Problem XIX. To conftruct a Scale of Natural Sines, Tangents, and Secants; alfo a Line of Chords, Rhumbs, and Longitudes. Practice. Defcribe a Circle which cross with two Diameters on the Point erect the Perpendicular parallel to which continue the Diameter infinitely towards then graduate the quadrantal Arch and thro' cach Divifion, from the Center be the Tangent-Line, gra- Then with one Foot of the Compaffes in transfer the Divifions of the Tangent with the other Point to the Line fo fhall that be a Line of 10. 20. 30. 40. &c. 10. 20. 30. 40. &c. AB CB } 10. 20. 30. &c. E AB AC Then from the faid Divifions of the Arch let fall Perpendiculars to the Radius fo fhall that Radius be a Line of Right-Sines, graduated in the Points. Also, if from the Point you draw Lines to the Divifions in they will graduate the Line for a Line of half Tangents, in 10. 20.30. 40. &c. H Next Next graduate the Arch and transfer thofe Divifions to the Chord fo fhall that be a Line of Chords graduated in the Points Divide the Semidiameter into 60 equal Parts, in AE · AE 10. 20. 30. 40. &c. from which drop Perpendiculars to the Arch ED transfer thofe Divifions in the Arch, to the Chord fo fhall that Chord be the Line of Longitudes, graduated in the Points Laftly, divide the quadrantal Arch into Eight equal Parts, in CE 10. 20.30. &c. } ED 10. 20. 30. &c. BD 1. 2. 3. 4. &c. BD then transfer thofe Divifions to Chord fo shall that Chord be a Line of 1. 2. 3. 4.830. Rhumbs, graduated in the Points Thofe Lines being thus made and graduated, are to be placed appofitely on any Inftrument of Pafteboard, Wood, or Brafs, &c. and this is what is commonly called the Plain-Scale; which may be made of any Size according to the Radius or Semidiameter of the Circle. N.B. This Problem ought to be well understood by every one who would have a rational Knowledge of Trigonometry, the Conftruction of this Scheme being the Grounds and Rationale of every Kind there of CHAP. CHA P. V. Various Methods of Constructing a Canon of Natural Sines, Tangents, and Secants: Alfo of the Logarithmetic Ca non. T HE firft Method for making Natural Sines, &c. is deduced from the foregoing Theorems. For, by Theorem XVII, the Chord of 60 Degrees is equal to the Radius or Semidiameter of the Circle; confequently, half the Radius is the Sine of 30 Degrees; and therefore if Radius be fuppofed equal to 10000000 Parts, the Sine of 30 Degrees fhall be equal to 5000coo of the faid Parts. Now, the Sine of the Arch of 30 Degrees being given, the Sine of the Arch of 15 Degrees is given alfo (or may be found) by Theorem XXII; By the fame Theorem alfo, the Sine of 15 Degrees being given, we fhall have the Sine of 7 Deg. and 30 Min. and this Sine being known, we fhall find the Sine of 3 Deg. 45 Min. by the fame Theorem; and thus proceeding, till 12 Bifections being made, we come to the Sine of an Arch of 52′′ 44′′′′′ 03′′"" 45′ whose Co-Sine is nearly equal to Radius. In fuch a Cafe, 'tis evident by Theorem XXV, that Arches are proportional to their Sines; and therefore, H2 therefore, As the Arch of 52" 44" 03""" 45""""" is to an Arch of one Minute; So is the Sine of the Arch of 52" 44"" 03""" 45"""" to the Sine of an Arch of one Minute; which therefore is thus known. The Sine of an Arch of one Minute being found, we find, by Theorem XXI, its Co-Sine, or Sine of 89° 59', and the Sine of an Arch of two Minutes will become known, by Theorem XXIII. and its CoSine alfo (or Sine of 89° 58') as beforefaid. The Sines of the Arches of 1' and 2', as alfo their Co-Sines, viz the Sines of 89° 59′ and 89° 58′ being thus found; if, in the Scheme to Theo. XXVII. we make each of the Arches AB, BC, CD, &c. equal to 2'; then ÷AB=1'; and AC=2′; and AD=3'; &c. and let the Co-Sine of one Minute, viz. 89° 59' be called K; then the Sines of all the other Minutes of the Quadrant may be found by Corol. to the faid Theorem XXVII, and Corol. 1. to Theorem XXIII. Thus the Proportion will be; As R: 2K:: Sine of 2': Sine of i' + Sine of 3'; thus the Sine of 3' is found. Alfo R: 2K:: S. 3': S. 2'+S. 4'; hence the Sine of 4' is found. Again, As R: 2K:: S.4' : S. 3'+S. 5'; wherefore the Sine of 5' is thus known. Alfo, As R 2K :: S. 5′ S. 4′+S. 6'; and fo the Sine of 6' will be had, and in like manner will all the other Sines be obtain'd. But, because the Radius is to Double Co-Sine of an Arch of one Minute, as 1 is to 2, (for the Sine of 89° 59' is equal to Radius, to a Number of Figures exceeding the Tables) therefore the above Proportions will stand thus, As I 2 S. 2' : S. 1' + S. 3'; As1:2::S. 3': S. 2' + S. 4', &c. That is, if from the Double of any given Sine, be fubducted the Sine of the Minute next before, there will remain the Sine of the Minute next following. For |