440221
[Commentary:
At the end of Chapter XIX of Variorum responsorum liber VIII (1593), under the heading 'ALIUD', Viète listed sixteen propositions connecting sines, tangents, and secants
(Viete 1593d, Chapter 19, . In the first edition of the Responsorum, the pages were numbered only on the recto side. However, the pagination went badly wrong, so that in Chapter XIX we have the sequence: 37, 38, 39, 38. This sometimes makes it difficult to follow Harriot's references correctly. Here it seems that he has seen the number '38' on the right-hand (recto) page, and thus inferred that the left-hand (verso) page must be 37v, whereas it is in fact 39v. In the 1646 edition of Viète's Opera mathematica
the sixteen propositions are to be found on pages
On this and the following pages, Harriot worked through the sixteen propositions systematically. On this page he lists the first six. Note that for Harriot, as for Viète, trigonometrical relationships arose from astronomy. Thus the concepts of sine, tangent, and secant related not to angles defined by a pair of lines meeting at a point, but to arcs of a circle with a given radius, and therefore only by implication to the angles subtended by
At the top of the page Harriot listed the relevant quantities: sine, tangent, secant, radius (or whole sine), and the symbols he regularly used for them. The equivalent names used by Viète were sinus, prosinus, transinuosa, totus. Harriot had no words for cosine, cotangent, or coseceant. Where we would use 'cosine', for example, he spoke of the sine of the complement. Thus he wrote BC for sine(arc BC) but BC for the sine of the complement of BC, that is, cosine(BC). ]
On this and the following pages, Harriot worked through the sixteen propositions systematically. On this page he lists the first six. Note that for Harriot, as for Viète, trigonometrical relationships arose from astronomy. Thus the concepts of sine, tangent, and secant related not to angles defined by a pair of lines meeting at a point, but to arcs of a circle with a given radius, and therefore only by implication to the angles subtended by
At the top of the page Harriot listed the relevant quantities: sine, tangent, secant, radius (or whole sine), and the symbols he regularly used for them. The equivalent names used by Viète were sinus, prosinus, transinuosa, totus. Harriot had no words for cosine, cotangent, or coseceant. Where we would use 'cosine', for example, he spoke of the sine of the complement. Thus he wrote BC for sine(arc BC) but BC for the sine of the complement of BC, that is, cosine(BC). ]
vieta in lib. 8. respons.
pag. 37.
[Translation: Viète, in Responsorum liber VIII, page 37, ]
pag. 37.
[Translation: Viète, in Responsorum liber VIII, page 37, ]
sinus
tangens
secans
[Translation: sine
tangent
secans
]
tangens
secans
[Translation: sine
tangent
secans
]
1. Sinus peripheriæ, Radius: Radius, Secans complementi.
2. Sinus comp. peripheriæ. Radius. Radius. Secans peripheriæ.
3. Tangens peripheriæ. Radius. Radius. Tangens
[Translation: 1. Sine of the arc : Radius = Radius : Secant of the complement.
2. Sine of the compplement of the arc : Radius = Radius : Secant of the arc.
3. Tangent of the arc : Radius = Radius : Tangent of the ]
2. Sinus comp. peripheriæ. Radius. Radius. Secans peripheriæ.
3. Tangens peripheriæ. Radius. Radius. Tangens
[Translation: 1. Sine of the arc : Radius = Radius : Secant of the complement.
2. Sine of the compplement of the arc : Radius = Radius : Secant of the arc.
3. Tangent of the arc : Radius = Radius : Tangent of the ]
Ergo: cum proponuntur duæ peripheriæ
4. Sinus peripheriæ primæ. Sinus secundæ. Secans. compl. secundæ. Secans compl. primæ.
5. Sinus compl. primæ. Sinus com. secundæ. Secans 2æ. Secans primæ.
6. Tangens primæ. Tangens 2æ. Tangens comp. 2æ. Tangens compl.
[Translation: Therefore, when there are given two arcs:
4. Sine of the first arc : Sine of the second = Secant of the complement of the second : Secant of the complement of the first.
5. Sine of the complement of the first : SIne of the complement of the second = Secant of the second : Secant of the first.
6. Tangent of the first : Tangent of the second = Tangent of the complemnt of the second : Tangent of the complement of the first.
4. Sinus peripheriæ primæ. Sinus secundæ. Secans. compl. secundæ. Secans compl. primæ.
5. Sinus compl. primæ. Sinus com. secundæ. Secans 2æ. Secans primæ.
6. Tangens primæ. Tangens 2æ. Tangens comp. 2æ. Tangens compl.
[Translation: Therefore, when there are given two arcs:
4. Sine of the first arc : Sine of the second = Secant of the complement of the second : Secant of the complement of the first.
5. Sine of the complement of the first : SIne of the complement of the second = Secant of the second : Secant of the first.
6. Tangent of the first : Tangent of the second = Tangent of the complemnt of the second : Tangent of the complement of the first.
Menda in Vieta.
[Translation: Wrong in Viète.
Correction.
[Translation: Wrong in Viète.
Correction.
sinus
tangens
secans
tangens
secans