Likeness, or Proportion to 4, As 4 doth to 2. viz. As 2 is to 4:: So is 4 to 8.

Lemma 1.

If three Numbers are proportional, the Rectangle or Product of the two Extreams; viz. of the first and laft Terms will be equal to the Square of the Mean or middle Term. (20 Eucl. 7.) As in thefe 2: 4:48. Here 8 x 216 the Product of the Extreams.

And 4 x 416 the Square of the Mean. Ergo 8 x 2 = 4 × 4.

Corol. 1.

Hence it follows, that if the Product of any two Numbers be equal to the Square of a third Number; thofe three Numbers will be in Proportion.

Lemma 2.

If four Numbers are proportional, the Product of the two Extreams will be equal to the Product of the two Means (19 Euclid 7.)

As in thefe, 2:4:8:16. Here 16x2=32.
And 8 x 4 = 32. Confequently 16 x 28 x 4.

Corol. 2.

From hence it follows, that if the Product of any two Numbers, be equal to the Product of any other two Numbers, those four Numbers are Proportionals.

And from these two Lemma's it will be eafy to conceive, that if never so many Numbers are in continued Proportion; the Pro duct of the two Extreams, will be equal to the Product of any two Means, that are equally diftant from the Extreams.

64. &c.

And if the Number of

As in these 2 4. 8. 16. 32 Here 64 x 232 × 4 = 16 × 8. &c. Terms be odd.

As in thefe 2. 4. 8. 16. 32 64. 128. &c.
Then 128 x 264×4=32x8 = 16 x 16.

Note, The Character made Use of to fignify continued Preportionals is.

In

In every Series of ÷ (viz. of continual Proportionals) that Number which is compared to another, is called the Antecedent of the Ratio; and that Number to which it is compared, is called it's Confequent.

As in thefe, 2: 4:48. Here 2 is the Antecedent, and 4 is the Confequent; and 4 the middle Term is an Antecedent to 8 it's Confequent: whence it follows, that in every Series of all the middle Terms between the first and laft are both Antecedents and Confequents.

For 2: 44: 8:: 8:16:: 16: 32:: 32: 64 &c. So that all the Terms except the laft are Antecedents. all the Terms except the first are Confequents.

Lemma 3.

16.32.

And

If never fo many Numbers are proportional, it will be: As any one of the Antecedents is to it's Confequent: So will the Sum of all the Antecedents be; to the Sum of all the Confequents. (12 Euclid 5.)

That is, in the foregoing Series. 2:4:2+4+8+16+32:4+8+16+32+64.

For it is evident, that 4+8+16+32+64 the Sum of all the Confequents, is double to 2 +4 +8 +16+ 32 the Sum of all the Antecedents; as 4 is to 2, according to the Ratio, and would have been Triple, or Quadruple, &c. had the Ratio been 3, or 4, &c.

Note, In every Series of the Ratio is found by dividing any of the Confequents by it's Antecedent.

As in these 2:6:: 6:18::18:54

54: 162. Here 2) 6 (3 the Ratio. Or 6) 18 (3 &c.

From the fecond and third Lemma's may be raised two general Theorems or Rules, for finding the Sum of any Series in without a continued Addition of all the Terms.

Let the Series 2 4 8 16. 32.64. 128. be given, to find it's Sum.

Theorem 1.

512-4

4-2

In Words at length thus,

From the Product of the fecond and laft Terms fubtract the Square of the first Term, and that Remainder being divided by the fecond Term less the firft, will give the Sum of all the Series.

Or if the firft Term, the common Ratio, and the laft Term be only given.

Theorem 2.

Then,

Multiply the last Term into the Ratio, and from their Product fubtract the first Term; divide that Remainder by the Ratio lefs Unity or 1, and it will give the Sum of all the Series.

For 4%

22=512—4. As above.

Confequently 2x-2=256-2 viz. the laft divided by z.

Let 2. 6. 18. 54. 162. 486. be the given Series. Here 2 is the firft Term, 3 is the Ratio, and 486 the laft Term.

But 486 x 31458. And 1458-21456.
Then 3-12) 1456 (728 the Sum required.
That is, 728=2+6+18+34+162+486.

Since in either of thefe Theorems it is required to have the laft, Term known, (the which in a long Series of will be very tedious to come at by a continued Multiplication) it will therefore be convenient to fhew how to obtain either the laft Term or any other Term, whofe Place is affigned, without producing all the Terms.

In order to that, it will be neceffary to premife the Coherence or Similitude that is betwixt Numbers in Arithmetical Progreffion and thofe in Geometrical Proportion.

If to any Series of Numbers in when the firft Term is not an Unit or 1, there be affigned a Series of Numbers in Arithmetical Progreffion, beginning with an Unit or 1, and whofe common Difference is 1. called Indices or Exponents.

Then will the Addition or Subtraction of any two of those Indices (or Numbers in Arithmetical Progreffion) directly correfpond with the Product, or Quotient of their refpective Terms in the Series of.

That is, {

As 3+4=7

So 8 x16128 the feventh Term in ÷÷÷
As 6+4=10.

Again, So 64x16=1024. the tenth Term in

As

-

Or, 250 128316. Or, {As 6-24 }

So

So 64÷4=16

} &c.

But if the Series of begin with an Unit, the Indices muft begin with a Cypher.

Now by the help of the Indices, and a few of the firft Terms in any Series of, it is plain that any Term whofe Place or Diftance from the firft Term is affigned, may be speedily obtained without producing the whole Series.

EXAMPLE 1.

A Man bought a Horfe, and was to give a Farthing for the first Nail, two for the fecond, four for the third, &c. in, the Number of Nails was to be 7 in every Shoe, viz. 28 Nails in all. What must he have paid for the Horse?

Which is here to be accounted the 28th and laft Term. Because the first Term in the Series is 1, which doth neither multiply nor divide.

Now this 134217728 being the Number of Farthings to be paid for the last Nail, by it the common Ratio which is 2, and the firft Term which is 1, may be found the Sum of all the Series, per Theorem 2.

Viz.

134217728

2

268435456 From this Product fubtract 1. 268435456-1-268435455. Then 2-1-1 the Divifor. Confequently 268435 455 is the Sum of all the Series, or Price of the Horfe in Farthings, which being brought into Pounds, (See page 46) will be 279620 l. 5 s. 3 d. 3 grs.

EXAMPLE 2.

A cunning Servant agreed with a Master (unskilled in Numbers) to ferve him Eleven Years without any other Reward for his Service but the Produce of one Wheat Corn for the first Year; and that Product to be fowed the fecond Year, and fo on from Year to Year until the End of the Time, allowing the Increase to be but in a ten-fold Proportion.

It is required to find the Sum of the whole Produce.

I

2

Firft

3 4

.5

Indices or Years. IO. 100 1000. 10000. 100000 Wheat Corns in÷÷

Then So 1000 x 100 = 1000000. the 6th Year's Produce

And {

6+5=11

1000000 x 100000 1000oooooooo. The eleventh of laft Years Produce.

Then (either by Theorem 1, or 2) the Sum of all the Series will be 111111111 Corns. Now it may be computed from Pages 31 and 34, that 685 Wheat Corns, round and dry out of the middle of the Ear, will a Statute Pint. If fo,

Then 7680) 110 (14467592 Pints, but 64 Pints are contained in a Bufhel.

Therefore 64) 14467592 (226056 Bufhels. be fold for 3 Shillings the Bufhet;"

Then

226056

3

Shillings 67816833908 7. 8. 41 d. Recompence for Eleven Years Service

Suppofe it to

A very good

There are feveral pretty Questions refolved by Numbers in Arithmetical Progreffion; and by thofe in, which the ingenious Learner will eafily perceive hereafter; viz. When we come to the Solution of Queftions relating to Intereft and Annuities, c