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6 Admit 100 stones were laid 2 yards distant from each other in a right line, and a Basket placed 2 yards from the first stone-I demand how many miles a man shall go in gathering them singly into the Basket? Ans. Il miles, 3 furlongs, 180 yards.
7 A merchant sold 1000 yards of linen at 2 pins for the first yard, 4 for the second, and 6 for the third, &e. increasing 2 pins for every yard--I demand how much the Linen produced, when the pins were afterwards sold at 12 for a farthing also whether the said merehant gained or lost by the sale thereof, and how much supposing the said linen to have been bought at 60 per yard? Ans.
The Linen produced 861 17s 10d.
Ă. When the two extremes, and the number of Terms in any series of numbers in Arithmetical Progression are given, and the common Difference of all the Terms in that series are required, then
Divide the Differenee between the two Extremes, by the number of Terms, less one; the Quotient will be the common difference.
1 There are 21 men, whose ages are equally distant from each other in Arithmetical Progression; the youngest is 20 years old, and the eldest is 60—1 demand the common difference of their ages, and the age of each man? Ans. the common difference is two years—therefore,
60 is the age of the First man. 60-_=58 is the age of the Second. 58—2=56 is the age of the Third,
56—2=54 is the age of the Fourth, &c. 2 A debt is to be discharged at 16 several payments in : Arithmetical Proportion, the first payment is to be ! 41. the last 100l. what is the whole debt, and what must each payment be? Ans. the whole debt is 9121. the common Difference is 5l 14s 8d. therefore,
141. Us. Od. 1st. payment. 141 0$ od +5l. 14s. 8d.=19 14 8 2d. 19 14 8 +5 14 8 =25 9.4 3d. 25. 94 +5 14 8 =31 4 0 4th, &c.
A man is to travel from York to a certain place in. 12.
days, and go but 3 miles the first day, increasing every day's journey by an equal excess, so that the last day's journey may be 36 miles; what will each day's journey be, and how many miles is the place he goes to distant from York? Ans. the common difference is 3; therefore,
3 is the first day's journey.
The whole distance is 234 miles. 4. A running footman, on a wager, is to travel from London northward, as follows; that is to say, he is to go 4 miles the first day, and 40 miles the last day, and to go. the whole journey in 10 days, increasing every day's journey by an equal excess: I demand the number of miles he travelled each day, and the length of the whole journey Ans. the common difference is 4: therefore,
4 is the first day's journey.
The whole journey is 220 miles.
OF GEOMETRICAL PROGRESSION. Q. What is Geometrical Progression ?
M. When any rank or series of numbers increases by one common multiplier, or decreases by one common divisor, those numbers are continued in Geometrical Progression; as 3, 6, 12, 24, increase by the multiplier 3and 24, 12, 6, 3, decrease by the divisor 2.
Note 1. If any pumber of terms be continued in Geometrical Progression, the product of the two extremes will be equal to the product of any two means equally distant from the extremes, as in 3, 6, 12, 24; where 3x24, are=6x12=72, and so of any larger number of terms.
2. If the number of terms be odd, the middlemost supplies the place of two terms; as in 3, 6, 12, where 3 x12 are=6X6=36.
3. The common multiplier, and the common divisor, are called Ratios.
Q. How is the sum of any series in Geometrical Progression obtained
A. 1. When all the terms alone are given, then from the Product of the second and last terms, subtract the square
of the first term; that remainder being divided by the second term less the first will give the sum of all the terms.
2. When the two extremes and the ratio are only given, then multiply the last term into the ratio, and from that product subtract the first term: that remaindei divide by ihe ratio less an unit or 1, the quotient is the sum of all the terms.
Note 1. As the last term in a long series of numbers is very tedious to come at by continual multiplication—it would be necessary for the readier finding it out, to have a series of numbers in Arithmetical Proportion, called Indices, beginning with an Unit whose common dif: ference is one-Also, whatsoever number of Indices you make choice of, let as many numbers ( in such Geometrical Proportion as are given in the question ) be placed under them. 1, 2, 3, 4, 5, 6,
7, Indices [Proportion. T
2, 4, 8, 16, 32, 64, 128, Numbers in Geometrical 2. But if the first term in Geometrical Proportion be different from the Ratio, the Indices must begin with a cypher.
[Proportion. 1, 2, 4, 8, 16, 32, 64, Nunibers in Geometrical 3. When the Indices begin with a cypher, the sum of the Indices made choice of must always be one less than the number of terms given in the question because l in the Indices stands over the second term, and 2. in the Indices stands over the third term, &c.
-4. Add any two of these Indices together, and that sum will directly correspond with the product of their respective terms.
5. By the help of these Indices, and a few of the first terms, in any series of Geometrical progression, any term whose distance from the first term is a trigned, though it were never so far, may speedily be oba tained without producing all the terms.
Thus < ; 2; 2; 3; & 5, 6, Indices
1. A man bought a horse, and by agreement was to give · a farthing for the first nail, two for the second, four for the third, &c. there were 4 shoes and 8 Dails in each shoe
I demand what the horse was worth at that rate ? Answer, 44739241. 55. 3d. 3qrs.
2. A merchant sold 15 yards of sattin, the first yard for Is, the second for 2s. the third for 4s. the fourth for 83. &c. I demand the price of the 15 yards? Ans. 16381. 78.
3. A draper sold 20 yards of superfine cloth, the first yard for 3d. the second for 9d. the third for 27d. d•c. in tripple Proportion Geometrical—I demand the price of the cloth ? Ans. 217924021. 10s.
4. A goldsmith sold ilb. of gold, at a farthing for the first ounce, a penny for the second, 4d. for the third, &c. in quadruple Proportion Geometrical demand what he
sold the whole for: also how much he gained by the salo thereof, supposing he gave for it 41 per ounce ?
He sold it for 58251 8s 5d Iqr. Ans.
5777 8 5 1. 5 A crafty servant agreed with a farmer (ignorant in numbers) to serve him 12 years, and to have nothing for his service but the produce of a wheat-corn for the first year; and that product to be sowed for the second year; and so on from year to year, until the end of the said time -I demand the worth of the whole produce, supposing the increase to be but in a tenfold proportion, and sold out at 45 per bushel? Answer 4521121 4s. rejecting remainders.
Note 1. 7680 Wheat or 'barley-corns are supposed to make a pint, and 64 pints a bushel.
2. If the first tèrm in any series be either greater or less than the ratio (except unity) then multiply any two terms together, and their product divide by the first terms-that quotient will exactly correspond with the sum of their indices.
6 A thresher worked 20 days at a farmer's, and received for the first day's work 4 barleycorns, for the second 12: barley-corns, for the third 36 barley-corns, and so on in triple proportion geometrical-1 demand what the 20 days labour came to, supposing the whole quantity to be sold for 2s 6d per bushel ? Answer 17731 7s 6d rejeeting remainders.
7 A merchant sold 30 yards of fine velvet, trimmed with gold very curiously, at 2 pins for the first yard, 6 pins for the second, 18 pins for the third, &c. in triple proportion, geometrical- demand how muck the velvet produced, when the pins were afterwards sold at 100 for a farthing; also, whether the said merchant gained or lost by the sale thereof, and how much, supposing the said velvet to have been bought at 500 per yard ?
The velvet produced Answ. }
21446992921 13s. 0.
21 44697792 13 oz.
OF PERMUTATION. Q. What is Permutation? 1. Changing the Order of things.
Q. How do you find all the variations any number of things is capable of going through?
A. Multiply all the given terms one into another con"tinually; the last product is the number of changes required.
1. I demand how many changes may be rung upon 12 bells-and also how long they would be in ringing but once over, supposing 24 changes might be rung in one minute, and the year to contain 365 days, 6 hours ? Ans. the number of changes is 479001600, and the time is 37 years, 49 weeks, 2 days, 18 hours.
2. Seven gentlemen who were travelling met together by chance, at a certain inn upon the road, where they were so well pleased with their host, and each other's company, that in a frolick they offered him 301 to stay at that place so long as they, together with him, could sit every day at dinner in a different order : the host thinking that they could not sit in many different positions, because there were but few of them, and that himself would make no considerable alteration; he being but one, imagined that he should make a good bargain, and readily, (for the sake of a good dinner, "and better company) entered into an agreement with them, and so made himself the eighth person; I demand how long they staid at the said inn, and how many different positions they sat in? Ans. the number of positions were 40320; and the time that they staid was 110 years, 142 days; allowing the year to consist of 365 days. 6 hours.
Note. There is one thing in progression, and in varying the order of things, which is well worth our observation; and that is, the power of numbers, which is surprisingly great, and beyond common belief ; and is no ways conceivable by a common practitioner, hardly by a very good artist; it being (in appearance ) not so much against reason as above it. The first example in geometrical progression discovers what a prodigious sum of money a horse sold after that manner would produce, viz no less than four million, four hundred and seven. ty-three thousand nine hundred and twenty-four pounds; whereas if the same horse had been sold at the same rate, and but a 4th part of the nails, he would have brought to his owner no more than 58. 3d. three farthings. The second example in Permutation does likewise discover the impossibility of the innkeeper's performing his promise: and in both, the simplicity of two men, who thinking they have got very good bargains, do, instead thereof, find themselves severe sufferers. And although at the first appearance each question seems to produce a trifle; yet upon a mature consideration, there would not be found a man in the kingdom able to purchase the one, or long-lived enough to stand to the agreement with the other. llence observe the great possibility of a man's being imposed on in this way by sharpers, without a careful examiration into the a fair, before any contract is made.