27. The first term of a geometrical progression is 5, and the ratio 3; required the 13th term? Ans. 2657205. 301. PROMISCUOUS EXAMPLES FOR PRACTICE. 1. Nine sea officers divide a prize, the first receives 201. the second 60l. and so on in triple proportion; what sum will the Admiral (who has the largest share) receive? Ans. 1312201. 2. Bought 12 pigs, and paid a farthing for the first, a halfpenny for the second, and so on, doubling continually the price of the last; what did they cost me? Ans. 41. 5s. 3d.. 3. A servant agreed with his master for 12 months, to receive a farthing for the first months' service, a penny for the second, 4d. for the third, &c. what sum did his wages amount to? Answer, 58251. 8s. 5d. 4. 4. The profits of a certain trading company, which has been established 12 years, have increased yearly in geometrical progression; the gain of the first year was 51. and that of the year just expired 8857351. required the ratio of increase, and the sum of the profits? Ans. the ratio 3. the sum 1328600l. 5. A person of property in Ireland agreed with Government to exert his influence, to procure seamen for the navy; the first month he sent over 1 man, the second 2 men, the third 4, and so on in geometrical progression; what number did he send over in 15 months, and how many in the last month of that time? Ans. sent in all 32767 men: in the last month 16384. 6. Suppose a laceman agrees to sell 22 yards of lace at the rate of 2 pins for the first yard, 6 for the second, and so on in triple proportion; what sum will he receive for the whole, allowing the pins to be worth a farthing a hundred? Answer, 3268861. Os. 9d. 7. What sum would a horse sell for that has 4 shoes on, with 8 nails in each shoe, at 1 farthing for the first nail, 2 for the second, 4 for the third, and so on? And what would be the price of another horse, having only two shoes, on the same conditions? Ans. 44739241. 5s. 3d. the first: and 681. 5s. 3d. the last. PART II. LOGARITHMS. HISTORICAL INTRODUCTION. 1. LOGARITHMS are a series of numbers in arithmetical progression, adapted to another series in geometrical progression, in such sort that O in the former series always corresponds to 1 in the latter, and the succeeding terms of the former to the succeeding terms of the latter, each to each. 2. The use of Logarithms is to lessen the labour and time which long calculations performed by common numbers necessarily require, addition and subtraction by Logarithms performing multiplication and division by numbers, &c. so that an operation may be performed in a few minutes by Logarithms, which would sometimes require as many hours by common arithmetic. 3. But the advantages attending the use of Logarithms would be very limited, if these useful numbers were exclusively confined to a geometrical progression; the common numbers not being in geometrical, but in arithmetical progression: this defect has been happily supplied by an admirable contrivance, which will be explained in its proper place, whereby Logarithms are extended to the entire algorithm of numbers, every number, whether integral or fractional, having its proper Logarithm. The word Logarithm is derived from the Greek λoyos, ratio, and agiduos, number, and implies either the ratio of numbers, or number of ratios, both interpretations being descriptive of the nature of Logarithms. 4. The fundamental property of Logarithms is this; if an arithmetical progression be applied to a geometrical one, in the manner above stated, the terms of the former will be indices to those of the latter: now if any two of these indices be added together, the sum will be the index of the product of the two numbers corresponding to those indices: if one index be subtracted from another, the remainder will be the index of the quotient which arises by dividing the number corresponding with the former, by that corresponding with the latter: if an index be multiplied by any number, the product will be the index of the term which is the power denoted by that number; and if an index be divided by any number, the quotient will be the index of the root denoted by that number. This property of the two progressions was known to the ancients, and treated of by Euclid and Archimedes. Stifelius in his Arithmetica Integra, printed at Nuremberg in 1544, explains it at large, shewing its use and application in a great variety of instances; so that it seems this author was in possession of the general idea of Logarithms, although under another name: and the reason assigned for his not computing Tables is, that he was not under a necessity of performing those long and troublesome calculations, which require the aid of Logarithms. Justus Byrgius, and Longomontanus, are re Justus Byrgius was a French Mathematical Instrument Maker, and assistant Astronomer to the Landgrave of Hesse; the invention of the Sector is ascribed to him, as was that of Logarithms, but the latter has never been proved: he flourished in the latter part of the 16th century. c Christian Longomontanus was born at a village of the same name in Denmark, in 1562; his father's name was Severin, and it is remarkable, that notwithstanding the obscurity of his father and his birth-place, he has contrived to dignify and eternize them both, by stiling himself in the title-page of some of his works, Christianus Longomontanus, Severini Filius. His parents being very poor, obliged him to work for his daily support, but he occasionally took lessons of the parish priest; at length he eloped, and went to the College at Wyburg, where he spent eleven years, being obliged to work for his living, and study ported to have known and constructed these numbers; but the person to whom the world is indebted for the first publication of them was, John Lord Napier, Baron of Merchiston, in Scotland, in a work intitled, Mirifici Logarithmorum Canonis Descriptio, printed in 1614. This work contains Tables of the Logarithms of Numbers, and of the Logarithmic Sines, Tangents, and Secants, for every minute of the quadrant, with definitions, description of the Tables, &c. but the Author chose to omit giving the method of construction, until the opinion of the learned concerning his invention should be ascertained. This discovery immediately excited the attention of mathematicians, and a translation of Napier's Book into English was made by Mr. Edward Wright, the inge : alternately he afterwards spent eight years as a very useful assistant to the celebrated Astronomer, Tycho Brahe; and at length obtained the Professorship of Mathematics at Copenhagen, where he died in 1647. His principal work is entitled, Astronomica Danica, 4to. 1621. and fol. 1640. John Lord Napier, (or Neper, as he is sometimes called,) was born in 1550, and was educated at the University of St. Andrews; he made the tour of Europe, and after his return applied himself closely to literature and science. Mathematics, especially astronomy, appear to have been his favorite study; and the numerous and intricate calculations requisite in the latter branch, put him upon various contrivances for shortening the work, which proved the source of the noble invention of Logarithms. He was the inventor of the instrument called Napier's Bones, consisting of five rulers of bone, wood, pasteboard, or ivory, whereby the arithmetical operations of multiplication, division, &c. may be performed mechanically with great ease; a full description of which he published in 1617, in a work entitled Rabdologia, seu Numerationis per Virgulas libri duo. Napier likewise invented the rule for the five Circular Parts in spherical Trigonometry; he died at Merchiston, in 1617. d Edward Wright, Esq. lived at the end of the 16th, and beginning of the 17th century. He was deeply skilled in mathematics and mechanics, and is justly celebrated as the inventor of what is erroneously called Mercator's Chart, having first discovered the true method of dividing the meridian line, on which Mercator's projection is founded; the principles of which he clearly shewed in The Correction of certain Errors in Navigation; a work, which, although written many years before, was not published till 1699. Every thing valuable in the celebrated maps of Jodicus Hondius was derived from the instructions given him on the subject by Wright; nevertheless the former with unheçom |