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Of Mr. Dodson's Anti-logarithmic Canon. The only remaining considerable work of this kind published, that I know of, is the Anti-logarithmic Canon of Mr. James Dodson, a very ingenious mathematician, which work he published in folio in the year 1742; a very great performance, containing all logarithms under 100000, and their corresponding natural numbers to it places of figures, with all their differences and the proportional parts; the whole arranged in the order contrary to that used in the common tables of numbers and logarithms, the exact logarithṁs being here placed first, and increasing continually by 1, from 1 to 100000, and their corresponding nearest numbers in the columns opposite to them; and by means of the differences and proportional parts, the logarithm to any number, or the number to any logarithm, cach to ut places of figures, is readily found. This work contains also, besides the construction of the natural numbers to the given legarithms, "precepts and examples, Phewing some of the uses of logarithms, in facilitating the moft difficult operations in common arithmeric, cases of interest, annuities, mensuration, &c; to which is prefixed an introduction, containing a short account of logarithms, and of the most considerable improvements made, since their invention, in the manner of constructing them.”

The manner in which these numbers were constructed, confifts chiefly in imitations of some of the methods before described by Briggs, and is nothing more than generating a scale of 100000 geometrical proportionals from the least term to 10 the greatest, each continued to 11 places of figures; and the means of effecting this are such as easily flow from the nature of a series of proportionals, and are briefly as follows. First between 1 and 19 are interposed-9 mean proportionals; then between each of these 11 terms there are interpored 9 other means, making in all 101 terms; then between each of these a 3d set of 9 means, making in all 1001 terms; again between each of these a 4th set of 9 means, making in all 10001 terms; and lastly between each two of these terms, a 5th set of 9 means, making in all 100001 terms, including both the 1 and the 10. The first four of these 5 sets of means, are found each by one extraction of the 10th root of the greater of the two given terms, which root is the least mean, and then multiplying it continually by itself according to the number of terms in the fection or set; and the 5th or last section is made by interposing each of the 9 means by help of the method of differences before taught. Namely, putting 10 the greatest term = A, ATO

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B, B1 =C, CS = D, DI's = E, and Et* = F; now extracting the 10th root of A or 10, it gives 1,2589254118 = B = A' for the least of the ift set of means; and then multiplying it continually by itself, we have B, B2, B', B4,

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&c to B10 = A for all the 1o terms: 2dly, the oth root of 1,2589254118 gives 1,0232929923 = C = B7' = Ardo for the least of the 2d class of means, which being continually multiplied gives C, CP, C?, &c to C100 = B10 = A for all the 2d class of 100 terms: 3dly, the 10th root of 1,0232929923 gives 1,0023052381 = D = с = BT60 = ATS:s for the least of the 3d class of means, which being continually multiplied gives D, D2, D3, &c to D1000 = 100 = Bio = A for the 3d class of 1000 terms: 4thly, the 10th root 1,0023052381 gives 1,0002 302850 = E=

Dio CIÓ - Bross =A Todot for the least of the 4th class of means, which being continually multiplied gives E, E2, E3, &c to E 10000 = D1000 = C100 = B10 = A for the 4th class of 10000 terms. Now these 4 classes of terms thus produced, require no less than 11110 multiplications of the least means by themselves ; which however are much facilitated by making a small table of the first 10 or even 100 products of the constant multiplier, and from thence only taking out the proper lines and adding them together: and these 4 clasles of numbers always prove themselves at every oth term, which must always agree with the corresponding successive terms of the preceding class. I he remaining 5th class is constructed by means of differences, being much easier than the method of continual multiplication, the ift and 2d differences only being used, as the 3d difference is too small to enter the computation of the sets of 9 means between each two terms of the 4th class. And the several ad differences for each of these sets of 9 means, are found from the properties of a set of proportionals 1,5, r2, 13, &c, as Terms ist dif. 2d dif. i 3d dif. 1 & disposed in the ift column of the annexed 1 X 1r

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*] table, and their feveral orders of differences as in the other columns

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&c is evident that each cocolumn, both that of the given terms of the progression, and those of their orders of differences, forms a scale of proportionals, having the same common ratior; and that each horizontal line or row forms a geometrical progression having all the same .common ratio r- 1, which is also the ist difference of each set of means; so r- 112 is the ist of the ad differences, and which is constantly the same, as the 3d differences become too small in the required terms of our progression to be regarded, at least near the beginning of the table : hence, like as 1, r-1,

and -I are the ift term with its ist and 2d differences; so y, mn. r-I, and mn. r-112 are any other term with its ist and 2d differences. And by this rule the ist and ad differences are to be found for every set of 9 means, viz. multiplying the is

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term of any class (which will be the several terms of the series E, E, E', &c, or every joth term of the series F, F2, F3, &c) by r- 1 or F-for the ift difference, and this multiplied by F-i again for the true ad difference at the beginning of that class. Thus the both root of 1,0002302850 or E gives 1,000023026116 for F or the ift mean of the lowest class, therefore FI=r-I= ,000023026116 is its ist difference, and the square of it is nie =,0000000005302 its ad difference; then is ,00002 3026116F100 or ,000023026116Er the ist difference, and ,0000000005302F201 or ,0000000005302E2, is the ad difference at the beginning of the nth class of decades. And this ad difference is used as the constant 2d difference through all the 10 terms, except towards the end of the table where the differences increase fast enough to require a small correction of the ad difference, and which Mr. Dodson effects by taking a mean ad difference among all the ad differences in this manner; having found the series of ist differences F-1.E”, F-1. En +!, F-1. En+2, &c, take the differences of these, and of them will be the mean ad differences to be used, namely

F-1 Ex+1-ET, En + 2 — Enti, &c are the mean 2d differences. And this is not only the more exact but also the easier 'way. The common ad difference and the successive ist differences are then continually added through the whole decade, to give the succeffive terms of the required progreffion.

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LTHOUGH the nature and construction of logarithms have been

pretty fully treated in the preceding history of such numbers, where the more learned and curious reader will find abundant fatisfaction, I shall here give a brief, easy, and familiar idea of these matters, for the practical use of young students in this subject.

The Definition and Notation of Logarithms. Logarithms are the indices, or arithmetical series of numbers, adapt. ed to the terms of a geometrical series, in such sort that o corresponds to, or is the index of, 1 in the geometricals.

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1, 10, 100, 1000, 10000, 100000, &c. geometric series. Where the same indices serve equally for any geometric series; and from which it is evident that there may be an endless variety of systems of logarithms to the same common numbers, by varying the 2d term 2, or 3, or 10, &c, of the geometric series; as this will change the original series of terms whose indices are the whole numbers, 1, 2, 3, &c; and by interpolation the whole system of numbers may be made to enter the geometrical series, and receive their proportional logarithms, whether integers or decimals.

Ör, the logarithm of any number is the index of that power of some other number, which is equal to the given number. So if N be = rm, then the logarithm of N is n, which may be either positive or negative, and r any number whatever, according to the different systems of logarithms. When N is i, then n = o, whatever the value of ris ; and consequently the logarithm of 1 is always o in every system of logarithms. When n is = 1, then N is = r; confequently r is always the number whose logarithm is i in every system. When r is = 2.718281828459 &c, the indices are the hyperbolic logarithms, such as in our 7th table; so that n is the hyperbolic logarithm of 2-718&c.". But in the common logarithms

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is = 10; so that the common logarithm of any number (101) is (n) the index of that power of 10 which is equal to the said number. So 1000, being the 3d power of 10, has 3 for its logarithm ; and if 50 be = 101.69897, then is 1.69897 the common logarithm of 50. And hence it follows that this decimal series of terms 104, 10), 102, 10", 10°, 10, 10, 10 -3,

10 4 or 10000, 1000, 100

I, 'I, or , '001, have 4 3, 2 I

-3, respectively for their logarithms.

The logarithm of a number comprehended between any two terms of the first series, is included between the two corresponding terms of the latter, and therefore that logarithm will consist of the fame index (whether positive or negative) as the less of those two terms, together with a decimal fraction, which will always be positive. So the number 50, falling between 100 and 10, its logarithm will fall between 2 and I, and is = 1:69897, the index of the . less term together with the decimal •69897 : also the number '05, falling between the terms •i and .oi, its logarithm will fall between - 1 and — 2, and is indeed = –2 + •69897, the index of the less term together with the decimal •69897. The index is also called the characteristic of the logarithms, and is always an integer, either positive or negative, or else = 0; and it shews what place is occupied by the first fignificant figure of the given number, either above or below the place of units, being in the former case + or pofitive, in the latter - or negative.

When the characteristic of a logarithm is negative, the sign - is commonly set over it, to distinguish it from the decimal part, which being the logarithm found in the tables, is always positive: so - 2 + 69897, or the logarithm of .05, is written thus 2:69897. But on some occafions it is convenient to reduce the whole expression to a negative form ; which is done by making the characteristic figure less by I, and taking the arithmetical complement of the decimal, that is, beginning at the left hand, subtract each figure from 9, except the last fignificant figure, which subtract from 10; so thall the remainders form the logarithm intirely negative. Thus the logarithm of .05, which is 2.69897 or — 2 + .69897, is also expressed by – 1.30103, which is wholly negative. It is also sometimes thought more convenient to express such logarithms wholely as positive, namely by only joining to the tabular decimal the complement of the index to 10; in which way the above logarithm is expressed by 8.69897 ; which is only increasing the indices in the scale by io. It is also convenient, in many operations with logarithms, to take their arithmetical complements, which is done by beginning at the left hand, and subtracting every figure from 9, but the last figure from 10: so the arithmetical complement of 1.69897 S and of 2.69897 / where the index - 2, being negais 8.30103, ( it is 11 30103, Š tive, is added to g, and makes 11.

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