20,460254766 +21,11481e+185,252368710

382,35783+ 197,2951ee +25,47588985278,87272e+ 81,5960ee + 0,881647759- 3,63941e+

Viz. 191,149651515

=190,836110259

5,6337ee

Hence it will be 443,75515e-284,5248ee =0,313541256
And 1,55963e-ee-,0011019821 D

r-e=a=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. {the Radius: to the Tangent of the fame Arch.

As the Cofine of any Arch is to the Sine of that Arch:: fo is

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, the other may be found, thus, VBC-CA-BA. Or VBC-BA=CA. But, if either the Sine or Co-fine be given, the Tangent may (I prefume) be more easily found by the following Theorems.

B

A

D

Let

Let BC

1. CA=S. BA=x and FD=T. Then, if

S be given, T may be found by this

xx

Let the Sine of 90°. 13. (before found) be given, viz. 0,3291415 =S, to find T the Tangent of the fame Arch. First 0,3291415 ,3291415=0,108334127 SS. Again 10,108334127 =0,8916658731-SS. Then 0,891665873) 0,108334127 (0,1214963253 and 0,12149632530,3485632 = 7, 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 fhould 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 ufed, is in a great Measure overcome; which, I think, I may justly claim as my own.

AN

ΑΝ

INTRODUCTION

T

TO THE

Mathematicks.

PART IV.

CHA P. I.

Definitions of a Tone, and its cations.

HERE are feveral Definitions given of a Cone: The
Learned Dr. Barrow, upon Euclid, hath it thus:

"A Cone (faith he) 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 shall, 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 be

gan, the Superficies that is made by fuch a Line is call'd a Co"nical Superficies; and the folid Figure contain'd within that Su"perficies and the Circle is call'd a Cone. The Point remaining till is the Vertex of the Cone, &'c."

Altho' both thefe Definitions are equally true, and, with a little Confideration, may be pretty easily understood; yet I shall here propose 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 please, 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 just 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 pafs from the Vertex, or Point V, to the Center 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, because 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

V

G

Any folid Cone may be cut by Plains (which I shall all-along hereafter call Right Lines) into five Sections.

Sect. I.

If a Right Cone be cut directly thro' its Axis, the Plain or Superficies of that Section will be a plain Ifofceles Triangle, as HVG Fig. 2, viz. the Sides (HV and VG) of the Cone will be the Sides of the Triangle, the Diameter (HG) of the Cone's Bafe will be the Bafe of the Triangle, and (VC) its Axis will be the perpendicular Height of the Triangle.

Sect. 2.

If a Right Cone be cut (any where) off by a Right-line parallel to is Bafe, as hg (it will be eafy to conceive, that) the Plain of that Section will be a Circle, because the Cone's Bafe is fuch: wherein one Thing ought to be clearly understood, which may be laid down as a Lemma, to demonftrate the Properties of the following Sections.

Lemma.

If any two Right Lines, infcrib'd within a Circle, do cut or cross each other (as hg doth bb in the annexed Figure) the Rectangle made of the Segments of one of the Lines will be equal to the Rectangle made of the Segments of the other Line. (See Theorem 15. Page 315.)

V

B

H

Sect. 3.

B

If a Right Cone be (any where) cut off by a Right Line that cuts both its Sides, but not parallel to its Bafe (as TS in the following Figure) the Plain of that Section will be an Ellipfis (vulgarly called an Oval) viz. an oblong or imperfect Circle, which hath feveral Diameters, and two particular Centers. That is,

1. Any Right Line that divides an Ellipfis into two equal Parts is call'd a Diameter; amongst which the longest and the forteft are particularly diftinguifh'd from the reft, as being of most general Ule; the other are only applicable to particular Cafes.