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



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For, beginning with 1, there will be {
Next, beginning with 2, there will be {2 :3 :}

Again, beginning with 3 it will be 53
Which in all make 6 or 3 Times 2, viz. 1x2x3=6
Suppose four Things are proposed to be varied ;
Then they will admit of 24 several Changes, as to their Order
of different places.

3 4

4 3 3

4 For beginning the Order with i it will be

3 4 2 Here is six different Changes.



4 3 2 And for the same 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=1*2*3*4. 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, &c, As in this following Table.


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2 X 3

6 x 4

I 20 X

6 7 8 9 10 II 12 &c.

= 24 24 x 5 = ( 20

6 = 720 720 x 7

= 5040 5040 X 8

= 40320 40320 x 9 = 362880 362880 x 10 = 3628800 3628800 x II 39916800 39916800 x 12 = 479001600 &c.

&c. M 2


These may be thus continued on to any assigned Number. Suppose to 24 the Number of Letters in the Alphabet, which will admit of 620448401733239439360000 several Variations.

From these Computations may be started several pretty, and indeed, very strange, Questions.


Six Gentlemen, that were travelling, met together by Chance at a certain Inn upon the Road, where they were so pleased with their Host, 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 Hoft, 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 Hoft, will admit of 5040 different Positions ; but 5040 being divided by 3654 (the Number of the Days in one Year) will give 13 Years and 291 Days. A very pretty Frolick indeeed.,

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 Cheapside, 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 singing but once over.

First, 1x2x3 x4x5x6x7x8x9x10x!Tx!2=479001600, the Number of Changes.

Then supposing there might be rung 10 Changes in one Minute: viz. 12*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 (9ı Years and 26 Days. So long would those 12 Bells have been continually ringing without any Intermiffio:1, before all their different Changes could have been truiy sung but once over. It is ftrange, and seems almot incredible, that a few Things thould produce such Varieties.

But that which seemis yet more strange and surprising (yea, even im pullible to thoie who are not versed in the Power of Numbers)

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is, that if two Bells more had been added to the aforesaid 12 they would have advanced the Number of Changes (and consequently the Time) beyond common Belief. For 14 Bells would require (at the same 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.


Of Proportion Disjunct; commonly called the Golden


PRoportion Disjunei, or the Golden Mule, is either Direct or

Reciprocal, called Inverse. And those are both Simple and Compound.

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Irect 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. Consequently, 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 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 4 ::6:6x4. Or As 3 : 3 x 5 :: 6:6x5.
But 2 x 6 x 4 = 2*4*6. Or 3 * 6*5= 3 * 5 * 6.

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


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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 x 12 = 3= 18 = 3 * 12. Viz. 18 *4= 6 12.
Consequently 2 . 8. 6

· 24. And 18

6 I 2 • 4

are true Proportionals, per Corol. 2. page 77.

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 Division; their Product will be equal to the Number divided. See page 21.

As in these 2 :8::6; 24. Here 8 x 6 = 48 = 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.

6 8

: 2,

Esc. These Variations being well understood, will be of na small Use 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 Dificulty (if there be any) will be in the right ftating the Question, or abstracting the Numbers out of the Words in the Qucftion, and placing them down in their

Or 24 :

proper Order.

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

As for instance; if 3 Yards of Cloth cost 9 Shillings: What will 6 Yards cost at the same 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 cot?

N. B.

N.B. The Term, which moves the Question, hath generally some of those Words before it ; viz. What will : How many? How long : How far: or How much : &c.

Then (carefully observe this; viz.) the first Term in the Supposition must always be of the fame Kind and Denomination with that Term which moves the Question. And the Term fought will always be of the same Kind and Denomination with the second Term in the Suppofition.

Thus, { vas bil. yds Bil



Then All Questions in direct Proportion may be answered by three several Theorems.


Multiply the second and third Terms together, and Theorem 1. divide their Produ&t by the first Term; the Quotient

will be the Answer required.
yds. phil. yds. mil.
Thus 3:9 :: 6 : 18. The Answer.

S because the second Term 3) 54 (18 Shillings, was Shillinge.

Divide the second Term by the first, then multiply the Theorem 2. Quotient into the third Term, and their Product

will be the Answer required.
yds. fil. yds. fil.
3 : 9::

6 : 18.
Thus 3) 9 =3. Then 3 x6 =18, as before.


Theorem 3.

Divide third Term by the first, then multiply the

Quotient into the second Term, and their Prodk?? will be the Answer.

yds. fhil. yds. mill.

3:9:: 6 : 18.
Thus 3) 6 (=2. And 9 x2= 18, as before.

Here you see that all the three Theorems are equally true ; but the first is most general, and usually practifed. Yet the two last may be readily performed, when either the second or third Term can be divided by the first; and will be found of fingular Use in the Rules of Fellowship, &c. as will appear further cn.



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