134217728

2

268435456 From this Product fubtract 1.

Viz. 268435456—1=268435455. Then 2-11 the Divifor. Confequently 268435455 is the Sum of all the Series, or Price of the Horfe in Farthings, which being brought into Pounds, (See page 46) will be 2796201. 5s. 3 d. 3 qrs.

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 second 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.

First {

Then

ΙΟ 100.1000. 10000. 100000
As 4+2=6

Indices or Years. Wheat Corns in

{So 10000 x 100 1000000 the 6th Year's Produce.

And {1000000 x 100000 1000c0000000 the eleventh or

laft Year's Produce.

Then (either by Theorem 1. or 2) the Sum of all the Series will be 11111110 Corns. Now it may be computed from Pages 31 and 34, that 7680 Wheat Corns, round and dry out of the middle of the Ear, will fill a Statute Pint. If so,

Then 7680) 111111111110 (14467592 Pints, but 64 Pints are contained in a Bushel.

Therefore 64) 14467592 (226056 Bufhels. Suppose it to be fold for 3 Shillings the Bufhel;

Then { 226056+

3

Shillings 678168 339081. 8 s. 4 d. A very good Recompence for Eleven Years Service.

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

There is alfo a third Kind of Proportion, called Mufical, which being but of little or no common Ufe, I fhall therefore give but a fhort Account of it.

Mufical Proportion or Habitude is, when of three Numbers, the first hath the fame Proportion to the third, as the Difference between the firft and second hath to the Difference between the fecond and third.

As in thefe, 6. 8. 12 viz. 6: 12:: 8-6: 12-8 If there are four Numbers in Mufical Proportion; The first will have the fame Proportion to the fourth, as the Difference between the first and fecond hath to the Difference between the third and fourth.

The Method of finding out Numbers in Mufical Proportion, is beft expreffed by Letters; as fhall be fhewed in the Algebraick Part.

Sect. 3. How to Change or Vary the Order of Things, &c.

THI

HIS being a Thing not treated of in any common Books of Arithmetick, (that I have had the Opportunity of perufing) made me think it would be acceptable to the young Learner, to know how oft it is poffible to vary or change the Order or Position of any propofed Number of Things.

As how many feveral Changes may be rung upon any propofed Number of Bells; or how many feveral Variations may be made of any determined Number of Letters, or any other Things propofed to be varied.

The Method of finding out the Number of Changes is by a continual Multiplication of all the Terms in a Series of Arithmetical Progrefbons, whofe first Term and common Difference is Unity or 1. And the loft Term the Number of Things propofed to be varied, viz. 1 × 2 × 3 × 4 × 5x6x7, &c. As will appear from what follows.

1. If the Things propofed to be varied are only two, they admit of a double Pofition (as to Order of Place) and no more.

And if three Things are propofed to be varied, they may

be changed fix feveral Ways (as to their Order of Place) and no

more.

For, beginning with I, there will be 1.2.3

Next, beginning with 2, there will be

1 3 2

2

22

I •

3

3 I

3.1.2 7 3 2 I ×2×3=6.

Again, beginning with 3, it will be Which in all make 6 or 3 Times 2, viz. Suppofe four Things are propofed to be varied; Then they will admit of 24 feveral Changes, as to their Order of different Places,

For beginning the Order with 1, it will be
Here are fix different Changes.

=

3

And for the fame Reafon there will be 6 different Changes, when 2 begins the Order, and as many when 3 and 4 begins the Order; which in all is 24 1 x 2 x 3x 4. And by this Method of proceeding, it may be made evident, that 5 Things admit of 120 feveral Variations or Changes; and 6 Things of 720, &c. As in this following Table.

The Number The Manner how The different Changes or Vari

These may be thus continued on to any affigned Number. Suppofe to 24 the Number of Letters in the Alphabet, which will admit of 620448401733239439360000 feveral Variations. From thefe Computations may be started feveral pretty, and indeed, very ftrange, Questions.

EXAMPLES.

Six Gentlemen, that were travelling, met together by Chance at a certain Inn upon the Road, where they were so pleased with their Hoft, and each other's Company, that in a Frolick they made a Contract to stay at that Place, fo long as they, together with their Hoft, could fit every Day in a different Order or Pofition at Dinner; which by the foregoing Computations will be found near 14 Years. For they being made 7 with their Hoft, will admit of 5040 different Pofitions; but 5040 being divided by 365 (the Number of the Days in one Year) will give 13 Years and 291 Days. A very pretty Frolick indeed.

I have been told, that before the Fire of London (which happened Anno 1666) there were 12 Bells in St Mary Le Bow's Church in Cheapfide, London. Suppofe it were required to tell how many feveral Changes might have been rung upon thofe 12 Bells; and at a moderate Computation how long all thofe Changes would have been ringing but once over.

First, 1×2×3×4×5×6×7×8×9×10x11x12=479001600, the Number of Changes.

Then fuppofing there might be rung 10 Changes in one Minute: viz. 12 x 10 = 120 Strokes in a Minute, which is 2 Strokes in a Second of Time: Now according to that Rate there must be allowed 47900160 Minutes to ring them once over in all their different Changes; viz. 10) 479001600 (47900160.

In one Year there is 365 Days, 5 Hours, and 49 Minutes; which, being reduced into Minutes, is 525949.

Then 525949) 47900160 (91 Years and 26 Days.

So long would thofe 12 Bells have been continually ringing without any Intermiffion, before all their different Changes could have been truly rung but once over. It is ftrange, and feems almoft incredible, that a few Things fhould produce fuch Varieties.

But that which feems yet more ftrange and furprizing (yea, even impoffible to those who are not verfed in the Power of Numbers)

is, that if two Bells more had been added to the aforesaid 12 they would have advanced the Number of Changes (and confequently the Time) beyond common Belief. For 14 Bells would require (at the fame Rate of ringing as before) about 16575 Years to ring all their different Changes but once over.

And if it were poffible to ring 24 Bells in Changes (and at the fame rate of 10 Changes in a Minute, which is 2 Strokes in one Second) they would require more than 117000000000000000 Years to ring them but once over in all their different Changes; as may easily be computed from the precedent Table.

CHAP. VII.

of Propoztion Disjunct; commonly called the Golden Rule.

PRoportion Disjunct, or the Golden Rule, is either Direct or Reciprocal, called Inverfe. And thofe are both Simple and Compound,

SECT. I.

Irect Proportion is, when of four Numbers, the first bears the fame Ratio or Proportion to the fecond; as the third doth to the fourth.

As in these 28: 6:24.

Confequently, the greater the fecond Term is, in refpect to the firft; the greater will the fourth Term be, in respect to the third. That is, as 8 the fecond Term is 4 Times greater than 2 the firft Term: So is 24 the fourth Term, 4 Times greater than 6 the third Term.

Whence it follows, that if four Numbers are in Direct Proportion, the Product of the two Extreams will always be equal to the Product of the two Means, as well in Disjunct as in continued Proportion; according to Lemma 2. page 77.

For As 2: 2 x 46: 6 x 4. Or As 3: 3×5:: 6:6 x 5.
But 2 x 6x42x4x6. Or 3x6x5=3×5x6.

That is, the Product of the Extreams is equal to that of the
Means.