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inventor of the celebrated scale, called Gunter's Scale. Briggs, in 1624, produced his Arithmetica Logarithmica, which contained the Logarithms of 30,000 natural numbers, viz. from 1 to 20000, and from 90000 to 100000, to 14 places of figures besides the index, together with the difference of the Logarithms. The computation of the 70 intermediate Chiliads was afterwards performed by Adrian Vlaq, and published at Gouda, in Holland. Briggs himself lived to complete a table of logarithmic sines and tangents for the hundredth part of every degree, to 14 places of figures, besides the index, together with a table of natural sines for the same parts to 15 figures, and the tangents and secants for the same to 10 places, with the construction of the whole. These were published in 1633, at Gouda, under the supervision of Adrian Vlaq. The author was not spared to publish the explanation of their use, but entrusted the execution of this duty, while dying, to Henry Gellibrand, who added a preface and the application of Logarithms to Plane and Spherical Trigonometry.

Logarithms continued to be cultivated by the English mathematicians with zeal, nor were they neglected on the Continent. The aid which they brought to the astronomer, rendered them to him infinitely valuable. By shortning his labor, he had time for new discoveries, and, in fact, produced the same effect as if the duration of his existence had been prolonged. The industrious Kepler, as might be expected, early directed his attention to this interesting subject. Peter Cruger, a Danish astronomer, also published in 1634, tables of Logarithms. In Italy they were cultivated by Cavalleri. In 1668, Mercator published his Logarithmotechnia, a work of great merit, in which he treated of Log. arithms upon principles purely arithmetical. In the same year, James Gregory published his Exercitationes Geometricæ, and immediately after, the Logarithmotechnia.

At this period the theory of Logarithms was sufficiently developed, and every difficulty removed; in fact, the facile methods of constructing them came too late to be of any considerable use in the computation of the tables; and at the present day, new methods of constructing them are only interesting as improvements in analysis. We may, therefore, here conclude their history.

Note. For a more full history of Logarithms, Hutton's introduc. tion to his tables may be consulted with advantage. See also Rees' and the Edinburgh Encyclopædia, and especially Maseres' Scriptores Logarithmici, in 6 vols. quarto, in which is contained a vast collection of interesting and useful matter. The early editions of Logarithmic tables which are valuable have become scarce. Some that have been mentioned in the history are highly valuable for their correctness, especially the Arithmetica Logarithmica, 1624, and the subsequent edition of Vlaq. The best modern tables are Hutton's 8vo., Taylor's 4to. and Callet's Tables Portatives, Paris, 1795. The editions of Gardiner are generally very incorrect. It will, perhaps, be interesting to some, to know that while the copies of Vlaq, the most complete of their kind, had become very scarce in Europe, and no one would run the risk of bringing out a new edition, they were reprinted in the royal palace of the Emperor of China, in 3 vols. folio, with Chinese characters. We said that Gardiner's tables were incorrect. We add in explanation, that the remark is not to be applied to all the editions. The octavo edition published at London in 1706, and especially the one published in 1742, may be recommended with entire confidence. But most of the subsequent editions, and especially the 5th, published in 1787, is so incorrect, that no dependence can be placed upon it. More may be learned of them from Hutton's Introduction.

THEORY AND CONSTRUCTION OF LOGARITHMS.

There are various methods of explaining the theory of Logarithms, two or three of the most important of which will receive particular attention when we enter upon the treatise of Day. But, to place the subject within the reach of those who may not be very familiar with Algebra and Geometry, we shali give an explanation which can easily be understood by those whose mathematical knowledge does not extend beyond the operations of Arithmetic.

An Arithmetical Progression is a series of numbers which increase or decrease by a common difference. The series of natural numbers, 1, 2, 3, 4, &c. is of this kind; so is also the series of numbers, 1, 3, 5, 7, 9, 11, &c. Here the common difference is 2; but it may be any number, whole or fractional. It is a property of this series, that double of any term is the sum of the two adjoining terms, so that an arithmetical mean between two numbers is half their sum. A geometrical progression is a series of numbers which increase by a common multiplier, or decrease by a common divisor, called their ratio. The series 1, 2, 4, 8,

16, 32, 64, &c. is of this kind. Here the ratio, is 2, but it may be any number, whole or fractional. In a geometrical series, the square of any term is the product of the 2 adjoining terms: and hence, a geometrical mean between two numbers, is the square root of their product.

To proceed with our theory, let us suppose the terms of any continued geometrical progression, which begins with unity, and the terms of any continued arithmetical progression which begins with 0, to be arranged in respect to each other as follows, in which for the sake of a particular example, the ratio of the geometrical series is 2, and the common difference of the arithmetical is unity. The 1st row repre sents the geometrical, and the 2nd, the arithmetical series. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 Supposing the two series to be continued to any extent, the numbers of the arithmetical series are called the logarithms of the corresponding terms in the geometrical series. From this definition, it will be easy to verify, by induction, the truth, of the two following properties of logarithms. 1. The sum of the logarithms of any two terms of the geometrical series, is the logarithm of that term which is their product. 2. The difference of the logarithms of any 2 terms of the series is the logarithm of that term which is the quotient of the greater divided by the less. For example, the sum of 5 and 7, the logarithms of 32 and 128, is 12, the logarithm of the term 4096, which is the product of 32 and 128. Again, the difference of 10 and 6, the logarithms of the terms 1024 and 64, is 4, the logarithm of 16, which is the quotient of 1024 divided by 64. By these properties, multiplication or division of any terms of the geometrical progression may be readily performed, by the addition or subtraction of their logarithms and the inspection of the tables. Thus, to find the product of the terms 16 and 64, we add their logarithms 4 and 6, and opposite to 10, the sum in the arithmetical series we find 1024, the answer. Again, to divide 2048 by 256, we subtract 8 the logarithm of the latter, from 11, the logarithm of the former, and opposite to 3, we find 8, the quotient.

We may suppose a term to be interpolated between each 2 adjoining terms of both series, by taking the square root of

their product in the geometrical series, and half their sum in the arithmetical-series; each would then consist of double the number of terms, the new terms in the geometricel series would be approximate values of the square roots of the numbers 2=1×2,8=2×4, 32=4×8, and so on; and those of the arithmetical series or the logarithms of the others would be .5 (0+1)÷2, 1.5=(1+2)÷2, 2.5=(2+3)÷2, The 1st 7 terms would then be as follows:

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We might go on in this way, interpolating terms continually; every new interpolation would change the ratio and common difference; but still the two series would have the same relative properties as atfirst.

The terms of the geometrical progression which have the series of numbers 1, 2, 3, 4, &c. for their logarithms, proceed, in Brigg's system, in a decuple ratio, thus:

Geom. Series. 1, 10, 100, 1000, 10000, 100000, &c.
Arith. Series. 0, 1, 2, 3,

4,

5, &c.

If we conceive a new term to be interpolated between each adjoining pair in either series, as explained in the last article, and this operation to be again repeated on the resulting series, and so on continually, the number of terms in the two series will continually increase, and each succeeding interpo. lation will produce a new geometrical series, having smaller differences between its terms than those of the preceding series. If, taking a particular case, we suppose the extreme terms of the geometrical series to be 1 and 100000, as above, a term interpolated between each adjoining pair would increase the number to 11, a 2d interpolation would produce 21, and a 3d, 41. Proceeding in this way, after 20 repetitions of the process, there would be produced 2 corresponding series, each having 5242881 terms, and the last two in the geometrical series would be 99999.78 and 100000, dif. fering only by the small fraction. .22 and as the terms go on increasing, no other adjoining two would differ by so much ;

therefore some one or other would be nearly equal to every whole number from 1 to 1000000, and the corresponding terms of the arithmetical series would be the logarithms of the numbers, according to Brigg's system.

We shall now show how, according to this mode of conceiving logarithms, they may be actually calculated; and as a particular example, we shall find Briggs' logarithm of 5. 1. The number 5 is between 1 and 10, the logarithms of which we already know to be 0 and 1 : Let a geometrical mean be found between the former two, and an arithmetical mean between the latter two. The geometrical mean will be the square root of the product of the numbers 1 and 10, which is 3.162277, and the arithmetical mean will b half the sum of the logarithms 0 and 1, which is 0.5, therefore the logarithm of 3.162277 is 0.5. But as the mean thus found is not sufficiently near to the proposed number, we must proceed with the operation as follows. 2. The number 5, whose logarithm is sought, is between 3.162277, the mean last found, and 10, the logarithm of which we know to be 0.5 and 1; we must now find a geometrical mean between the former two, and an arithmetical mean be. tween the latter two. The one of these is √(3.162277× 10)=5 623413, and the other is (1+0.5)÷2=0.75; there fore the logarithm of 5.623413 is 0.75. 3. We have now obtained two numbers, namely 3.162277, and 5.623413, one on each side of 5, together with their logarithms 0.5 and 0.75, we therefore proceed exactly as before, and accordingly we find the geometrical mean, or √(3.162277× 5.623413) to be 4.216964, and the arithmetical mean, or (0.5+0,75)÷2, to be 0.625; therefore the logarithm of 4.216964 is 0.625. We proceed in the same manner with the numbers 4.216964 and 5.62341, (one of which is less and the other greater than 5,) and their logarithms 0.625 and 0.75, and find a new geometrical mean, viz. 4.86974, and its corresponding arithmetical mean, or logarithm, 0.6875.

We must go on in this way, till we have found 22 geometrical means, and as many arithmetical terms, or logarithms. And that we may indicate how these are found from each other, let the numbers 1 and 10 be denoted by A and B, and their geometrical means taken in their order by C, D, E,

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