Quando una peripheria est maior quadranti
<lb/>
ut
<math>
<mstyle>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
est
<emph style="super">in</emph>
3,
<emph style="super">a</emph>
et 4,
<emph style="super">a</emph>
diagrammati; summatur
<lb/>
eius residuum ad semicirculum. Et tum
<lb/>
operatio erit secundum primum vel secundum casum.
<lb/>
Quare hoc modo sunt duo tantummodo
<lb/>
<lb/>
[
<emph style="bf">Translation: </emph>
Note.
<lb/>
When one arc is greater than a quadrant, as
<math>
<mstyle>
<mi>b</mi>
<mi>c</mi>
</mstyle>
</math>
is in the 3rd and 4th diagrams, there are taken their residuals from a semicircle. And then the operation will be as the first or second case.
