There is also a third Kind of Proportion, called Mufical, which being but of little or no common Ufe, I shall therefore give but a fhort Account of it. Mufical Proportion or Habitude is, when of three Numbers; the firft hath the fame Proportion to the third, as the Difference between the first and fecond hath to the Difference between the fecond and third. If there are four Numbers in Mufical Proportion; The firft will have the fame Proportion to the fourth, as the Difference between the firft and fecond hath to the Difference between the third and fourth. As in these 8 14 21 84. Here 8 84 14-86: 84-2163. That is, 8 84: 6 63. 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 Uary 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 propofed 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 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 Progreffions, whofe firft Term and common Difference is Unity or 1. And the left Term the Number of Things propofed to be varied, viz. 1 × 2 × 3 × 4 × 5 × 6×7, &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. 2. And if three Things are propofed to be varied, they may I be be changed fix feveral Ways (as to their Order of Place) and no $3.1.2 23 2 I Which in all make 6 or 3 Times 2, viz. I x 2x3=6 Suppofe four Things are propofed to be varied; Then they will admit of 24 feveral Changes, as to their Order of different Places. 4 2 3 4. 3 4 2 3 2 4 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 241x2x3x4 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. Thefe 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 several pretty, and indeed, very ftrange, Questions. Six Gentlemen, that were travelling, met together by Chance at a certain Inn upon the Road, where they were fo 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 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. 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, 1×2×3×4×5×6×7×8×9×10×11×12=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 muft 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 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 furprifing (yea, even impoffible to thole 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 1170009000000ÿÿ¤¶¤ Years to ring them but once over in all their different Changes; as may eafily 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 Inverfe. And thofe are both Simple and Compound. SECT. I. Direct Proportion is, when of four Numbers, the first bearing the fame Ratio or Proportion to the fecond; as the third doth to the fourth. As in these 2: 8 :: 6 : 24. Confequently, the greater the fecond Term is, in refpect to the firft; the greater will the fourth Term be, in refpect 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 4:: 6: 6 x 4. Or As 3: 3 × 5:: 6: 6 x 5. That is, the Produce of the Extreams is equal to that of the 1 Again, the lefs the fecond Term is, in respect to the first; the lefs will the fourth Term be in respect to the third. As in thefe 18: 6:: 12: 4. That is, 18: 183 :: 12: 123. But 18 x 123=18÷÷3 × 12. Viz. 18 x 4 = 6 x 12. Confequently 2. 8. 6. 24. And 18. 6. 12. 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 fecond Number multiplied into the third, be equal to the first multiplied into the fourth, it is eafy to conceive, that if the Product of the fecond and third be divided by the firft, the Quotient muft 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 thefe 2: 8 :: 6 : 24. Here 8 X 6 =48=24× 2. Note, Any four Numbers in direct Proportion may be varied feveral Ways. As in thefe. Viz. If 2: 8: : 6 : 24. Then 2 : 6: 8 And 6 24 2: 8. Or 24: 6 ::8 : 2, &c. Thefe Variations being well underflood, will be of no fmall Use in the ftating of any Queftion 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 ftating the Queftion, or abftracting the Numbers out of the Words in the Queftion, and placing them down in their proper Order. Now this will be very eafy, if it be truly confidered, that always two of the three given Terms, are only fuppofed, and affigned or limit the Ratio or Proportion. The third moves the Question; and the fourth gives the Anfwer. As for inftance; if 3 Yards of Cloth coft 9 Shillings': What will 6 Yards coft at the fame Rate or Proportion? Here 3 Yards, and 9 Shillings, are two fuppofed Numbers that imply the 'Rate; as appears by the Word [if] viz. If 3 Yards coft 9 Shillings (then comes the Queftion) What will 6 Yards coft? N. B. |