demand the age of each person? Answer, B was 20, C 60, D 8 years of age. 5. A man lying at the point of death, left to his 3 sons all his estate in money, viz. to F half wanting 50l. to G one third. and to H the rest, which was less than the share of G-1 demand the sum left, and each man's part? Ans. the sum left was 3601. whereof F had 130, G 120l. H 110/ 6. A certain man having drove his Swine to the market, viz. Hogs, Sows and Pigs, received for them all 50l. being paid for every Heg 18s. for every Sow 16s. for every Pigs. There were as many Hogs as Sows, and for every Sow there were three Pigs-1 demand how many were of each sort? Answ. 25 Hogs, 25 Sows, 75 Pigs. there 7. A surly old fellow being demanded the ages of his four children, answered, you may go and look: but if you must needs know, my first son was born just 1 year after I was married to his mother, who, after his birth, lived 5 years, and then died in child-bed, with my second son; four years after that I married again, and within two years had my third and fourth sons at a birth: the sum of whose two ages is now equal to that of the eldest-I demand their several ages? Answer, the first son was 22 years old, the second 17, the third 11, and the fourth 11 years old. OF COMPARATIVE ARITHMETIC. Q: WHAT is Comparative Arithmetic? A. It is such as answers Questions by Numbers, having Relation one to another. Q. Wherein does this relation consist? A. It consists either in Quantity or Quality. Q. What is the Relation of Numbers in Quantity? A. They are always two, the Antecedent and the Consequent. Q. In what does relation of numbers in quantity consist? A. It consists in the Difference, or else in the Rate or Reason that is found between the Terms propounded. Note. The difference of any two Numbers is the Remainder? but the Rate or Reason is the Quotient of the Antecedent divided by the Consequent. Q. What is relation of numbers in quality or progres sion? A. Progression or Proportion is the respect that the Reason of Numbers have one to another: Q. How many must the terms be? A. Three, or more, but never less, because less than $ will not admit of a comparison of reason or differences. OF PROGRESSION. Q. How many kinds of Progression are there? OF ARITHMETICAL PROGRESSION. A. Arithmetical Progression is when several Numbershave equal differences-as 1, 2, 3, 4, differ by 1, or 2, 4, 6, 8 differ by 2. Note 1. If any number of terms differ by Arithmetical Progression, the sum of the two extremes will be equai to the sum of any two means equally distant from the extremes. As in 2, 4, 6, 8,: where 2×8 are=4×6=10, 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 1, 2, 3; where 1×3 are 2×24. CASE 1. = Q. What do you observe in this first case? A. When the two Extremes, and the number of terms in any series of numbers in arithmetical progression are given, and the sum of all the terms is required, then multiply the sum of the two extremes by half the number of terms: Or, Multiply half the sum of the extremes by the whole number of terms, the product is the total of all the terms. EXAMPLES. 1 How many Strokes does the Hammer of a Clock strike in 12 hours? Answer 78. 2 A merchant hath sold 100 yds. of superfine cloth, viz. the 1st yard for 1s. the 2d 2s. the 3d for 3s, &c. I demand how much he received for the said cloth? Ans. 2521 10s. 3 Bought 19 yards of Shalloon, and gave id for the first yard, 3d for the second, 5d for the third, &c. increasing 2d every yard-1 demand what I gave for the 19 yards? Ans. 1l 10s 1d. 4 A Mercer sold 20 yards of silk, at 3d for the first yard, 6d for the 2d, 9d for the 3d, &c. increasing 3d every yard-I demand what he sold the 20 yds. for? Ans. 2l 12s 6d. 5 A Butcher bought 100 head of Cattle, viz. Oxen, and gave for the first Ox 1 Crown, for the second Ox 2 Crowns, for the third Ox 3 Crowns, &c.—I demand what the Cattle cost him? Ans. 1262 10s.. 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. 11 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, &c. 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 merchant gained or lost by the sale thereof, and how much, supposing the said linen to have been bought at 6d per yard? Ans. The Linen produced 86/ 17s 10d. CASE 2. Q. What do you observe in this second Case? A. 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 Difference between the two Extremes, by the number of Terms, less one; the Quotient will be the common difference. EXAMPLES. 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-I demand the common difference of their ages, and the age of each man ?. Ans. the common difference is two years-therefore, Years. 60 is the age of the First man.. 60-258 is the age of the Second.. 2 A debt is to be discharged at 16 several payments in Arithmetical Proportion-the first payment is to be 141. the last 100l. what is the whole debt, and what must each payment be? Ans. the whole debt is 9127. the common Difference is 57 14s 8d. therefore, 141 Os Od+51. 14s. 19 14 8 5 14 25 9 4 +5 14 141. Os. Od. 8d.19 14 8 8 =25 9 4 8 =31 4 0 3d. 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, Miles. 3 is the first day's journey. 3+3 6 is the second. 6+3=9 is the third. 9+3=12 is the fourth, &c. 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 jour ney 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, Miles. 4 is the first day's journey. 4+4=8 is the second. 8+4=12 is the third, &-c. The whole journey is 220 miles. OF GEOMETRICAL PROGRESSION. Q. What is Geometrical Progression ? A. 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 number 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 3×24, are=6×12=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 × 12 are 6×6=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 se cond 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 remainder divide by the 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 difference is one-so, 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. Thus {2; 2; 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. Thus (0, 1, 2, 3, 4, 5, 6, Indices [Proportion. 1, 2, 4, 8, 16, 32, 64, Numbers 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 1 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 assigned, though it were never so far, may speedily be obtained without producing all the terms. EXAMPLES. 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 nails in each shoe -I demand what the horse was worth at that rate? Answer, 4473924l. 5s. 3d. 3qrs. 2. A merchant sold 15 yards of sattin, the first yard for 1s. the second for 2s. the third for 4s. the fourth for 8s. &c. I demand the price of the 15 yards? Ans. 1638l. 7s. 3. A draper sold 20 yards of superfine cloth, the first yard for 3d. the second for 9d. the third for 27d. §c. in tripple Proportion Geometrical-I demand the price of the cloth? Ans. 217924021. 10s. 4. A goldsmith sold 1lb. of gold, at a farthing for the first ounce, a penny for the second, 4d. for the third, &c. in quadruple Proportion Geometrical-I demand what he |