But, if the given Arch be greater than 45 Degrees, you must take its Complement to 90°. viz. fubtract it from 90 Degrees, and reduce the Remainder into Minutes, as before. Then multiply the Square of these Minutes, into this conftant Multiplicator, 0,000000084616 calling their Product p, and putting a = the Sine fought, as before. Then will a++ 28a3 + 195aa + 36 paa +108pa-28a= 196 — 81p Example. Suppofe it were required to find the Sine of 75°. 32. or (which is the fame Thing) to find the Co-fine of 14°. 28. =868', whofe Square 753424 X 0,000000084616 = 0,06375172518 = p. Hence the Equation in Numbers will be aaaa + 28aaa + 197, 295062aa 21,114814a 190,8361102588. - Note, I here take r = I because the Arch is so near to 90°, and therefore I make it r — e = a. Viz. 205,1808-461,48e+287,29ee190,8361 D Theorem {1,606—e Operation. 1,606).,049930 (0,031 = e — e = 0,031 47I for a fecond Operation; which, being involv'd as before, will pro duce thefe following Numbers. 190,836110259 Hence it will be 443,75515e-284,5248ee0,313541256 rea=0,9682932 the Sine of 75°. 32. as was required. Having found the Sine and Co-fine of any Arch, the Tangent is ufually found by this Proportion; Viz. { As the Cofine of any Arch: is to the Sine of that Arch : : so is the Radius: to the Tangent of the fame Arch. For fuppofing BC= BD Radius, AC the Sine of the Arch CD. Then BA is the Co-fine, and FD the Tangent of the fame Arch. But BA: CA:: BD: FD, &c. Now by this Proportion there is required to be given both the Sine and Co-fine of the Arch, to find the Tangent. 'Tis true, if the Radius, and either the Sine or the Co-fine be given, B A D Or the other may be found, thus, ✔BC-CA-BA. OBC-BA=CA. But, if either the Sine or Co-fine be given, the Tangent may (I prefume) be more eafily found by the following Theorems. Let Let BC 1. CAS. BA=x and FD=T. Then, if S be given, T may be found by this Theorem {√ SS =T Let the Sine of 90°. 13. (before found) be given, viz. 0,3291415 =S, to find the Tangent of the fame Arch. Firft 0,3291415X T ,32914150,108334127 SS. Again 1-0,108334127 =0,891665873 ISS. Then 0,891665873) 0,108334127 (0,1214963253 and ✔ 0,1214963253 = 0,3485632 = T, the Tangent, of 19° 13'. As was required. And fo you may proceed to find T= the Tangent, when x the Co-fine is given. = Perhaps it may here be expected, that I should have fhew'd and demonftrated (or at least have inferted) the Proportions from whence the foregoing Equations for making Sines were produced; but I have omitted that, as alfo their Ufe in computing the Sides and Angles of plain Triangles by the Pen only (viz.without the Help of Tables) for the Subject of my Difcourfe hereafter, if Health and Time permit. In the mean Time, what is here done may fuffice to fhew, that the making of Sines by fuch a laborious and operofe Way, as was. formerly used, is in a great Measure overcome; which, I think, I may juftly claim as my own. AN AN INTRODUCTION T TO THE Mathematicks. PART IV. CHA P. I. Definitions of a Cone, and its Scdions. HERE are feveral Definitions given of a Cone: The "A Cone (faith ke) is a Figure made when one Side of "a Rectangled Triangle, (viz. one of thofe Sides that contain the "Right Angle) remaining fix'd, the Triangle is turn'd round * about, 'till it return to the Place from whence it first mov'd: "And if the fix'd Right Line be equal to the other which con"taineth the Right Angle, then the Cone is a Rectangled Cone; "but if it be lefs, 'tis an Obtufe angled Cone; if greater, an "Acute angled Cone. The Axis of a Cone is that fix'd Line "about which the Triangle is mov'd: The Bafe of a Cone is the "Circle, which is defcrib'd by the Right Line mov'd about." (Defin. 18, 19, 20. Euclid. 11.) Sir Jonas Moor, in his Treatife of Conical Sections (taken out of the Works of Mydorgius) defines it thus: "If a Line of fuch a Length as fhall be needful fhall, upon a "Point fix'd above the Plain of a Circle, fo move about the Cir"cle, until it return to the Point from whence the Motion began, the Superficies that is made by fuch a Line is call'd a Co"nical Superficies; and the folid Figure contain❜d within that Superficies and the Circle is call'd a Cone. The Point remaining "ftill is the Vertex of the Cone, &c." 66 A a a Altho Altho' both thefe Definitions are equally true, and, with a little Confideration, may be pretty easily understood; yet I shall here propofe one very different from either of them; and, as I prefume, more plain and intelligible, especially to a Learner. If a Circle defcrib'd upon ftiff Paper (or any other pliable Matter) of what Bignefs you pleafe, be cut into two, three, or more Sectors, either equal or unequal, and one of those Sectors be fo roll'd up, as that the Radii may exactly meet each other, it will form a Conical Superficies. That is, if the Sector HVG be cut out of the Circle, and fo roll'd up as that the Radii V H and VG may juft meet each other in all their Parts, it will form a Cone, and the Center will become a Solid Point, call'd the VERTEX of the Cone; the Radius VH, being every where equal, will be the Side of the Cone, and the Arch HG will become a Circle, whofe Area is call'd the Cone's Bafe. H A Right Line being fuppos'd to país from the Vertex, or Point V, to the Genter of the Cone's Bafe, as at C, that Line (viz. VC) will be the AXIS, or perpendicular Height of the Cone. If a Solid be exactly made in fuch a Form, it will be a compleat or perfect Cone; which I fhall all-along call a Right Cone, because its Axis VC ftands at Right Angles with the Plain of its Bafe HG, and its Sides are every where equal. Any Cone, whofe Axis is not at Right Angles with the Plain of its Bafe, may be properly call'd an imperfect Cone, becaufe its Sides are not every where equal (as in the annexed Figure.) Now, fuch an imperfect Cone is ufually call'd a Scalene, or Oblique Cone. H ν Any folid Cone may be cut by Plains (which I shall all-along hereafter call Right Lines) into five Sections. 1 |