Sect. 3. How to Change or Mary 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 perusing) made me think it would be acceptable to the young Learner, to know how oft it is poflible to vary or change the Order or Position of any proposed Number of Things. As how many several Changes may be rung upon any proposed Number of Bells; or how many several 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, whore firft Term and common Difference is Unity or 1. And the last Term the Number of Things proposed to be varied, viz. 1 x 2 x 3 x 4x5 x6 x7, &c. As will appear from what follows. : 1. If the 'Things proposed to be varied are only two, they ad. mit of a double Polition (as to Order of Place) and no more. 2. And if three Things are proposed to be varied, they may be be changed fix several Ways (as to their Order of Place) and no Dore. For, beginning with 1, there will be { 1 : li. 3. 2 Next, beginning with 2, there will be Again, beginning with 3 it will be 13.2.1 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. PI. 2. 3. 4. 1.2.4.3 For beginning the Order with i it will be 11. 3 · 4 · 2 Here is fix different Changes. 11.4 · 2 · 3 ·li.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=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, &c, As in this following Table, The Number | The Manner how | The different Changes or Variof Things their several ations every one of the propoproposed to Pariations are fed Numbers can admit of. be varied. produced. = 1 = 2 2 X 3 6 6 x4 = 24 = 120 -39916800 x 12 479001600 &c. These II 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 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 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. 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 Tine: Now according .o 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 Minuttsy 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 almost incredible, that a few Things should produce such Varieties. But that which seems yet more itrange and surprising (yea, even izpollibie to thosc who are not verled 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 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 117009900000000000 Years to ring them but once over in all their different Changes; 28 may easily be computed from the precedent Table. CHAP VII. Of Proportion Disjunct ; 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. 1. 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. Consequently, the greater the second Term is, in respect to the forst; 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. Means, |