Now, in order to fhew the Ufe of the foregoing Table by an Example, let it be required to find the Probability that a Perfon of 36 lives 30 Years longer, or attains to the Age of 66 Years: Lock in the Table against 36 Years and 66 Years, and correfponding thereto, you will find the Numbers 331 and 93 refpectively; fhewing, that out of 331 Perfons living of 36 Years of Age, only 93 of them arrive to the Age of 66: Therefore, feeing the whole Number of Perfons living at the Beginning of this Term, is to the Number remaining alive at the End of it, in the Ratio of 331 to 93; the Number of Chances that a Perfon of 36 Years of Age has to live 30 Years longer, will be to the Number of all the Chances that he has both to live beyond, and die within 30 Years, in the fame Ratio of 331 to 93; and therefore is the Measure of the Probability required; the Probability of the Happening of any Event being always to be confidered as the Ratio of the Chances which that Event has to happen, to all the Chances which it has both to happen and fail.

This being understood, fuppofe it were now required to find the Value of an Annuity of 1001. for a Life of 20 Years of Age, Interest at 4 per Cent. Because the present Value of rool. due at the End of one Year (Discount being allowed) is 96.15, it is plain, that so much would be the Value of the first Year's Rent, was the Purchaser fure to receive it; but the Probability of his living one Year appearing from the Table to be only, the aforefaid Sum 96.15, in order to make a just Deduction out of it for the Contingency of his dying before the End of one Year, ought to be diminished in the Ratio of 462 to 455, or multiplied by, which will reduce it to 947, equal to the true Value of the first Year's Rent. After the fame Manner may the Value of the fecond Year's Rent be calculated; for, fince the Probability of receiving this Rent, or of living two Years, is 44, let this be multiplied into 92.45, the prefent Value of 100l. to be received at the End of two Years, and the Product 89.65 will be the true Value of the fecond Year's Rent.

And, by a like way of proceeding, the Values of the third, fourth, fifth, &c. Year's Rent, to the utmost Extent of Life, may be determined; and the Sum of all these will be the required Value of the Annuity; which will be found to come out 14801. very near.

Those who are defirous of feeing these Investigations extended to finding the Value of two Lives, and then of three Lives, &c. may confult the late Mr. Thomas Simpfon's Treatife upon

this Subject; from whence the foregoing-Table, and Explication thereof, was tranfcribed.

Set. 4. Of Purchasing Free-hold, or Real Estates, at Compound Intereft.

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

Thus, let P, u, R, denote the fame Data as in the last Section, and fo on

u

น

u

u

R' RR' R3' R4' Rs'

Then the Series will be,
in until the laft Termo.
Sum of all the Antecedents.

Then will P-o (viz. P) be the

น

And P

will be the Sum of all

R

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

=

=

In this Question there is given u 75. R= 1,06 to find P. Ter Theorem 2. Thus R1 0,06) 75 = u(1250. P. the Anjwer required. And fo on for any of the reft, as Occafion requires. But if the Rent is to be paid, either by Quarterly, or Half yearly Payments ;

Then R
And R= √√1,06 for Quarterly
SR==
1,08 for Yearly

1,06 for Half-yearly Payments at 6
}

Or

{{

per

Cent.

for Half-yearly Payments at 8 per Cent.

R = √1,08
R=1,08 for Quarterly

The like is to be underflood for any other proposed Rate of Intereft, either greater or less than 6 per Cent.

The Application of thefe Theorems to Practice is fo very eafy, that it's needlefs to infert more Examples.

AN

A N

INTRODUCTION

TO THE

Mathematicks.

A

PART III.

CHAP. I.

Of Geometrical Definitions, &c.

Sect. 1. Of Lines and Angles.

POINT hath no Parts: That is, a Mathematical Point is not any Quantity, but only an affignable Place in any Quantity, denoted by a Point: 1.

As at A. and B.

B.

Such a Place may be conceived so infinitely fmall, 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 Dimenfion, because it may have any fuppofed Length, but no Breadth nor Thickness, being made or reprefented 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 defcribe a Line either freight or crocked 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 freight betwixt thofe Points that limit its Length, being the forteft Line that can be drawn between any Two

Points. As the Line AB.

"}

A

B.

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.

Of thefe Kinds of Lines there are various Sorts; but thofe of the Circle, Parabola, Ellipfis, and Hyperbla

C

F

D

G

are of most general Ufe in Geometry; of which a particular Account fhall be given further on.

5. PARALLEL LINES, or thofe that lie equally diftant from one another in all their Parts, viz. fuch Lines as being infinitely extended (upon the fame Plane) will never meet: As the Lines AB and ab or CD and cd.

d

4

B

d

B

6 LINES not PARALLEL, but INCLINING (viz. leaning) one towards another, whether they are Kight Lines, or Circular Lines, will (if they are extended) meet, and make an Angle; the Point where they meet is called the Angular Feint, as at A. And according as fuch Lines fland nearer or further off each other, the Angle is faid to be lef. or greater, whether the Lines that include the Angle be long or fhort. That is, the

C

d

B

A

C

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

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

S. A RIGHT ANGLE is that which is included betwixt Tw Lines, that meet one another Perpendicularly,

That

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

Angle. Such is the Angle inclu

ded between the Lines AC and

CB.

A

C

10. An ACUTE ANGLE is that which is less than a Right Angle: As the Angle included between the Lines CB and CD.

Thefe Two Angles are generally called OBLIQUE Angles.

Sect. 2. Of a Circle, &c.

B

-D

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

1. A SUPERFICIES or SURFACE is the Upper, or very Out-fide of any visible Thing. But by Superficies in GEOMETRY, is meant only fo 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 defigned; and it is produced or formed by the Motion of a Line, as a Line is defcribed by the Motion of a Point; thus:

Suppofe the Line A B were equally' moved (upon the fame Plane) to C D; then will the Points at A and B defcribe the Two Lines AC and BD; and by fo doing they will form (and inclofe) the SUPERFI

B

D

CIES or Figure ABCD, 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,

2

Note,