6734
[Commentary:
Further work on spherical triangles.
As on Add MS f. , there is reference to Clavius, Triangula rectilinea, atque sphaerica (Clavius 1586, Prop . ]
As on Add MS f. , there is reference to Clavius, Triangula rectilinea, atque sphaerica (Clavius 1586, Prop . ]
(9.
Casuum Limitatio cum designatione
[Translation: Determination of cases with a useful ]
Casuum Limitatio cum designatione
[Translation: Determination of cases with a useful ]
Animadvertendum quod in superioribus investigationibus analogiarum
sive casuum, ubi latera et signantur \le \le; debeat etiam
intelligi signata \ge \ge. Quoniam si sint eiusdem affectionis
sive utrique minora quadrantibus, sive maiora; non variatur
inde illatio neque casus.
Hinc in 6 notatis casibus positi trianguli, sunt 9 variationes
signorum ut
[Translation: It is to be noted that in the above investigations of ratios, or cases, where the sides and are marked with \le, \le, respectively, it is also to be understood that they could be marked with \ge, \ge, respectively. Because they have the same relationship, whether both less than a quadrant, or greater; therefore the result does not vary, nor the cases.
Here in the 6 denoted cases of the supposed triangle, there are 9 ]
sive casuum, ubi latera et signantur \le \le; debeat etiam
intelligi signata \ge \ge. Quoniam si sint eiusdem affectionis
sive utrique minora quadrantibus, sive maiora; non variatur
inde illatio neque casus.
Hinc in 6 notatis casibus positi trianguli, sunt 9 variationes
signorum ut
[Translation: It is to be noted that in the above investigations of ratios, or cases, where the sides and are marked with \le, \le, respectively, it is also to be understood that they could be marked with \ge, \ge, respectively. Because they have the same relationship, whether both less than a quadrant, or greater; therefore the result does not vary, nor the cases.
Here in the 6 denoted cases of the supposed triangle, there are 9 ]
Sunt præter illas novem, tres aliæ variationes; et non dantur
plures: sed istæ sunt impossibiles
[Translation: Besides those nine, there are three other variations; more are not given, but these are ]
plures: sed istæ sunt impossibiles
[Translation: Besides those nine, there are three other variations; more are not given, but these are ]
Una et ultima impossibilium probatur per Clavium pro: 27 de Sphæricis.
et a me magis perspicus in notis de conversione triangulorum:
reliquæ duæ conseqununtur per
[Translation: One, the final impossibility, is proved by Clavus in Proposition 27 of De sphærica; and by me more clearly in notation in the conversion of triangles; the remaining two follow by paraplerosis.
et a me magis perspicus in notis de conversione triangulorum:
reliquæ duæ conseqununtur per
[Translation: One, the final impossibility, is proved by Clavus in Proposition 27 of De sphærica; and by me more clearly in notation in the conversion of triangles; the remaining two follow by paraplerosis.
Alia designatio analogiarum et casuum,
usui magi
[Translation: Another specification of ratios and cases, more convenient in ]
usui magi
[Translation: Another specification of ratios and cases, more convenient in ]
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