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Enclosure: Thomas Jefferson’s Notes on Étienne Bézout’s Cours de Mathématiques, [before April 1815]

Enclosure

Notes on Étienne Bézout’s Cours de Mathématiques

[before Apr. 1815]

To reduce angles observed on an inclined plane to that of the horison. Bezout. trigonom. 320. cours d’Artillerie.

let d.g.i be the plane of the base of a hill, and a.b.c. points on it’s side at different heights.

 ° 
suppose the ∠b.a.c. found by observation to be 62–37
the ∠ of inclination b.a.d (a.d. being vertical) 88– 5
the ∠ of inclination of c.a. to wit c.a.d. 78–17

it is required to find the ∠g.d.i. in the plane of the horizon.

let a.b. and a.c be prolonged until they meet the horizontal plane d.g.i. at g. & i.

consider a.d. as the radius of the tables, then d.g. & d.i. become tangents of the tables,
and a.g. & a.i. their secants; turn to the tables.
the ∠g.a.d. = 88°–5′ has for it’s tabular secant 29.90 neglecting subseqt fractions
it’s tangent 29.88
the ∠i.a.d. = 78–17 has for it’s tabular secant 4.92
tangent 4.82

1st operation

 in the △ a.g.i. given a.g. = 29.9  } required g.i
a.i =  4.92 
∠a = 62°37′1
 ag + ai : ag − ai :: ti + g/2 : ti − g/2 = t 49°–43′
then Si. ∠g : ai :: Si. ∠a. : gi = 28.

2d opern

in the right angled △ a.i.d. given the ∠a. = 78°–17′
the hypoth. a.i = 4.92. reqd d.i.
Rad. : 4.92 :: Si. 78°–17′ : d.i. = 4.8175

in the right angled △ a.g.d. given the ∠a = 88°–5′
the hypoth. a.g = 29.9. reqd d.g.
Rad : 29.9 :: Si. 88°–5′ : d.g. = 29.883

3d opern

 in the △ g.d.i. given d.g = 29.883 } required the ∠g.d.i.
d.i =  4.92
g.i = 28
 dg : gi + di :: gi − di : gk − dk = 25.456
and di : Rad. :: kd : Si. ∠dik which is complement to the ∠gdi = 62°–39′–20″2
 
Suppose a the summit of a mountain: ad it’s height, b & c points at it’s foot, c being in the level of it’s base, or of the common horison, & b above that level.
°  
that the ∠ aCb is observed to be 63–20–40
Required ad the height of mountn bg the height of station b.3 & the whole △ cdg, being the reduction of the points abc to the plane of the horizon. aBc 66–39–20
aCd  7–59–36
aBl  7–48– 0
bCg  0–28–39
& that bc is by admeasuremt 6000. feet.
△ acd  °  
for ac. Si. bAc = 50° : bc 6000 :: Si. aBc = 66–39–20 : ac = 7191.27
for cd Rad. : ac = 7191.27 :: S. cAd = 82– 0–24 : cd = 7121.4
for ad Rad. : ac. :: S. aCd =  7–59–36 : ad = 1000.
△ abc
for ab S. bAc = 50° : bc = 6000 :: S. aCb = 63–20–40 : ab = 7000.
△ abl
for bl Rad. : ab = 7000 :: S bAl = 82–12– 0 : bl = 6935.23
 
for al Rad. : ab. :: S. aBl =  7–48– 0 : al =  950.
△ bcg
for cg Rad. : bc = 6000 :: S. cBg = 89–31–21 : cg = 5999.8
for bg Rad. : bc. :: S. bCg =  0–28–39 : bg =   50.
△ cdg
  dg : cd + cg :: cd − cg : dk − gk.
  6935.23 : 13121.2 :: 1121.6 : 2122.04
  then dk = 4528.63 and gk = 2406.59
△ cdk  °  
for dCk cd : Rad. :: dk. : S. dCk = 39–29–17
  then cDg = 50–30–43
△ cGk
for gCk cg : Rad. :: gk : S. gCk = 23–38–52
  then cGd = 66–21– 8
  then the ∠dCg = dCk + gCk = 63– 8– 9
we have then the height of the mountain ad = 1000.f
height of the station b. = 50.f.
and the whole △ cdg, to wit Note. the plane of the theodolite being adjusted to that of the horizon, will render truly (and without needing reduction) the horizontal angles at the two stations, whether they be in the same, or different horizontal planes; because, if different, as b. and c. their planes being parallel, the degrees on them will be coincident. but in this case the base measured on the inclined plane b.c. must be reduced to the horizontal one g.c.
 The above diagram supposes the plane of the theodolite adjusted to the inclined plane a.b.c. which renders necessary the reduction of it’s angles as well as lines to the horizontal plane d.g.c.
cd = 7121.4
cg = 5999.8
dg = bl = 6935.23
°  
∠cDg = 50–30–43
∠cGd = 66–21– 8
∠dCg = 63– 8– 9

MS (MHi); filed with TJ’s Weather Memorandum Book, 1802–16; written entirely in TJ’s hand on both sides of a small sheet; undated, but composed prior to, and probably much earlier than, shipment of TJ’s library to Washington.

TJ based these notes in part on section 320 of Étienne Bézout, Cours de Mathématiques, a l’Usage du Corps Royal de l’Artillerie (Paris, 1788; Sowerby, description begins E. Millicent Sowerby, comp., Catalogue of the Library of Thomas Jefferson, 1952–59, 5 vols. description ends no. 3681), 1:334–6.

1Equation interlined.

2Page ends here.

3Preceding six words interlined.

Index Entries

  • altitude; mathematical source for calculating search
  • Bézout, Étienne; Cours de Mathématiques, a l’Usage du Corps Royal de l’Artillerie search
  • books; on mathematics search
  • Cours de Mathématiques, a l’Usage du Corps Royal de l’Artillerie (É. Bézout) search
  • Jefferson, Thomas; Writings; Notes on Étienne Bézout’sCours de Mathématiques search
  • mathematics; books on search
  • mathematics; trigonometry search
  • scientific instruments; theodolites search
  • theodolite search