5327
[Commentary:
Work on spherical triangles.
On this page there are references to both Regiomontanus and Clavius, who both gave a version of the theorem given here.
The reference to Regiomontanus is to his De triangulis omnimodis libri quinque (Regiomontanus [1464], 1533, 1561, Prop V.2). (For another reference to the same proposition, see also Add MS 6782 f. .)
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
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
The reference to Clavius is to his Triangula sphærica in Triangula rectilinea, atque sphaerica (Clavius 1586, Prop .
Theorema 56, Propositio 58.
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,
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
Harriot translates Clavius's statement into symbols for the particular triangle shown in his diagram. He then goes on to prove that the versed sine of is greater than the versed sine of .
For another version of this page, see Add MS f. . ]
On this page there are references to both Regiomontanus and Clavius, who both gave a version of the theorem given here.
The reference to Regiomontanus is to his De triangulis omnimodis libri quinque (Regiomontanus [1464], 1533, 1561, Prop V.2). (For another reference to the same proposition, see also Add MS 6782 f. .)
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
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
The reference to Clavius is to his Triangula sphærica in Triangula rectilinea, atque sphaerica (Clavius 1586, Prop .
Theorema 56, Propositio 58.
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,
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
Harriot translates Clavius's statement into symbols for the particular triangle shown in his diagram. He then goes on to prove that the versed sine of is greater than the versed sine of .
For another version of this page, see Add MS f. . ]
(5.
Analogia per sinus versos, et universalis ad triangula
sphærica cuiuscunque
[Translation: Ratio by versed sines, and generally for a spherical trianlge under any ]
Analogia per sinus versos, et universalis ad triangula
sphærica cuiuscunque
[Translation: Ratio by versed sines, and generally for a spherical trianlge under any ]
Demonstratur a Regiomontano
lib. 5o. pr. 2, de triang.
A clavio pr. 58. de sphæricis
Ab alijs Trigonistis.
Et a nobis alibi in
[Translation: Demonstrated by Regiomontanus in De triangulis, Book 5, Proposition 2.
by Clavius in Proposition 58 of De sphæricis triangulis
By other triangulists.
And by me elsewhere in ]
[Commentary: The page 'elsewhere' referred to here is probably Add MS f. .
lib. 5o. pr. 2, de triang.
A clavio pr. 58. de sphæricis
Ab alijs Trigonistis.
Et a nobis alibi in
[Translation: Demonstrated by Regiomontanus in De triangulis, Book 5, Proposition 2.
by Clavius in Proposition 58 of De sphæricis triangulis
By other triangulists.
And by me elsewhere in ]
[Commentary: The page 'elsewhere' referred to here is probably Add MS f. .
Dico quod: […] (catolicos
[Translation: I say that (generally)
[Translation: I say that (generally)
Consectarium
ponatur quod:
In triangulo , datis duobus lateribus , ; cum angulo
queratur .
per superiorem analogiam, sit data quarta proportionalis, .
[…]
datur igitur,
[Translation: Consequence
It is supposed that, in triangle , from given two sides and with the angle , there is sought .
by the above ratio, let the given fourth proportional be
therefore is given
ponatur quod:
In triangulo , datis duobus lateribus , ; cum angulo
queratur .
per superiorem analogiam, sit data quarta proportionalis, .
[…]
datur igitur,
[Translation: Consequence
It is supposed that, in triangle , from given two sides and with the angle , there is sought .
by the above ratio, let the given fourth proportional be
therefore is given
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