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Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.Various

History

Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Name: Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.
Author: Various

Various

Science -- Periodicals

But it is demonstrated by Sir Isaac Newton, and by other authors, that
the force of all the particles, or of all the matter in the whole
spheroid AP _ap_, to turn _it_ about its center, is equal to ⅕th of
the force of a quantity of matter, placed at A, equal to the excess
of the matter in the whole spheroid above _that_ in the inscribed
sphere, whose axis is P_p_. Now this excess (assuming the ratio of
π to 1, to express _that_ of the area of a circle to the square of
the radius) will be truly represented by (4π/3) × OP × (OA² - OP²);
and, consequently, the force of all the matter in the whole earth, by
(_3tt_/_TT_) × (AK/OA) × (OK/OA) × (4π/15) × OP × (OA²- OP²). Let,
therefore, this quantity be now substituted for _F_, in the general
formula _F_/_ncf_, writing, at the same time, (4π/3) × OA² × OP,
and ⅖, in the place of their equals _c_ and _n_; by which means we
have (here) (_F_/_ncf_) = (_3tt_/_2TT_) × ((OA² - OP²)/OA²) × ((AK ×
OK)/OA²). Put the given quantity (_3tt_/_2TT_) × ((OA² - OP²)/OA²) =
_k_; and let the angle EA_e_ represent the horary alteration of the
position of the terrestrial equator, arising from the force _F_ (here
determined), and let the arch E_e_ be the regress of the equinoctial
point E, corresponding thereto: then, in the triangle EA_e_ (considered
as spherical) it will be sin. _e_ ∶ sin. AE (∷ sin. EA_e_: sin. E_e_) ∷
EA_e_ ∶ E_e_ (= (sin. AE x EA_e_)/sin. E) = _k_ × (sin. AE/sin. E) ×
((AK × OK)/OA²) = _k_ × ((sin. AE × cos. AH × sin. AH)/sin. E). But in
the triangle EHA, right-angled at A (where HA is supposed to represent
the sun’s declination, AE his right ascension, and HE his distance from
the equinoctial point E[207]) we have (_per spherics_)

   sin. AE   ∶ 1 (rad.)   ∷ co-t. E   ∶ co-t. AH,
  (sin. AH)² ∶ (sin. EH)² ∷ (sin. E)² ∶ 1² (rad.²)

From whence we get, sin. AE × co-t. AH × (sin. AH)² = (sin. EH)² ×
co-t. E × (sin. E)². But co-t. AH × sin. AH = co-s. AH × 1 (rad.), and
co-t. E × sin. E = co-s. E × 1 (rad.): therefore sin. AE × co-s. AH ×
sin. AH = (sin. EH)² × co-s. E × sin. E; and, consequently, _k_ × (sin.
AE × co-s. AH × sin. AH)/sin. E = _k_ × co-s. E × (sin. EH)² (= E_e_).

Public-domain text, read in full here on John Shaqi.

Opening the text

History

Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Various

Science -- Periodicals

But it is demonstrated by Sir Isaac Newton, and by other authors, that
the force of all the particles, or of all the matter in the whole
spheroid AP _ap_, to turn _it_ about its center, is equal to ⅕th of
the force of a quantity of matter, placed at A, equal to the excess
of the matter in the whole spheroid above _that_ in the inscribed
sphere, whose axis is P_p_. Now this excess (assuming the ratio of
π to 1, to express _that_ of the area of a circle to the square of
the radius) will be truly represented by (4π/3) × OP × (OA² - OP²);
and, consequently, the force of all the matter in the whole earth, by
(_3tt_/_TT_) × (AK/OA) × (OK/OA) × (4π/15) × OP × (OA²- OP²). Let,
therefore, this quantity be now substituted for _F_, in the general
formula _F_/_ncf_, writing, at the same time, (4π/3) × OA² × OP,
and ⅖, in the place of their equals _c_ and _n_; by which means we
have (here) (_F_/_ncf_) = (_3tt_/_2TT_) × ((OA² - OP²)/OA²) × ((AK ×
OK)/OA²). Put the given quantity (_3tt_/_2TT_) × ((OA² - OP²)/OA²) =
_k_; and let the angle EA_e_ represent the horary alteration of the
position of the terrestrial equator, arising from the force _F_ (here
determined), and let the arch E_e_ be the regress of the equinoctial
point E, corresponding thereto: then, in the triangle EA_e_ (considered
as spherical) it will be sin. _e_ ∶ sin. AE (∷ sin. EA_e_: sin. E_e_) ∷
EA_e_ ∶ E_e_ (= (sin. AE x EA_e_)/sin. E) = _k_ × (sin. AE/sin. E) ×
((AK × OK)/OA²) = _k_ × ((sin. AE × cos. AH × sin. AH)/sin. E). But in
the triangle EHA, right-angled at A (where HA is supposed to represent
the sun’s declination, AE his right ascension, and HE his distance from
the equinoctial point E[207]) we have (_per spherics_)

   sin. AE   ∶ 1 (rad.)   ∷ co-t. E   ∶ co-t. AH,
  (sin. AH)² ∶ (sin. EH)² ∷ (sin. E)² ∶ 1² (rad.²)

From whence we get, sin. AE × co-t. AH × (sin. AH)² = (sin. EH)² ×
co-t. E × (sin. E)². But co-t. AH × sin. AH = co-s. AH × 1 (rad.), and
co-t. E × sin. E = co-s. E × 1 (rad.): therefore sin. AE × co-s. AH ×
sin. AH = (sin. EH)² × co-s. E × sin. E; and, consequently, _k_ × (sin.
AE × co-s. AH × sin. AH)/sin. E = _k_ × co-s. E × (sin. EH)² (= E_e_).

Public-domain text, read in full here on John Shaqi.

Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Philosophical transactions, Vol. L. Part I. For the year 1757.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

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