. (For another reference to the same proposition, see also Add MS 6782
<ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/HSPGZ0AE&start=980&viewMode=image&pn=980">
f. </ref>
.) </s>
<lb/>
<quote xml:lang="lat">
<s xml:space="preserve">
V.II In omni triangulo sphaerali ex arcubus circulorum magnorum constante,
proportio sinus uersi anguli cuislibet ad differentiam duorum sinuum uersorum,
quorum unus est lateris eum angulum subtendentis:
alius uerò differentiae duorum arcuum ipsi angulo circumiacentium
est tanquam proportio quadrati sinus recti totius ad id,
quod sub sinibus arcuum dicto angulo circumpositorum continetur rectangulum</s>
</quote>
<lb/>
<quote>
<s xml:space="preserve">
In all spherical triangles composed from great arcs of circles, the ratio of the versed sine of any angle
to the difference of two versed sines, one of which is the side subtending the angle,
the other the difference of the two arcs adjacent to the angle,
is the proportion of the the square of the whole sine
to the product of the sines of the surrounding arcs by which the said angle is </s>
</quote>
<lb/>
<s xml:space="preserve">
The reference to Clavius is to his
<emph style="it">Triangula sphærica</emph>
in
<emph style="it">Triangula rectilinea, atque sphaerica</emph>
In omni triangulo sphærico, cuius duo arcus sint inæquales;
quadratum sinus totius ad rectangulum sub sinubus rectis duorum arcuum inæqulium contentum,
eandem proportionem habet, quam sinus versus anguli a dictis arcubus comprehensi
ad differentiam duorum sinuum versorum, quorum vnus differentiæ eorundem arcuum debetur,
alter vero tertio arcui, qui prædicto angulo oppostitus est, </s>
</quote>
<lb/>
<quote>
<s xml:space="preserve">
In all spherical triangles, whose two arcs are unequal,
the square of the whole sine to the product of the sines of the two unequal arcs
is in the same ratio as the versed sine of the angle between the said arcs
to the difference of two versed sines, one of which is of the difference of the arcs,
the other corresponding to the third arc, which is opposite the aforesaid </s>
</quote>
<lb/>
<s xml:space="preserve">
Harriot translates Clavius's statement into symbols for the particular triangle
<math>
<mstyle>
<mi>d</mi>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
shown in his diagram. He then goes on to prove that the versed sine of
<math>
<mstyle>
<mi>a</mi>
<mi>d</mi>
<mo>-</mo>
<mi>d</mi>
<mi>b</mi>
</mstyle>
</math>
is greater than the versed sine of
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
.
<lb/>
For another version of this page, see Add MS
<ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/MAH52R5E&start=100&viewMode=image&pn=101">
f. </ref>
. </s>
<s xml:space="preserve">]</s>
</p>
</div>
<head xml:space="preserve" xml:lang="lat">
(5.
<lb/>
Analogia per sinus versos, et universalis ad triangula
<lb/>
sphærica cuiuscunque
<lb/>
[
<emph style="bf">Translation: </emph>
Ratio by versed sines, and generally for a spherical trianlge under any ]</head>
<p xml:lang="lat">
<s xml:space="preserve">
Demonstratur a Regiomontano
<lb/>
lib. 5
<emph style="super">o</emph>
. pr. 2, de triang.
<lb/>
A clavio pr. 58. de sphæricis
<lb/>
Ab alijs Trigonistis.
<lb/>
Et a nobis alibi in
<lb/>
[
<emph style="bf">Translation: </emph>
Demonstrated by Regiomontanus in
<emph style="it">De triangulis</emph>
, Book 5, Proposition 2.
<lb/>
by Clavius in Proposition 58 of
<emph style="it">De sphæricis triangulis</emph>
<lb/>
By other triangulists.
<lb/>
And by me elsewhere in ]
<lb/>
[
<emph style="bf">Commentary: </emph>
The page 'elsewhere' referred to here is probably Add MS
<ref target="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/MAH52R5E&start=50&viewMode=image&pn=53">
f. </ref>
. </s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
Dico quod: […] (
<foreign xml:lang="gre">catolicos</foreign>
<lb/>
[
<emph style="bf">Translation: </emph>
I say that (generally) </s>
</p>
<p xml:lang="lat">
<s xml:space="preserve">
Consectarium
<lb/>
ponatur quod:
<lb/>
In triangulo
<math>
<mstyle>
<mi>a</mi>
<mi>d</mi>
<mi>b</mi>
</mstyle>
</math>
, datis duobus lateribus
<math>
<mstyle>
<mi>a</mi>
<mi>d</mi>
</mstyle>
</math>
,
<math>
<mstyle>
<mi>d</mi>
<mi>b</mi>
</mstyle>
</math>
; cum angulo
<math>
<mstyle>
<mi>d</mi>
</mstyle>
</math>
<lb/>
queratur
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
.
<lb/>
per superiorem analogiam, sit
<emph style="super">data</emph>
quarta proportionalis,
<math>
<mstyle>
<mi>y</mi>
</mstyle>
</math>
.
<lb/>
[…]
<lb/>
datur igitur,
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
<lb/>
[
<emph style="bf">Translation: </emph>
Consequence
<lb/>
It is supposed that, in triangle
<math>
<mstyle>
<mi>a</mi>
<mi>d</mi>
<mi>b</mi>
</mstyle>
</math>
, from given two sides
<math>
<mstyle>
<mi>a</mi>
<mi>d</mi>
</mstyle>
</math>
and
<math>
<mstyle>
<mi>d</mi>
<mi>b</mi>
</mstyle>
</math>
with the angle
<math>
<mstyle>
<mi>d</mi>
</mstyle>
</math>
, there is sought
<math>
<mstyle>
<mi>a</mi>
<mi>b</mi>
</mstyle>
</math>
.
<lb/>
by the above ratio, let the given fourth proportional be
