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· 20,460254766 + 21,114812 + 185,252368710 — 382,357838 + 197,29514e + 25,475889852 - 78,87272e + 81,596022

+ 0,881647759 - 3,639410 + 5,6337ee Viz. 191,149651515 — 443,755154 + 284,5248ee

= 190,836 110259 Hence it will be 443,755151 - 284,5248ee =0,313541256 And 1,55963e - ee =,0011019821 =D

Then

tn { 1,55963– Operation. 1,55963) 0,001101982i (0,0007068=

me= 0,00070 109123 1. Divifor 1,5589

1075210 2. Divisor 1,55893

935358
1398520

1247144 &c. Last r= 0,969

-e = 0,0007068 rce=a=0,9682932 the Sine of 75o . 32. as was required.

Having found the Sine and Co-fine of any Arch, the Tangent is usually found by this Proportion ; v: S As the Cofine of any Arch: is to the Sine of that Arch :: fois

W? the Radius: to the Tangent of the fame Arch.

For fuppofing BC=BD Radius, AC the Sine of the Arch CD. Then B A is the Co-fine, and FD the Tangent of the fame Arch. But B A: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 .

10. B either the Sine or the Co-fine be given, B

A D the other may be found, thus, V OBCOCA=B AOr V OBC-a BA=CA. But, if either the Sine or Co-fine be given, the Tangent may (I presume) be more easily found by the following

Theorems.

Let

Let BC=1. CAES. BA=x and FD=T. Then, if S be given, I may be found by this

Theorem { vis s=?
Or if x be given, T may be found by this

hcorem {v - =1

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AN

INTRODUCTION

Mathematicks.

TO THE

PART IV.

W

CH A P. I.
Definitions of a Tona,' and its mcdions.
H ERE are several 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, (uiz, one of those 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 less, 'tis an Obruse-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 Base of a Cone is the “ Circle, which is describ'd by the Right Line mov'd about." (Defin. 18, 19, 20. Euclid. 11.)

Sir Jonas Moor, in his Treatise of Conical Sections (taken out of the Works of Mydorpius) defines it thus:

" If a Line of such a Length as shall be needful Thall, upon a " Point fix'u above the Plain of a Circle, so move about the Cir« cle, until it return to the point from whence the Motion be“ gan, the Superficies that is made by such a Line is callid a Coc so nical Superficies; and the folid Figure contain’d within that Su" perficies and the Circle is calld a Cone. The Point remaining ftill is the Vertex of the Cone, &c."

A 2a

Alcho

Altho' both these Definitions are equally true, and, with a little Consideration, may be pretty easily understood; yet I shall here propose one very different from either of them; and, as I presume, more plain and intelligible, especially to a Learner.

If a Circle describ'd upon stiff Paper (or any other pliable Mate ter) of what Bigness you please, be cut into two, three, or more Sectors, either equal or unequal, and one of those Sectors be so rolld 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 so rollid 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 V will become a Solid Point, call’d the VERTEX of the Cone; the Radius V H, being every where equal, will be the side of the Cone, and the Arch HG will become a Circle, whose Area is callid the Cone's Base.

A Right Line being suppos'd to pafs From the Vertex, or Point V, to the Center of the Cone's Base, as at C, that Line (viz. V C) will be the AXIS, or perpendicular Height of the Cone.

If a Solid be exactly made in such a Form, it will be a compleat or perfect Cone; which I shall all-along call a Right Cone, because its Azis VC stands at Right Angles with the Plain of its Base HG, and its Sides are every where equal.

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Cone, which I haja compleade

Any Cone, whose Axis is not at Right Angles with the Plain of its Bale, may be properly callid an imperfect Cone, because its Sides are not every where equal (as in the annexed Figure.) Now, such an imperfect Cone is usual. ly callid a Scalene, or Oblique Cone.

• Any solid Cone may be cut by Plains (which I fall all-along hereafter call Right Lines) into five Sections,

Sca. Sect. 1. Ifa Right Cone be cut directly thro' its Axis, the Plain or Superficies of that Section will be a plain Iloceles Triangle, as HVG Fig. 2, viz. the Sides (H V and VG) of the Cone will be the Sides of the Triangle, the Diameter (HG) of the Cone's Base will be the Base of the Triangle, and (VC) its Axis will be the perpendicular Height of the Triangle,

Seft. 2. If a Right Cone be cut (any where) off by a Right-line parala lel to is Base, as (it will be easy to conceive, that) che Plain of that Section will be a Circle, because the Cone's Base is such : wherein one thing ought to be clearly understood, which may be laid down as a Lemma, to demonstrate the Properties of the following Sections.

r If any two Right Lines, inscrib'd within a Circle, do

cut or cross each other (as hg doch bb in the annexed

į Figure) the Rectangle made of ihe 'Segments of one ad. of the Lines will be equai to the Roelangle made of

the Segments of the other Line. (See Theorem 15. | Page 315.)

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Seet. 3:

If a Right Cone be (any where) cut off by a Right Line that cuts both its Sides, but not parallel to its Base (as TS in the following Figure) the Plain of that Section will be an Ellipsis (vulgarly called an Oval) viz. an oblong or imperfect Circle, which hath several 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 portes are particularly distinguish'd from the rest, as being of most general Ule; the other are only applicable to particular Cases.

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2. The

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