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Thomas Jefferson: Trigonometry Exercise, 4 December 1815

Trigonometry Exercise

1815. Dec. 4.  The 2. cases of all the ∠s or all the sides of a spheric. △ being given. tried by the formula supplementy to Nepier’s

 °  } suppose ag. & aeg the greater segments
a =  49–54 T32 × Td2 of segm. of mid. p. = T52 × Td2 of oppos. parts.
d =  37–19 t 31°–12′–30″ × taeg − deg2 = t 43°–36′–30″ × t 6°–17′–30″
e =  62–25
? ae =  14–45–42  °  
de  t. 43–36–30 9.9788944  31–12–30
ag  + t 6–17–30 9.0423948  ±  9–50–43
19.0212892  41–  3–13 = aeg
see No 91. dg  t 31–12–30 9.7822155  21–21–47 = deg.
= t aeg − deg2 =  9.2390737  =  9°–50′–43″
for ae.  Rad × Cos. ae =  Cot. a × Cot aeg
  Rad
 °  
Cot. a = 49–54– 0   9.9253524
+ Cot. aeg = 41– 3–13 10.0600716
R = Cos. ae  <1>9.9854240  = 14°–45′–42″

for de.  Rad ×  Cos de =  Cot. d × Cot deg
  Rad
 °  
Cot. d 37–19– 0   10.1178993
+ Cot. deg 21–21–47 10.4076544
R. = Cos de  10.5255537.  impracticable.

suppose dg & deg the greater segments

 °  
 then (as above) tdeg − aeg2 =  9°–50′–43″ and  41– 3–13 = deg
21–21–47 = aeg
for ae.  Rad ×  Cos. ae =  Cot. a × Cot. aeg
  Rad
 °  
Cot a 49–54– 0   9.9253524
+ Cot aeg 21–21–47 10.4076544
R. = Cos. ae  10.3330068  impracticable.

No 89.
given   found  °  
d = 63–45  dfg =  56–26–37  df =  70–53–30
e = 71–49  efg =  38–25–23  ef =  65–32–21.  or 63°–7′–30″
f = 94–52   de =  82–17
dg =  51–56–34 51–56–34
eg =  34–27 or 33–40
86. 23. 34 or 85–36–34  or 82°–17′
No 80  °   by Nepier  by Minto
given  a =  59–47  abc = 42–12–30  44– 2–19
b =  77–39–23   cbd = 35–26–53  33–37– 3
d =  64–15.  ab = 50– 3  52–58– 0
found  bd = 47–21  43–29–22
 ad = 56–15  

No 81  °    found by Nepier  by Minto
given  a =  112–40–45  cad =   65–43–45  64–31
b =   66– 7 bac =   46–57  48– 9–44
d =   59–50 a =  112–45–45  112–40–44
?  ab =   65–34  66–38
ad =   74–20  73–54–59
bd =   76–19–17  78–261
No 86.  °   found in ye opern
given  de =  79–56 af =  34–37–54″   eg =  62–34– 8
ef =  66–29 ad =  68–52–21 dg =  17–21–52
df =  34–14–27 aef =  38–17–58 fg =  29–59–24
   aed =  71–19–40    
  originally given  by Bezout  by Minto
reqd d  =  62–40  62–39– 6  62–39–47
e  =  33– 2  33– 1–42  33– 1–46
f  =  72–32– 6  72–30–39  107–28–41 ?
ae =  61– 1– 0 {  32– 0–56 = dfg ?
 75–27–45 = efg ?
give d. e. & f. as above
to find de, ef, df
} by Minto, the results are of  ef = 16– 1–34 by Bez. & Nepier
df = 59–53–30
eg = 13–33–37
dg. impossible

there must be something vicious in the data.

MS (CSmH: JF); written entirely in TJ’s hand on both sides of a single sheet.

In setting up these problems, TJ drew on examples published in John Potter, A System of Practical Mathematics (London, 1753; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3667), 277–8, 284–6, 289–90, 293. What TJ here calls the supplementary formula for solving spherical triangles when either all of the angles or all of the sides are known comes from David Steuart Erskine and Walter Minto, An Account of the Life, Writings, and Inventions of John Napier, of Merchiston (Perth, 1787), 97–103. For TJ’s explanation of Napier’s (nepier’s) Theorem, see TJ to Louis H. Girardin, 18 Mar. 1814, and Notes on Napier’s Theorem, printed above at 18 Mar. 1814. Étienne bezout discussed methods for solving spherical triangles given the values for the three sides in his Cours de Mathématiques, a l’usage Des Gardes du Pavillon et de la marine (Paris, 1764–69; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3680) pt. 2, 345–6, 349–50.

1Recto ends here.

Index Entries

  • An Account of the life, writings, and inventions of John Napier, of Merchiston (D. S. E. Buchan and W. Minto) search
  • A System of Practical Mathematics (J. Potter) search
  • Bézout, Étienne; Cours de Mathématiques, a l’usage Des Gardes du Pavillion et de la marine search
  • books; on mathematics search
  • Buchan, David Steuart Erskine, 11th Earl of; An account of the life, writings, and inventions of John Napier, of Merchiston search
  • Cours de Mathématiques, a l’usage Des Gardes du Pavillion et de la marine (É. Bézout) search
  • Jefferson, Thomas; Writings; Trigonometry Exercise search
  • mathematics; books on search
  • mathematics; Napier’s theorem search
  • mathematics; trigonometry search
  • Minto, Walter; An Account of the Life, Writings, and Inventions of John Napier, of Merchiston search
  • Napier, John (of Merchiston); mathematical theorem of search
  • Potter, John (mathematician); A System of Practical Mathematics search