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Quest. 19. A person, looking on his watch, was asked wbat was the time of the day, who answered, It is between 5 and 6; but a more particular answer being required, he said that the hour and minute hands were then exactly together : What was the time ?
Ans. 27 is min. past 5. Quest. 20. If 20 men can perform a piece of work in 12 days, how many men will accomplish another thrice as large in one-fifth of the time?
Ans. 300. Quest. 21. A father devised is of his estate to one of his sons, and is of the residue to another, and the surplus to his relict for life. The children's legacies were found to be 5141 68 8d different: Then what money did he leave the widow the use of ?
Ans. 12701 18 9 #d. Quest. 22. A person, making his will, gave to one child
of his estate, and the rest to another. When these legacies came to be paid the one turned out 1200! more than the Other : What did the testator die worth?
Ans. 40001. Quest. 23. Two persons, A and B, travel between London and Lincoln, distant 100 miles, A from London, and B from Lincoln, at the same instant. After 7 hours they ineet on the road, when it appeared that a had rode li miles an hour more than B. Al what rate per hour then did each of the travellers ride ? Ans. A. 73%, and B 63 miles.
Quest. 24. Two persons, A and B, travel between London and Exeter. A leaves Exeter at 8 o'clock in the morning and walks at the rate of 3 miles an hour, without intermission; and b sets out from London at 4 o'clock the same evening, and walks for Exeter at the rate of 4 miles an hour constantly. Now, supposing the distance between the two cities to be 130 miles, whereabouts on the road will they meet?
Ans. 69.3 miles from Exeter. Quest. 25. One hundred
One hundred eggs being placed on the ground in a straight line, at the distance of a yard from each other: How far will a person travel who shall bring them one by one to a basket, which is placed at one yard from the
Ans. 10100 yards, or 5 miles and 1300 yds. Quest. 26. The clocks of Italy go on to 24 hours : Then how many strokes do they strike in one complete revolution of the index?
Ans. 300. QUEST. 27. One Sessa, an Indian, having invented the game of chess, she wed it to his prince, who was so delighted
with it, that he promised him any reward he should ask; on
Ans. 5 per cent.
Ans. A 445, $ 230, c 325.
Ans. 20 min. past 5. Quest. 32. Suppose that I have of a ship worth 12001; what part of her have I left after selling of of my share, and what is it worth?
worth 1851. QUEST. 33. Part 1200 acres of land among A, B, C ; so that B may have 100 more than A, and c 64 more than B.
Ans. A 312, B 412, C 476.
Quest. 36. What length must be cut off a board, 84 inches broad, to contain a square foot, or as much as 12 inches in length and 12 in breadth?
Ans. 16 inches. QUEST. 37 What sum of money will amount to 1381 2s 6d, in 15 months, at 5 per cent. per annum simple iaterest?
Ans. 1307 QUEST. 38. A father divided his fortune among his three sons, A, B, C, giving a 4 as often as B 3, and c 5 as often as
B 6; what was the whole legacy, supposing a's share was 40001.
Ans. 95001. Quest. 39. A young hare starts 40 yards - before a greyhound, 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 hold, and what ground will be run over, count. ing froin the outsetting of the dog?
Ans. 60z5sec. and 530 yards run. Quest. 40. Two young gentlemen, without private fortune, obtain commissions at the same time, and at the age of 18. One thoughtlessly spends 101 a year more than his pay; but, shocked at the idea of not paying his debts, gives his creditor a bond for the money, at the end of every year, and also insures his life for the amount; cach bond costs him 30 shillings, besides the lawful interest of 5 per cent. and to insure his life costs him 6 per cent
The other, having a proper pride, is determined never to run in debt; and, that he may assist a friend in need, perseveres in saving 101 every year, for which he obtains an interest of 5 per cent. which interest is every year added to bis savings, and laid out, so as to answer the effect of compound interest.
Suppose these two officers to meet at the of 50, when each receives from Government 4001 per annum; that the one, seeing his past errors, is resolved in future to spend no more than he actually has, after paying the interest for what he owes, and the insurance on his life.
The other, having now something before hand, means in future, to spend his full income, without increasing his stock.
It is desirable to know how much each has to spend per annum, and what money the latter has by him to assist the distressed, or leave to those who deserve it? Ans. The reformed officer has to spend 661 198 18.5389d
per annum. The prudent officer has to spend 4371 128 113043792
per annum. And the latter has saved, to dispose of, 7521 198 9-1896d.
END OF THE ARITHMETIC:
OGARITHMS are made to facilitate troublesome calcu. lations in numbers. This they do, because they perform · multiplication by only addition, and division by only subtracttion, and raising of powers by multiplying the logarithm by the index of the power, and extracting of roots by dividing the logarithm of the number by the index of the root. For, logarithms are numbers so contrived, and adapted to other Bumbers, that the sums and differences of the former shall correspond to, and show, the products and quotients of the latter, &c.
Or, more generally, logarithms are the numerical expo. nents of ratios; or they are a series of numbers in arith,
The invention of Logarithms is due to Lord Napier, Baron of Merchiston, in Scotland, and is properly considered as one of the most useful inventions of modern times. A table of these numbers was first published by the inventor at Erlinburgh, in the year 1614, in a treatise entitled Canon Mirificum Logarithmorum, which was eagerly received by all the learned throughout Europe. Mr. Henry Briggs, then professor of geometry at Gresham College, soon after the discovery, went to visit the noble inventor; after which, they jointly undertook the arduous task of computing new tables on this subject, and reducing them to a more convenient form than that which was at first thought of. But Lord Napier dying soon after, the whole burden fell upon Mr. Briggs, who, with prodigious labour and great skill, made an entire Canon, according to the new form, for all numbers from 1 to 20000, and from 90000 to 10100, to 14 places of figures, and published it at London, in the year 1624, in a treatise entitled Arithmetica Logas rithmica, with directions for supplying the intermediate parts.
metical progression, answering to another series of bumbers in geometrical progression.
So, 1, 2, 3, Thus,
4, 5, 6, Indices, or logarithms. 1, 2, 4, 8, 16, 32, 64, Geometric progression.
4, So, 1, 2, 3, 5, Or
6, Indices, or logarithms. 21, 3, 9, 27, 8i, 243, 729, Geometric progression.
So, 1, 2, 3, Or
5, Indices, or logs. 1, 10, 100, 1000, 10000, 100000, Geom progress. Where it is evident, that the same indices serve equally for any geometric series; and consequently there may be an
This Canon was again published in Holland by Adrian Vlacq, in the year 1628, together with the Logarithms of all the numbers which Mr Briggs had omitted ; but he contracted them down to 10 places of decimals. Mr. Briggs also computed the Logarithms of the sines,, tangents, and secants, to every degree, and centesm, or 100th part of a degree, of the whole quadrant; and annexed them to the natural sines, tangents, and secants, which he had before computed, to fifteen places of figures. These Tables, with their construction and use, were first published in the year 1633, after Mr. Briggs's death, by Mr. Henry Gellibrand, under the title of Trigonometria Britannica.
Benjamia Ursinus also gave a Table of Napier's Logs, and of sines, to every 10 seconds. And Chr. Wolf, in his Mathematical Lexicon, says that one Van Loser bad computed them to every single second, but his untimely death preventer their publication. Many other authors have treated on this subject ; but as their numbers are free quently inaccurate and incommodiously disposed, they are now gene. rally neglected. The Tables in most repute at present, are those of Gardiner in 4to, first published in the year 1742; and my own Tables in 8vo, first printed in the year 1785, where the Logarithms of all numbers may be easily found from 1 to 10000000; and those of the sines, tangents, and secants, to any degree of accuracy required.
Also, Mr. Michael Taylor's Tables in large 4to, containing the common logarithms, and the logarithmic sines and tangents to every second of the quadrant. And, in France, the new book of logarithms by Callet ; the 2d edition of which, in 1795, has the tables still farthe extended, and are printed with what are called stereotypes, the ypes in each page being soldered together into a solid mass or block.
Dodson's Antilogarithmic Canon is likewise a very elaborate work, and used for finding the numbers answering to any given logarithm,