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Philosophical transactions, Vol. L. Part II. For the year 1758.: 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 II. For the year 1758.: 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 II. For the year 1758.: 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

[156] Because -_ẋ_ ⁄ √(1 - _xx_) and -_Ẋ_ ⁄ √(1 - _XX_) are known to
express the fluxions of the circular arcs whose co-sines are _x_ and
_X_, it is evident, if those arcs be supposed in any constant ratio of
1 to _n_, that _nẋ_ ⁄ √(1 - _xx_) = _Ẋ_ ⁄ √(1 - _XX_), and consequently
that _nẋ_ ⁄ √(_xx_ - 1) (= _nẋ_ ⁄ (√-1 × √(1 - _xx_)) = _Ẋ_ ⁄ (√-1 ×
√(1 - _XX_)) = _Ẋ_ ⁄ √(_XX_ - 1). From whence, by taking the fluents,
_n_ × Log. (_x_ + √(_xx_ - 1)) (or Log. (_x_ + √(_xx_ - 1))_ⁿ_) = Log.
_X_ + √(_XX_ - 1); and consequently (_x_ + √(_xx_ - 1))_ⁿ_ = _X_ +
√(_XX_ - 1): whence also, seeing _x_ - √(_xx_ - 1) is the reciprocal
of _x_ + √(_xx_ - 1), and _X_ - √(_XX_ - 1) of _X_ + √(_XX_ - 1), it
is likewise evident, that (_x_ - √(_xx_ - 1))_ⁿ_ = _X_ - √(_XX_ - 1).
Hence, not only the truth of the above assumption, but what has been
advanced in relation to the roots of the equation _zⁿ_ - 1 = 0, will
appear manifest. For if _x_ ± √(_xx_ - 1) be put = _z_, then will _zⁿ_
(= (_x_ ± √(_xx_ - 1))_ⁿ_) = _X_ ± √(_XX_ - 1): where, assuming _X_
= 1 = co-s. 0 = co-s. 360° = co-s. 2 × 360° = co-s. 3 × 360°, _&c._
the equation will become _zⁿ_ = 1, or _zⁿ_ - 1 = 0; and the different
values of _x_, in the expression (_x_ ± √(_xx_ - 1)) for the root _z_,
will consequently be the co-sines of the arcs, 0 ⁄ _n_, 360° ⁄ _n_, (2
× 360°) ⁄ _n_, _&c._ these arcs being the corresponding _submultiples_
of those above, answering to the co-sine _X_ (= 1).----In the same
manner, if _X_ be taken = -1 = co-s. 180° = co-s. 3 × 180° = co-s. 5 ×
180°, _&c._ then will _zⁿ_ = -1, or _zⁿ_ + 1 = 0; and the values of _x_
will, in this case, be the co-sines of 180° ⁄ _n_, 3 × (180° ⁄ _n_), 5
× (180° ⁄ _n_), _&c._

[157] _Avellana purgatrix_; in French, _medicinier_.

[158] This refers to Mr. Baker’s having supposed, that old iron and old
brass may be mixt sometimes, and melted down together.

[159] Vide Wilkins’s real Character, p. 131. Bellon. aquat. p. 330.

[160] Some of the Pour-contrel kind have but one row of suckers on the
arms: such an one I have seen, whose arms were thirty inches long.

[161] Of this I gave an account some years ago, in my attempt towards a
Natural History of the Polype, chap. v.

[162] See Plate xxxi. Fig. 1.

[163] _De Num. quibusd. Sam. et Phœn. &c. Dissert._ p. 56-59. & Tab.
II. Oxon. 1750.

[164] _Marm. Palmyren. a Cl._ Dawk. _edit._ pass.

[165] Vid. Hadr. Reland. _Palæst. Illustrat._ p. 1014. Traject.
Batavor. 1714. Erasm. Frœl. ad _Annal. Compendiar. Reg. & Rer. Syr._
Tab. VIII. &c. Viennæ, 1754.

[166] _De Antiq. Hebræor. et Græcor. Lit. Libel._ Joan. Baptist.
Biancon. p. 31, 32. Bononiæ, 1748.

[167] 1. Maccab. i. 10.

[168] Hadr. Reland. _De Num. Vet. Hebr._ pass. Trajecti ad _Rhenum_,
1709.

[169] See Plate xxxi. Fig. 2.

[170] Honor. Arigon. _Num. Phœnic._ Tab. I. Num. 3, 6. Tarvisii, 1745.

[171] Nicol. Haym Roman. _Del Tesor. Britan._ Vol. i. p. 106. In
Londra, 1719.

[172] See Plate xxxi. Fig. 3.

[173] See Plate xxxi. Fig. 3.

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

Opening the text

History

Philosophical transactions, Vol. L. Part II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

Various

Science -- Periodicals

[156] Because -_ẋ_ ⁄ √(1 - _xx_) and -_Ẋ_ ⁄ √(1 - _XX_) are known to
express the fluxions of the circular arcs whose co-sines are _x_ and
_X_, it is evident, if those arcs be supposed in any constant ratio of
1 to _n_, that _nẋ_ ⁄ √(1 - _xx_) = _Ẋ_ ⁄ √(1 - _XX_), and consequently
that _nẋ_ ⁄ √(_xx_ - 1) (= _nẋ_ ⁄ (√-1 × √(1 - _xx_)) = _Ẋ_ ⁄ (√-1 ×
√(1 - _XX_)) = _Ẋ_ ⁄ √(_XX_ - 1). From whence, by taking the fluents,
_n_ × Log. (_x_ + √(_xx_ - 1)) (or Log. (_x_ + √(_xx_ - 1))_ⁿ_) = Log.
_X_ + √(_XX_ - 1); and consequently (_x_ + √(_xx_ - 1))_ⁿ_ = _X_ +
√(_XX_ - 1): whence also, seeing _x_ - √(_xx_ - 1) is the reciprocal
of _x_ + √(_xx_ - 1), and _X_ - √(_XX_ - 1) of _X_ + √(_XX_ - 1), it
is likewise evident, that (_x_ - √(_xx_ - 1))_ⁿ_ = _X_ - √(_XX_ - 1).
Hence, not only the truth of the above assumption, but what has been
advanced in relation to the roots of the equation _zⁿ_ - 1 = 0, will
appear manifest. For if _x_ ± √(_xx_ - 1) be put = _z_, then will _zⁿ_
(= (_x_ ± √(_xx_ - 1))_ⁿ_) = _X_ ± √(_XX_ - 1): where, assuming _X_
= 1 = co-s. 0 = co-s. 360° = co-s. 2 × 360° = co-s. 3 × 360°, _&c._
the equation will become _zⁿ_ = 1, or _zⁿ_ - 1 = 0; and the different
values of _x_, in the expression (_x_ ± √(_xx_ - 1)) for the root _z_,
will consequently be the co-sines of the arcs, 0 ⁄ _n_, 360° ⁄ _n_, (2
× 360°) ⁄ _n_, _&c._ these arcs being the corresponding _submultiples_
of those above, answering to the co-sine _X_ (= 1).----In the same
manner, if _X_ be taken = -1 = co-s. 180° = co-s. 3 × 180° = co-s. 5 ×
180°, _&c._ then will _zⁿ_ = -1, or _zⁿ_ + 1 = 0; and the values of _x_
will, in this case, be the co-sines of 180° ⁄ _n_, 3 × (180° ⁄ _n_), 5
× (180° ⁄ _n_), _&c._

[157] _Avellana purgatrix_; in French, _medicinier_.

[158] This refers to Mr. Baker’s having supposed, that old iron and old
brass may be mixt sometimes, and melted down together.

[159] Vide Wilkins’s real Character, p. 131. Bellon. aquat. p. 330.

[160] Some of the Pour-contrel kind have but one row of suckers on the
arms: such an one I have seen, whose arms were thirty inches long.

[161] Of this I gave an account some years ago, in my attempt towards a
Natural History of the Polype, chap. v.

[162] See Plate xxxi. Fig. 1.

[163] _De Num. quibusd. Sam. et Phœn. &c. Dissert._ p. 56-59. & Tab.
II. Oxon. 1750.

[164] _Marm. Palmyren. a Cl._ Dawk. _edit._ pass.

[165] Vid. Hadr. Reland. _Palæst. Illustrat._ p. 1014. Traject.
Batavor. 1714. Erasm. Frœl. ad _Annal. Compendiar. Reg. & Rer. Syr._
Tab. VIII. &c. Viennæ, 1754.

[166] _De Antiq. Hebræor. et Græcor. Lit. Libel._ Joan. Baptist.
Biancon. p. 31, 32. Bononiæ, 1748.

[167] 1. Maccab. i. 10.

[168] Hadr. Reland. _De Num. Vet. Hebr._ pass. Trajecti ad _Rhenum_,
1709.

[169] See Plate xxxi. Fig. 2.

[170] Honor. Arigon. _Num. Phœnic._ Tab. I. Num. 3, 6. Tarvisii, 1745.

[171] Nicol. Haym Roman. _Del Tesor. Britan._ Vol. i. p. 106. In
Londra, 1719.

[172] See Plate xxxi. Fig. 3.

[173] See Plate xxxi. Fig. 3.

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

Philosophical transactions, Vol. L. Part II. For the year 1758.: 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 II. For the year 1758.: Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

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