23. A man dying left his wife in expectation, that a child would be afterward added to the surviving family; and making his will ordered, that, if the child were a son, of his estate should belong to him, and the remainder to his mother; but if it were a daughter, he appointed the mother and the child the remainder. But it happened, that, the addition was both a son and a daughter, by which the widow lost in equity 2400l. more than if there had been only a girl what would have been her dowry, had she had only a son? Ans. 2100l. 24. A young hare starts 40 yards before a grey-hound, and is not perceived by him till she has been up 40 seconds; she scuds away at the rate of 10 miles an hour, and the dog, on view, makes after her at the rate of 18. How long will the course continue, and what will be the length of it from the place, where the dog set out? · Ans. 60 seconds, and 530 yards run. 25. A reservoir for water has two cocks to supply its by the first alone it may be filled in 40 minutes, by the second in 50 minutes, and it has a discharging cock, by which it may, when full, be emptied in 25 minutes. Now, supposing that these three cocks are all left open, that the water comes in, and that the influx and efflux of the water are always alike, in what time would the cistern be filled? Ans. 3 hours zo min. 26. A sets out from London for Lincoln at the very same time that B at Lincoln sets forward for London, distant 100 miles after 7 hours they met on the road, and it then appeared, that A had ridden 1 than B at what rate an hour did each : mile an hour more of them travel? Ans. A 71, B 611 miles. 27. What part of 3d. is a third part of 2d. ? Ans. 28. A has by him 1cwt. of tea, the prime cost of which was 961. sterling. Now, granting interest to be at 5 per cent. it is required to find how he must rate it per pound pound to B, so that by taking his negotiable note, payable at 3 months, he may clear 20 guineas by the bargain? Ans. 14s. 1d. sterling. 29. What annuity is sufficient to pay off 50 millions of pounds in 30 years, at 4 per cent. compound interest? Ans. 28915051. 30. There is an island 73 miles in circumference, and 3 footmen all start together to travel the same way about it; A goes 5 miles a day, B 8 and C 10 when will they all come together again? Ans. 73 days. 31. A man, being asked how many sheep he had in his drove, said, if he had as many more, half as many more, and 7 sheep and a half, he should have 20: how many had he? Ans. 5. 32. A person left 40s. to 4 poor widows, A, B, C and' D; to A he left, to B4, to C and to D, desiring the whole might be distributed accordingly what is the proper share of each ? Ans. A's share 14s. d. B's 10s. 6d. C's 85. 5d. D's 7s. d. 8d. 33. A general, disposing of his army into a square, finds he has 284 soldiers over and above; but increasing each side with one soldier, he wants 25 to fill up the square: how many soldiers had he? Ans. 24000. 34. There is a prize of 212l. 145. 7d. to be divided among a captain, 4 men, and a boy; the captain is to have a share and a half; the men each a share, and the boy of a share what ought each person to have ? Ans. The captain 541. 14s. d. each man 361. 9s. 44d. and the boy 121. 3s. 12d. 35. A cistern, containing 60 gallons of water, has 3 unequal cocks for discharging it; the greatest cock will empty it in one hour, the second in 2 hours and the third in 3 in what time will it be empty, if they all run togethAns. 32 minutes. er ? 8 G G 36. In 36. In an orchard of fruit trees, pears, of them bear apples, plumbs, and 50 of them cherries: how many trees are there in all ? Ans. 600. Ans. 5 days. 37. A can do a piece of work alone in 10 days, and B in 13; if both be set about it together, in what time will it be finished? 38. A, B and C are to share 10000ol. in the proportion of, and respectively; but C's part being lost by his death, it is required to divide the whole sum properly between the other two. I Ans. A's part is 571423, and B's 4285737 END OF ARITHMETIC. LOGARITHMS. LOGARITHMS are numbers, so contrived, and adapt ed to other numbers, that the sums and differences of the former shall correspond to, and shew, the products and quotients of the latter. Or, logarithms are the numerical exponents of ratios; or a series of numbers in arithmetical progression, answering to another series of numbers in geometrical progression. Thus Or Or I, 2, 3, 42 5, 6, indices, or logarithms. I, 2, 4, 8, 16, 32, 64, geometric progression. ९१. 6, indices, or logar. O, I, 2, 3, 4, 5 geometric progress. 5, ind. or log.. 0, I, 2, 3, 4, Where it is evident, that the same indices serve equally for any geometric series; and consequently there may be an endless variety of systems of logarithms to the same common numbers, by only changing the second term 2, 3, or or 10, &c. of the geometrical series of whole numbers and by interpolation the whole system of numbers may be made to enter the geometric series, and receive their proportional logarithms, whether integers or decimals. It is also apparent from the nature of these series, that if any two indices be added together, their sum will be the index of that number, which is equal to the product of the two terms in the geometric progression, to which those indices belong. Thus, the indices 2 and 3, being added together, make 5; and the numbers 4 and 8, or the terms corresponding to those indices, being multiplied together, make 32, which is the number answering to the index 5. In like manner, if any one index be subtracted from another, the difference will be the index of that number, which is equal to the quotient of the two terms, to which those indices belong. Thus, the index 6 minus the index 42; and the terms corresponding to those indices are 64 and 16, whose quotient 4; which is the number answering to the index 2. For the same reason, if the logarithm of any number be multiplied by the index of its power, the product will be equal to the logarithm of that power. Thus, the index or logarithm of 4, in the above series, is 2; and if this number be multiplied by 3, the product will be =6 which is the logarithm of 64, or the third power of 4. And if the logarithm of any number be divided by the index of its root, the quotient will be equal to the logarithm of that root. Thus, the index or logarithm of 64 is 6; and if this number be divided by 2, the quotient will be3 which is the logarithm of 8, or the square root of 64. The logarithms most convenient for practice are such, as are adapted to a geometrical series, increasing in a tenfold proportion, as in the last of the above forms; and are those, |