320161
[Commentary:
Here Harriot examines a statement that appears as Proposition VI of the 'Dati sexti', Chapter XIX of
Viète's Variorum responsorum liber VIII
(Viete 1593d, Chapter 19, DATI SEXTI, Prop . See also Add MS 6782
f. .
VI.
Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulares peripheriæ.
VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given
The reference to Pitiscus is Trigonometria: sive de solutione triangulorum tractarus brevis et perpsicuus (1595).
The reference to Lansberg is to Triangulorum geometriae libri quatuor (1591).
The reference to Regiomontanus is to De triangulis omnimodis libri quinque (Regiomontanus [1464], 1533, 1561, Prop IV.31). ]
VI.
Data summa vel differentia duarum perpheriarum, quarum sinus datam habeant rationem, dantur singulares peripheriæ.
VI. Given the sum or difference of two arcs, whose sines are in a given ratio, each arc is given
The reference to Pitiscus is Trigonometria: sive de solutione triangulorum tractarus brevis et perpsicuus (1595).
The reference to Lansberg is to Triangulorum geometriae libri quatuor (1591).
The reference to Regiomontanus is to De triangulis omnimodis libri quinque (Regiomontanus [1464], 1533, 1561, Prop IV.31). ]
Data differentia.)
Sit differentia duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut et .
[…]
sinus arcus
Datur igitur . nam .
Datur etiam . nam sinus complementi est vel dimidij arcus .
Cætera ut supra. et habetur sinus arcus.
Tum: . arcus minor
. arcus maior quæsitis
Tum etiam arcus maior
Sit differentia duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut et .
[…]
sinus arcus
Datur igitur . nam .
Datur etiam . nam sinus complementi est vel dimidij arcus .
Cætera ut supra. et habetur sinus arcus.
Tum: . arcus minor
. arcus maior quæsitis
Tum etiam arcus maior
12.) Data summa vel differentia duarum periferiarum,
quarum sinus datam habeant rationem: dantur singulares
[Translation: Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are ]
quarum sinus datam habeant rationem: dantur singulares
[Translation: Given the sum or difference of two arcs, for which the sines are in a given ratio, the individual arcs are ]
per Tangentes exhibit
Vieta in responsis pag. 37.
Pitiscus pag. 92.
Lansbergis. pag.
[Translation: Shown by tangents
by Viète in Responsorum, page 37,
Pitiscus, page 92,
Lansberg, page ]
Vieta in responsis pag. 37.
Pitiscus pag. 92.
Lansbergis. pag.
[Translation: Shown by tangents
by Viète in Responsorum, page 37,
Pitiscus, page 92,
Lansberg, page ]
Quæ modo usui accomodatior est, quam per
sinus solos, quando tangentibus ut
[Translation: Which method of use is more convenient than by sines alone, when by tangents, as one ]
sinus solos, quando tangentibus ut
[Translation: Which method of use is more convenient than by sines alone, when by tangents, as one ]
Sed quando non licet Tangentibus uti, modus per solos sinus (etsi laboriosior)
adhibendus est. Exhibatur a Regiomontano lib. 4. prop. 31. de triangulis
Modus ille hic apponitur paucis
[Translation: But when one does not want to use tangents, the method by sines alone (though more laborious) is shown. It is given by Regiomontanus, Book IV, Proposition 31 of De triangulis. That method set out here is explained a ]
adhibendus est. Exhibatur a Regiomontano lib. 4. prop. 31. de triangulis
Modus ille hic apponitur paucis
[Translation: But when one does not want to use tangents, the method by sines alone (though more laborious) is shown. It is given by Regiomontanus, Book IV, Proposition 31 of De triangulis. That method set out here is explained a ]
Data summa.)
Sit summa duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut ad .
fiat:
Datur ergo , nam . est sinus dimidij arcus .
Datur etiam . nam sinus complementi est vel dimidij arcus .
[…]
sinus arcus
Tum: arcus maior
arcus minor quæsita
Tum etiam: . arcus maior
[Translation: Given the sum.)
Let be the sum of the two arcs, and the ratio of the two sines of the sought arcs as to .
construct:
Therefore is given, for , and is the sine of half the arc .
Also is gien, for the sine of the complement is , or half the arc .
the sine of arc
Then , the greater arc.
the lesser arc sought.
Then also , the greater arc sought.
Sit summa duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut ad .
fiat:
Datur ergo , nam . est sinus dimidij arcus .
Datur etiam . nam sinus complementi est vel dimidij arcus .
[…]
sinus arcus
Tum: arcus maior
arcus minor quæsita
Tum etiam: . arcus maior
[Translation: Given the sum.)
Let be the sum of the two arcs, and the ratio of the two sines of the sought arcs as to .
construct:
Therefore is given, for , and is the sine of half the arc .
Also is gien, for the sine of the complement is , or half the arc .
the sine of arc
Then , the greater arc.
the lesser arc sought.
Then also , the greater arc sought.
Data differentia.)
Sit differentia duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut et .
[…]
sinus arcus
Datur igitur . nam .
Datur etiam . nam sinus complementi est vel dimidij arcus .
Cætera ut supra. et habetur sinus arcus.
Tum: . arcus minor
. arcus maior quæsitis
Tum etiam arcus maior
[Translation: Given the difference.)
Let be the difference of the two sought arcs, and the ratio of the sines of the sought arcs and .
sine of the arc
Therefore is given, for .
Also is given, for the sine of the complement is , or half the arc .
The rest as above. And we have the sine of arc .
Then , the lesser arc.
, the greater arc sought.
Then also , the greater arc sought.
Sit differentia duarum peripheriam.
ratio sinuum quæsitarum peripheriarum ut et .
[…]
sinus arcus
Datur igitur . nam .
Datur etiam . nam sinus complementi est vel dimidij arcus .
Cætera ut supra. et habetur sinus arcus.
Tum: . arcus minor
. arcus maior quæsitis
Tum etiam arcus maior
[Translation: Given the difference.)
Let be the difference of the two sought arcs, and the ratio of the sines of the sought arcs and .
sine of the arc
Therefore is given, for .
Also is given, for the sine of the complement is , or half the arc .
The rest as above. And we have the sine of arc .
Then , the lesser arc.
, the greater arc sought.
Then also , the greater arc sought.