6131
[Commentary:
The reference on this page is to Proposition 21 from Chapter 19 of Variorum responsorum liber VIII
(Viete 1593d, Chapter 19, Prop .
XXI.
Trianguli cujuslibet sphærici.
Datis duobus lateribus, & angulo cui unum ex illis lateribus opponitur, datur angulus cui alterum datorum laterum opponitur.
Vel,
Datis duobs angulis, & latere quod alteri datorum angulorum opponitur, datur latus reliquo oppositum.
Given two sides and the angle opposite one of those sides, the angle opposite the other is known.
Or,
Given two angles, and the side opposite one of the given angles, the side opposite the other is
Immediately after the statement of the proposition, Viète gave the following statement, under the heading Syntomon.
Quæ per factionem sub sinibus peripheriarum & adplicationem ad sinum totum exurgunt, eadem opere additionis vel subductionis præsto sunt.
Cum duæ peripheriæ angulum acutum componunt, est
Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia, minus sinu complementi composita.
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.
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
In modern notation this statement may be written as: . 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. ]
XXI.
Trianguli cujuslibet sphærici.
Datis duobus lateribus, & angulo cui unum ex illis lateribus opponitur, datur angulus cui alterum datorum laterum opponitur.
Vel,
Datis duobs angulis, & latere quod alteri datorum angulorum opponitur, datur latus reliquo oppositum.
Given two sides and the angle opposite one of those sides, the angle opposite the other is known.
Or,
Given two angles, and the side opposite one of the given angles, the side opposite the other is
Immediately after the statement of the proposition, Viète gave the following statement, under the heading Syntomon.
Quæ per factionem sub sinibus peripheriarum & adplicationem ad sinum totum exurgunt, eadem opere additionis vel subductionis præsto sunt.
Cum duæ peripheriæ angulum acutum componunt, est
Vt sinus totus ad sinum duplum primæ, ita sinus secundæ ad sinum complementi differentia, minus sinu complementi composita.
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.
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
In modern notation this statement may be written as: . 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. ]
Vieta lib. 8. resp.
pag. 39.
Syntomon
[Translation: Viète, Responsorum liber VIII.
]
Syntomon
[Translation: Viète, Responsorum liber VIII.
]
[???] in alia charta
[Translation: [???] in another sheet ]
[Translation: [???] in another sheet ]
Quæ per factionem sub sinibus peripherieriarum et adplicationem ad sinum totum exurgunt,
eadem opere additionis vel subductionis præsto sunt.
[Translation: 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.
[Translation: 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.
, una peripheria
, altera peripheria
, differentia
, aggregatum &
[Translation: is one arc, the other.
is the difference, the sum.
, altera peripheria
, differentia
, aggregatum &
[Translation: is one arc, the other.
is the difference, the sum.
Hæc quarta analogia est re eadem
cum secunda.
porro, prima et tertia analogiæ
reducuntur ad unam si quartus
terminus ita
[Translation: This fourth ratio is the same thing as the second.
Further, the first and third ratios are reduced to one if the fourth term is written ]
[Commentary: 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.
cum secunda.
porro, prima et tertia analogiæ
reducuntur ad unam si quartus
terminus ita
[Translation: This fourth ratio is the same thing as the second.
Further, the first and third ratios are reduced to one if the fourth term is written ]
[Commentary: 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.
Nota
Quando una peripheria est maior quadranti
ut est in3,a et 4,a diagrammati; summatur
eius residuum ad semicirculum. Et tum
operatio erit secundum primum vel secundum casum.
Quare hoc modo sunt duo tantummodo
[Translation: Note.
When one arc is greater than a quadrant, as 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.
Therefore by this method there are only two ]
Quando una peripheria est maior quadranti
ut est in3,a et 4,a diagrammati; summatur
eius residuum ad semicirculum. Et tum
operatio erit secundum primum vel secundum casum.
Quare hoc modo sunt duo tantummodo
[Translation: Note.
When one arc is greater than a quadrant, as 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.
Therefore by this method there are only two ]
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