Sect. 4. Of Purchasing Free-hold, or Real Estates,

at Compound Interest.

All Free hold or Real Estates, are supposed to be purchased or bought to continue for ever (viz. without any limited Time); therefore the Business of computing the true Value of fuch Estates is grounded upon a Rank or Series of Geometrical Proportionals continually decreasing, ad Infinitum.

Thus, let P, u, R, denote the fame Data as in the last Section.

น

Then the Series will be, R, RR' R' R' RS

and so on

in untill the laft Term = 0. Then will P-0 (viz. P) be the Sum of all the Antecedents. And Pwill be the Sum of all R the Confequents; therefore it will be u: :: P: P- which produces PR-u=P. This Æquation affords the following Theorems.

R

= P.

Example. Suppose a Free-hold Estate of 75 1. Yearly Rent were to be fold; what is it worth, allowing the Buyer 6 per Cent. &c. Compound Interest for his Money?

In this Question there is given u = 75. R = 1,06 to find P. Per Theorem 2. Thus R-1 = 0,06) 75 = u (1250 1.= P. the Answer required. And so on for any of the rest, as Occafion requires. But if the Rent is to be paid, either by Quarterly, or Half Yearly Payments;

Then R = 1,06 for Half Yearly
And R✓ ✓ 1,06 for Quarterly

Or

1,08

{

R = √ 1,08

for Yearly

7

Payments at 6 per Cent.

for Half Yearly Payments at 8 per Cent.

R = √ √ 1,08 for Quarterly

The like is to be understood for any other proposed Rate of In

tereft, either greater or less than 6 per Cent.

The Application of these Theorems to Practice is so very easy,

that it's needless to infert more Examples.

AN

Of Geometrical Definitions, &c.

Sect. 1. Of Lines and Angles.

A

POINT hath no Parts: That is, a Geometrical Point is
not any Quantity, but only an affignable Place in any
Quantity, denoted by a Point:
As at A. and B.

} 1.

B.

Such a Place may be conceived so infinitely small, as to be void of Length, Breadth, and Thickness; and therefore a Point may be faid to have no Parts.

2. A LINE is called a Quantity of one Dimension, because it may have any supposed Length, but no Breadth nor Thickness, being made or represented to the Eye, by the Motion of a Point.

That is, if the Point at A, be moved (upon the fame Plane) to the Point at B, it will describe a Line either right or circular (viz. trooked) according to its Motion.

Therefore the Ends or Limits of a Line are Points.

3. A RIGHT LINE, is that Line which lieth even or ftreight betwixt those Points that limit its Length, being the shortest Line

that can be drawn between any Two Points. As the Line AB.

}

AB.

Therefore, between any two Points, there can lie or be drawn but one

right Line.

4. A CIRCULAR, crooked or OBLIQUE Line, is that which

lies bending between those Points

which limit its Length, as the Lines

CD or FG, &c.

C

Of these Kinds of Lines there are

various Sorts; but those of the Circle,

F

Parabola, Ellipsis, and Hyperbola

D

G

are of most general Use in Geometry; of which a particular Account

shall be given further on.

5. PARALLEL LINES, are those that lie equally distant from one another in all their Parts, viz. such Lines as being infinitely extended (upon the fame Plane) will never meet: As the Lines A Bandab: or CD and cd.

6. LINES not PARALLEL, but INCLINING (viz. leaning) one towards another, whether they are Right Lines, or Circular Lines, will (if they are extended) meet, and make an Angle; the Point where they meet is called the Angular Point, as at A. And according as such Lines stand, nearer or further off each other, the

d

B

A

C

d

B

whether the Lines that include the Angle be long or short. That is, the

C

Lines Ad, and Af include the same Angle as A B, and AC doth; notwithstanding that AB is longer than Ad, &c.

7. All ANGLES included between Right Lines are called Rightlin'd Angles; and those included between Circular Lines are called Spherical Angles. But all Angles, whether Right-lin'd or Spherical, fall under one of these Three Denominations.

8. A RIGHT ANGLE is that which is included betwixt Two Lines, that meet one another Perpendicularly.

That

Perpendicular to each other.

D, C is to either or both of them.

Angle. Such is the Angle inclu

That is, AC, and CB, are Perpendicular to DC, as well as

9. An OBTUSE ANGLE is that which is greater than a Right

ded between the Lines AC and

CB.

10. An ACUTE ANGLE is that

A

B

D

C

which is less than a Right Angle:
As the Angle included between the Lines CB and CD.

These Two Angles are generally called OBLIQUE Angles.

Sect. 2. Of a Circle, &c.

Before a Circle and its Parts are defined, it will be convenient te give a brief Account of Superficies in general.

1. A SUPERFICIESOS SURFACE is the Upper, or very Out-fide of any visible Thing. But by Superficies in GEOMETRY, is meant only so much of the Out-fide of any Thing as is inclosed within a Line or Lines, according to the Form or Figure of the Thing designed; and it is produced or formed by the Motion of a Line, as a Line is described by the Motion of a Point; thus:

Suppose the Line A B were equally

moved (upon the same Plane) to C D; then will the Points at A and B defcribe the Two Lines AC and

BD; and by so doing they will

C

D

form (and inclose) the SUPERFI

CIES or Figure A BCD, being a Quantity of Two Dimensions, viz: it hath Length and Breadth, but not Thickness. Confequently the

Bounds or Limits of a Superficies are Lines.

Note,

Note, The Superficies of any Figure, is usually called its AREA.

2. A CIRCLE is a plain regular Figure, whose Area is bounded or limited by one continued Line, called the CIRCUMFERENCE or PERIPHERY of the Circle, which may be thus described or drawn.

Suppose a Right Line, as CB, to have one of its Extream Points, as C, fo fix'd upon any Plane, as that the other Point at B may move about it; then if the Point at B be moved round about (upon the fame Plane) it will defcribe a

B

Line equally difiant in all its Parts from the Point C, which will be the Circumference or Periphery of that Circle; the Point C, will be its CENTER, and the contained Space will be its Area, and the Right Line CB, by which the Circle is

C

thus described, is called RADIUS.

Confectary.

From hence'tis evident, that an infinite Number of Right Lines may be drawn from the Center of any Circle to touch its Periphery, which will be all equal to one another, because they are all Radius's.

And with a little Confideration it will be easy to conceive, that no more than two equal Right Lines can be drawn from any Point within a Circle to touch its Periphery, but from the Center only. (9. e. 3.)

3. EQUAL CIRCLES are those which have equal Radius's; for it's plain by the last Definition, that one and the fame Radius (as CB) muft needs describe equal Circles, how many foever they

5. A Semicircle (viz. Half a Circle) is a Figure included between the Diameter, and Half the Periphery cut off by the Diameter; as ADB,