1. EXAMPLES. A lady bought damask for a gown, at 8s. per yard, and lining for it, at 3s. per yard; the gown and lining contained 15 yards, and the price of the whole was 43 10s. How many yards were there of each? Suppofe 6 yards damafk, value 48s. Then fhe muft have 9 yards of lining, value 27s. Sum of their values=75s. So that the firft error is 5 too much, or +5 Again, fuppofe fhe had 4 yards of damafk, value 325. Then the muft have 11 yards of lining, value 335. Sum of their values=65s. So that the fecond error is 5 too little, or— 55. 65+ Then, X 5 yds. at 8s.2 10 yds. at 3s. I lo £3100 (Proof. Anf. 5 yards damafk, and 15-5=10 yds. Or, 6+4÷2=5 as before. (lining. 2. A and B have the fame income:- A faves of his; but B, by spending 30 per annum more than A, at the end of 8 years finds himfelf £40 in debt; What is their income, and, what does each spend per annum? Suppose { *80 120+ Anf. their income is £200 160 40+ (per annum. Alfo, A fpends £175 and (B 205 per annum. *Then, 80-10= 70, A's expenfe per annum; and 70+30= 100, B's expenfe per annum. Then 100 X 8 80x8=160, which fhould have been 40; therefore 160-40-120 more than it should be, for the firft error. In like manner proceed for the second error. 3. A and B laid out equal fums of money in trade : A gained a fum equal to of his ftock, and B loft £225; then A's money was double that of B; What did each. lay out; S300 225+ Anf. £600. 4. A labourer was hired for 60 days upon this condition, that, for every day he wrought, he should receive 3s. 4d. and for every day he was idle fhould forfeit Is. 8d. at the expiration of the time he received £3 155.; How many days did he work, and how many was he idle? Suppofe he worked 900 (2x Anf. He was employed 35 days, and was idle 25. 5. A gentleman has two horfes of confiderable value, and a carriage worth £100. Now, if the first horse be harneffed in it, he and the carriage together will be triple the value of the fecond; but if the fecond be put in, they will be 7 times the value of the first: What is the value of each horfe? 6. Suppofe 32 80 .44 X 160- Anj. One £20, and the other £40., There is a fifh, whofe head is 10 feet long; his tail is as long as his head and half the length of his body, and his body as long as the head and tail; What is the whole length of the fish? Head 7. What number is that, which, being increased by its, its, and 5 more, will be doubled? Suppofe { 8 3+ 16 1+ Anf. 20 8. A farmer having driven his cattle to market, received for them all, 80, being paid at the rate of £6 per ox, 4 per cow, and I 10s. per calf; there were as many sen as cows, and 4 times as many calves as cows: How many were there of each fort? Suppofe cows 6 Suppofe 12 X 16+ 112+ Anf. 5 oxen, 5 cows, and 20 calves. 9. A, B and C built a fhip, which coft them £1000-of which A paid a certain fum-B paid £100 more than A, and C 100 more than both; having finished her, they fixed her for fea with a cargo worth twice the value of the ship: The outfits and charges of the voyage amounted to of the fhip; upon the return of which, they found their clear gain to be of of the veffel, cargo and expenfes: Pleafe to inform me what the ship coft them, feverally; what fhare each had in her, and what, upon the final adjustment of their accompts, they had feverally gained? Suppofe it coft A 100 300- A owned of the fhip, which coft him I 175, and his fhare of the gain was £218 15s.-B owned 15, which coft £275, and his gain was £343 15.-C owned, which colt £550, and his gain was £687 10s. F F PERMUTATIONS PERMUTATIONS, and COMBINATIONS. The Permutation of Quantities is the fhewing how many different ways any given number of things may be changed. This is alfo called variation, alternation, or changes; and the only thing to be regarded here is the order they fland in; for no two parcels are to have all their quantities placed in the fame fituation. The Combination of Quantities is the fhewing how oftea lefs number of things can be taken out of a greater, and combined together, without confidering their places, or the order they ftand in. This is fometimes called election, or choice; and here every parcel must be different from all the reft, and no two are to have precifely the fame quantities or things. The Compofition of Quantities is the taking of a given number of quantities out of as many equal rows of different quantities, one out of every row, and combining them together. Here, no regard is had to their places; and it differs from Combination, only, as that admits but of one row of things. To find the number of permutations, or changes, that can be made of any given number of things, all different from each other.. RULE.-Multiply all the terms of the natural feries of numbers, from 1 up to the given number, continually together, and the last product will be anfwer required. EXAMPLES. * Any two things a and b are capable of two variations only; as ab, ba; whofe number is expreffed by 1×2. If there be three things, a, b, and c, then any two of them, leaving out the third, will have 1X2 variations; and confequently when the third is taken in, there will be 1X2X3 variations; and fo on, as far as you please. EXAMPLES. I. Christ Church, in Bofton, has 8 bells; How many changes may be rung on them? 1X2 X3 X4 X5 X6X7X8=40320 the Anf. 2. Nine gentlemen met at an inn, and were fo pleafed with their hoft, and with each other, that, in a frolic, they agreed to tarry fo long as they, together with their hoft, could fit every day in a different pofition at dinner; Pray how long, had they kept their agreement, would their frolic have lafted? Anf. 994133 years. 3. How many changes or variations will the alphabet admit of? Anf. 620448401733239439360000. Any number of different things being given-to find how many changes can be made out of them, by taking any given number of quantities at a time. RULE. Take a feries of numbers, beginning at the number of things given, and decreafing by 1, to the number of quantities to be taken at a time; the product of all the terms will be the answer required. EXAMPLES. I. How many changes may be rung with 4 bells out of 8 ? 8X7X6X5(4 terms)=1680 the Anf. 2. How many words can be made with 6 letters of the alphabet, admitting a number of confonants may make a word? - Anf. 96909120. PROBLEM 3. Any number of things being given--whereof there are feveral things of one fort, feveral of another, Sc. To find how many changes may be made out of them all. RULE 1. Take the feries I X2 X3 X4, &c. up to * the Any 2 quantities, a b, both different, admit of two changes; but if the quantities are the fame, or an, become aa, there will be only one alteration, which may be expreffed by 1X2 Any |