There is also a third Kind of Proportion, called Musical, which being but of little or no common Use, I shall therefore give but a short Account of it.

Musical Proportion or Habitude is, when of three Numbers; the first hath the same Proportion to the third, as the Difference between the first and second hath to the Difference between the second and third.

As in these, 6.8. 12. viz. 6:12:: 8-6:12-8 If there are four Numbers in Musical Proportion; The first will have the same Proportion to the fourth, as the Difference between the first and second hath to the Difference between the third and fourth.

Here 8: 84::14-8=6:84-21=63.
That is, 8: 84::6:63.

The Method of finding out Numbers in Musical Proportion, is best expressed by Letters; as shall be shewed in the Algebraick Part.

Sect. 3. How to Change or Aary the Order of Things, &c. THIS 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 Pofition of any proposed Number of Things.

As how many feveral Changes may be rung upon any proposed Number of Bells; or how many feveral Variations may be made of any determined Number of Letters, or any other Things proposed 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 Progreffions, whose first Term and common Difference is Unity or 1. And the last Term the Number of Things proposed to be varied, viz. Ix2x3x4x5 x 6 x 7, &c. As will appear from what follows.

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

2. And if three Things are proposed to be varied, they may be changed fix several Ways (as to their Order of Place) and no

more.

For, beginning with 1, there will be
Next, beginning with 2, there will be
Again, beginning with 3 it will be

Which in all make 6 or 3 Times 2, viz.
Suppose four Things are proposed to be varied;
Then they will admit of 24 several Changes, as to
of different Places.

For beginning the Order with I it will be

Here is fix different Changes.

I

3 23 I x 2x3=6

3.4 4.3 4

2

3. 3.4.2 4

2

1.4.3.

3 2

And for the fame Reason 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=1x2x3x4. And by this Method of proceeding, it may be made evident, that 5 Things admit of 120 several Variations or Changes; and 6 Things of 720, &, As in this following Table.

These may be thus continued on to any affigned Number. Suppose to 24 the Number of Letters in the Alphabet, which will admit of 620448401733239439360000 several Variations. From these Computations may be started feveral pretty, and indeed, very strange, 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, so long as they, together with their Host, could fit every Day in a different Order or Position at Dinner; which by the foregoing Computations will be found near 14 Years. For they being made 7 with their Host, will admit of 5040 different Positions; 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 indeced.

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. Suppose it were required to tell how many several Changes might have been rung upon those 12 Bells; and at a moderate Computation how long all those Changes would have been ringing but once over.

First, 1x2×3×4×5×6×7×8×9×10×11×12=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 Day, 5 Hours, and 49 Minutes; which, being reduced into Minutes, is 525949.

Then 525949) 47900160 (91 Years and 26 Days.

So long would those 12 Bells have been continually ringing without any Intermiffion, before all their different Changes could have been truly rung but once over. It is strange, and feems almoft incredible, that a few Things should produce fuch Varieties. But that which seems yet more strange and surprising (yea, even impossible to those who are not versed 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 same 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 Proportion Disjund; commonly called the Golden Rule.

Proportion Disjunct, or the Golden Kule, is either Direct or Reciprocal, called Inverse. And those are both Simple and Compound.

SECT. I.

Direct Proportion is, when of four Numbers, the first bearing the same Ratio or Proportion to the second; as the third doth to the fourth.

As in these 2:8::6:24.

Confequently, the greater the second Term is, in respect to the first; the greater will the fourth Term be, in respect to the third. That is, as 8 the second Term is 4 Times greater than 2 the first 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 4::6:6×4. Or As 3:3×5::6:6x5.
But 2 x 6 x 4 =2×4x6. Or 3 x 6 x 5 = 3 x 5 x 6.

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

Again, the less the second Term is, in respect to the first; the less will the fourth Term be in respect to the third.

As in these 18:6::12:4.

That is, 18:18 ÷ 3::12:12 ÷ 3.

But 18 × 12÷3=18÷3 × 12. Viz. 18 x 4 = 6 x 12.
Consequently 2 8 6. 24. And 18 6.12

true Proportionals, per Corol. 2. page 77.

4. are

From these Confiderations, comes the Invention of finding a fourth Number in Proportion to any three given Numders. Whence it is called the Rule of Three.

For if the second Number multiplied into the third, be equal to the first multiplied into the fourth, it is easy to conceive, that if the Product of the second and third be divided by the first, the Quotient must needs be the fourth Number. For if that Number, which divides another, be multiplied into the Quotient produced by that Divifion; their Product will be equal to the Number divided. See page 21.

As in these 2:8::6:24.

Here 8 x 648 = 24 x 2.

But if 24 x 2 = 48, then will 48 ÷ 2 = 24. Or 48 ÷ 24 = 2.

Note, Any four Numbers in direct Proportion may be varied several Ways. As in these.

Viz. If 2:8::6:24. Then 2:6::8:24.

And 6:24::2:8. Or 24:6 :: 8: 2, .

These Variations being well understood, will be of no small Ufe in the ftating of any Question in this Rule of Three.

When three Numbers are given, and it is required to find a fourth Proportional; the greatest Difficulty (if there be any) will be in the right stating the Question, or abstracting the Numbers out of the Words in the Question, and placing them down in their proper Order.

Now this will be very easy, if it be truly confidered, that always two of the three given Terms, are only supposed, and affigned or limit the Ratio or Proportion. The third moves the Question; and the fourth gives the Anfwer.

As for inttance; if 3 Yards of Cloth cost 9 Shillings: What will 6 Yards cost at the fame Rate or Proportion ?

Here 3 Yards, and 9 Shillings, are two supposed Numbers that imply the Rate; as appears by the Word [if] viz. If 3 Yards coft 9 Shillings (then comes the Question) What will 6 Yards cost?

N. B.