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Lalande's tables to 10,000 occupy no less than 110 pages, 18mo. and part of the 111th page. Mr. Bailey has given, in his astronomical tables, a table of Logarithms to 1000 contained on 3 pages, 4to. but only to 4 places of decimals; and subsequently he has given, as being more convenient for ordinary use, an Antilogarithmic table, of the same size and extent as the former, and besides these I am not aware of any smaller hitherto published.*
Having occasion some years ago to lay off the divisions on some sliding rules, I felt it was desirable to have at one view a table of Logarithms to 1000, that I might avoid the inconvenience of turning leaves with a beam compass in hand : I therefore wrote out the body of the first table nearly in its present form. It was immediately evident that if furnished with differences and proportional parts, the table would serve for most common purposes : accordingly the difference between the numbers in columns 4 and 5 having been written, the decimal parts were got by means of a sliding rule, with no more labour than writing them off.
The extension of the table beyond 1000 was partly to fill up the blank space, and also to avoid partly the inconvenience of the unequal differences at the beginning of the table. I found, however, that as it stood, even when the differences were most unequal, by attending to the actual difference between the columns, and allowing a proportional part for the difference between that and the difference here given, (and still more easily by allowing at sight for the difference between the proportional part on that and the next line according to the distance from the middle of the line,) I could get a result true, generally, within one, and always within two units in the last figure; and the same more conveniently by taking the proportional part of the actual difference by means of a sliding rule, which when engaged in calculation, I like to have always within reach.
The second table was made for the purpose of finding the number to a given Logarithm more readily than can be done by the other table. It has also the advantage of having more equal common differences. The utility of it for all ordinary purposes has been experienced by others besides myself.
* Dr. Maclear, for the use of the Cape Observatory, has remodelled Mr. Bailey's 2d Table, and printed it on two foolscap pages, with proportional parts, in a form not unlike that here given.
Each table may be made to do all that can be done by the other, hut not with equal convenience. The first table giving at sight the logarithm to a number of 3 figures, and the second giving at sight the number to a logarithm of 3 figures, the proportional parts for the 4th, and subsequent figures are additive: but if a number to a given logarithm be sought by means of the first table, or if the logarithm to a given number be sought by means of the second, one subtraction is required for each figure in the proportional part: and subtraction, though a simple operation, is by no means so short or so easy as addition, and hence the advantage of using both tables instead of either exclusively.
Great care has been taken to make these tables correct in every case to the nearest unit in the last figure. The first table was taken from Lalande, whose tables are known to be correct, and has been rigidly compared in every figure. The second table was made by means of the common table in Callet, and after having been written out, it was examined by reading out the number to every logarithm, using Babbage's tables. When it was uncertain in this way whether the number was more or less than 5 in the 6th place, it has been determined by a calculation carried to ten figures.
For finding the common differences and proportional parts the following method was used. Having determined to give these for the middle between columns 4 and 5 of each table, those for the first table were thus found. M denoting as usual the common modulus, and N the number in the first column of the Table, the common difference has been taken by the formula - ; and the proportional parts by its decimal products, taking care to make each true to the nearest unit in the 5th place of decimals.
For the second table the principal was the same, but the process vastly simpler. By the common differential formula d log N=M * N; (the same as that used above) j from this we have d N=N.du^N: in which M and d. log N. being constant, the only variable is N, and by the nature of the Table log N. in each successive line increases by unit in its 2nd figure. Hence the logarithm of d N being calculated for any one line, it is found for each succeeding line by adding one to its 2nd digit, and the common difference, or the number corresponding to this calculated logarithm being found more or less nearly in some column in the body of the table, the successive differences will be found in the corresponding part of the same vertical columns throughout the table. The proportional parts are found exactly in the same way; and thus, for finding the whole of the common differences and proportional parts, only ten calculations are needed ; the remaining labour being merely to transcribe from the columns in the body of the table, taking care to keep the numbers always true in the last figure.
The whole of the differences and proportional parts, after being written in, have been carefully re-examined, and I hope it will be found that these precautions have not been misapplied, so as to have failed as to the object intended.
These tables are of the minimum size, at least I do not see the possibility of making them any smaller without rendering them useless. Lalande " after 50 years' experience," at page 6, of the preface to his Tables, makes the following remarks on the advantage of smallness :—
"La plupart des calculs n'exigent que les minutes: les astronomes, 'les navigateurs, les militaires, les geographes, les arpenteurs, les archi'tectes, ont un besoin continuel de petites tables, bien plus rarement des 'grandes. Si Ton cherche les minutes dans un gros volume qui con'tient les secondes, on perd du terns. Le format de celui-ici n'exige 'rigoreusement que le terns necessaire a l'operation : d'ailleurs les vues 'basses ont de la peine avec les grandes tables; enfin plus le volume est 'mince, plus il est commode a l'usage ordinaire. Ainsi j'ai reduit 'celui-ci au pur necessaire.".