Edward Sang on his tables
These computations were designed and undertaken with the view to the change from the ancient subdivision of the quadrant to the decimal system, a change long desired, and destined inevitably to be made. One hundred years ago it was on the very point of being completed. Mathematicians were then engaged in the introduction of the decimal system into every branch of calculation and measurement; but for the introduction of this new system into the measurement of angles, it was necessary to have a new trigonometrical canon. The French Government deputed M Prony, with a large army of computers, to compile this new canon, and astronomers awaited with impatience the advent of this indispensable preparative. Laplace had, in anticipation, reduced all his data in the Mécanique Céleste to the new system, and instruments had been graduated suitably.
We can hardly doubt but that if this new canon had then been published, the decimal graduation of the quadrant would have been very generally adopted even at the beginning of the present century; by the end of the first decade of this century it might indeed have been universally adopted. But the new trigonometrical tables, though magniloquently described, never made their appearance ; and thus for something like seventy years the progress of the sciences thereon depending has been impeded.
Very few are old enough to remember the disappointment felt throughout the scientific world. About 1815, in our school, the boys were exercised in computing short tables of logarithms and of sines and tangents, in order to gain the right to use Hutton's seven-place tables; and well do I recollect the almost awe with which we listened to descriptions of the extent and value of the renowned Cadastre Tables.
In 1819 the British Government, at the instigation of Gilbert Davies, M. P., approached the French Government with a proposal to share the expense of publishing the Cadastre Tables, and a commission was appointed to consider the matter. The negotiations, however, fell through, for reasons which were never very publicly made known; but in the session 1820-21 the rumour was current amongst us students of mathematics in the University of Edinburgh, that the English Commissioners were dissatisfied of the soundness of the calculations - and so it was that the idea of an entire recalculation came into my mind.
In the year 1848, encouraged by the acquisition of a copy of that admirable work, Burckhardt's Table des Diviseurs up to three million, the idea took a concrete shape in my mind, and I resolved to systematise the work which before I had carried on in a desultory way. Necessarily the first step was to construct a table of logarithms sufficiently extensive to satisfy all the wants of computers in trigonometry and astronomy; and having many times felt the inconvenience of the loss of the details of the calculations made on separate papers, I resolved to record from the very beginning every important step. This plan of operation has many conveniences - it enables us to retrace and examine every case of doubt, and also to take advantage, in new calculations, of anything in the previous work which may happen to be applicable.
For all the ordinary operations of surveying and practical astronomy five-place logarithms, as M Lalande has stated, are perfectly sufficient; and for the higher branches of astronomy and geodetics the usual seven-place tables are enough. But for the purpose of constructing new working tables it becomes necessary to carry the actual work further, both in the extent of the arguments and in the number of decimal places, and therefore I determined on the formation of a table of logarithms to nine places for all numbers up to one million. But again, in order that such a table be true to the ninth place, the actual calculation must be carried still further - and to meet the cases in which the doubtful figures from, say, 4997 to 5003 might occur in one million of cases, it became prudent to carry the accuracy even to the fifteenth place. And this limit of accuracy was further defined by the circumstance that there the differences of the third order just disappear. Even then it may happen that the doubt as to the figures which are to be rejected may not be cleared up, and it follows that a still more minute criterion should be at hand for use, and therefore the order of the work came to be as follows.
In the first place, the computations of the logarithms of all numbers up to ten thousand, to twenty-eight (for twenty-five) places, was undertaken. At the outset each logarithm of a prime number was computed twice, but as the work proceeded, it was judged advisable to have three distinct computations of each. The whole of this work is distinctly recorded and indexed, so that every step in reference to any given number can at once be traced out.
The idea was entertained of this work being ultimately extended to one hundred thousand, and the logarithms of the composite numbers from ten to twenty thousand were computed, spaces being left for those of intermediate prime numbers.
By the addition of the logarithms thus obtained, those of the great majority of composite numbers from the limit one hundred thousand to one hundred and fifty thousand were computed, and the intervals were filled up by help of second differences. In this part of the work I was aided by my daughters. But, in all such separate additions, we are liable to sporadic errors, and in order to guard against these the whole of this work was redone by the use of the last two figures of the second differences; and thereafter the calculations were made by short interpolations of second differences an the way to three hundred and seventy thousand. Necessarily, on account of the occurrence of the minute final errors, the last, or fifteenth, figures cannot be trusted to within one or two units; and after a very severe examination of the whole, it was found that in a very few instances this accumulation of last-place inaccuracy extended even to five units; and thus we are warranted in expecting that no last-place error will be found reaching so far as to a unit in the fourteenth place - a degree of accuracy far, very far, beyond what can ever be required in any practical matter.
In the compilation of the trigonometrical canon the same precautions were taken for securing the accuracy of the results. In the usual way, by means of the extraction of the square root, the quadrant was divided into ten equal parts, and the sines of these computed to thirty-three, for thirty places. These again were bisected thrice, thus giving the sine of each eightieth part of the quadrant; all the steps of the process being recorded.
The quinquesection of these parts was effected by help of the method of the solution of equations of all orders, published by me in 1829; and the computation of the multiples of those parts was effected by the use of the usual formula for second differences. A table of the multiples of 2 ver. 00c 25' [note that Sang is using 1c to denote the 100th part of a right angle] was made to facilitate the work, and the sines, first differences, and second differences were recorded in such a way as to enable one instantly to examine the accuracy. The same method of quinquesection was again repeated, and the computation of the canon to each fifth minute was effected by help of a table of one thousand multiples of 2 ver. 00c 05', the record being given to thirty-three places, the verification being examined at every fifth place. In this work there is no likelihood of a single error having escaped notice.
For the third time this method of quinquesection was applied in order to obtain the sines of arcs to a single minute. A table of one thousand multiples of 2 ver. 00c 01' was computed to thirty-three places, but in the actual canon it was judged proper to curtail these, and the calculations were restricted to eighteen decimals on the scroll paper. In the actual canon as transcribed, only fifteen places are given. In all cases the function, its first difference, and its second difference are given in position ready for instantaneous examination; and the whole is expected to be free of error excepting in the rare cases where the rejected figures are 500 - these cases being duly noted.
For the computation of the canon of logarithmic sines the obvious process is to compute each one of its terms from the actual sine, by help of the table of logarithms; but this process does not possess the great advantage of self-verification, and attempts have been made to obtain a better one. Formulae indeed have been given for the computation of the logarithmic sine without the intervention of the sine itself, but when we come to apply these formulae to actual business we find that they imply a much greater amount of labour than the natural process does; and, after all, they are only applicable to the separate individual cases.
Napier, as is well known, arranged his computations of the logarithms from the actual sines in such a way as to lessen by one-half the amount of the labour. Napier's arrangement was therefore followed, and the work was begun from the sine of 100c down to 50c. The calculations were made by help of the fifteen-place table of logarithms from 100,000 to 370,000. If this table had been continued up to the whole million, the labour would have been greatly diminished, but we had to bring the numbers to within the actual range of our table by halving or doubling as the case might be. The results were then tested by first, second, and third differences, and in not a few cases the computation had to be redone, for the sake of some minute difference among the last figures. The log sines for the other half of the quadrant, that is from 50c to 0c, were deduced from the preceding by the use of first differences alone. The log tangents from 50c down to 0c were also deduced directly by help of the first differences alone. In this way the series of fundamental tables needed for the new system has been completed, so far as the limit of minutes goes.
While that work was in progress, a circumstance occurred which temporarily changed the order of procedure. Kepler's celebrated problem has ever since his time exercised mathematicians, and, sharing the ambition of many others, I also sought often, and in vain, for an easy solution of it. Accident brought it again before me, and this time, considering not the relations of the lines connected with it, but the relations of the areas concerned, an exceedingly simple solution was found. In order to give effect to this method it was necessary to compute a table of the areas of circular segments in terms of the whole area of the circle. That again rendered it necessary to calculate the sines measured in parts of the quadrant as a unit, instead of in parts of the radius, as usual. This computation was effected by using the multiples of twice the versed sine formerly employed. From this again the canon of circular segments for each minute of the whole circumference was readily deduced. The mean anomaly of a planet may be deduced from its angle of position, or as it is generally called, its excentric anomaly, by simple additions and subtractions of these circular segments. The converse problem is very easily resolved, particularly when the first estimate is a tolerably close one. In order to be able promptly to make this first estimate sufficiently near in every possible case, a table of mean anomalies from degree to degree of the angular position, and also from degree to degree of the angle of excentricity of the orbit, has been computed according to the decimal system.
The change to this system is inevitable. Each new discovery, each improvement in the art of observing, intensifies the need for the change, at the same time that each augmentation of our stock of data arranged in the ancient way adds to the difficulties. How much the change is needed may be estimated by an inspection of the Nautical Almanac. Every page in it cries out aloud in distress, 'Give us decimals.' For the sun's meridian passage, the usual difference columns are suppressed, and those titled 'var. in 1 hour' are substituted; and similarly for the moon's hourly place a column titled 'var. in 10m' is given; while for the interpolation of lunar distances, proportional logarithms of the difference are given. While artisans and physicists are using the ten-millionth part of the earth's quadrant as their unit of linear measure, astronomers are still subdividing the quadrant into 90, 60, 60, and 100 parts. The labour of interpolation is unnecessarily doubled at the very least, and that heavy burden is laid on the shoulders of all the daily users of the ephemeris. The trouble attending the reduction of observations tends to lead the navigator to shun the making of observations. The matter is not merely of national, it is of cosmopolitan interest - and this continuous waste of labour has much need to be ended.
The collection of computations above described contains all that is essentially needed for the change of system, as far as the trigonometrical department is concerned; the great desideratum being the Canon of Logarithmic Sines and Tangents. In addition to the results being accurate to a degree far beyond what can ever be needed in practical matters, it contains what no work of the kind has contained before, a complete and clear record of all the steps by which those results were reached. Thus we are enabled at once to verify, or, if necessary, to correct the record, so making it a standard for all time.
For these reasons it is proposed that the entire collection be acquired by, and preserved in, some official library, so as to be accessible to all interested in such matters; so that future computers may be enabled to extend the work without the need of recomputing what has been already done; and also so that those extracts which are judged to be expedient may be published.
Seeing that the Logarithmic Canon is useful in all manner of calculations, the printing Of the table of nine-place logarithms might be advantageously proceeded with at once. The publication of the corresponding Canon of Logarithmic Sines and Tangents would only be advisable in the expectation of its early adoption by astronomers.
But land-surveyors, when transporting the theodolite from one station to another, have to compute the new azimuth from the previously observed one. This is easily done by adding or subtracting 180c; yet in the hurry of business this occasionally gives rise to mistakes. On the other hand, with 400c on the azimuth circle, we should only have to add or subtract 200c, thus almost obviating the chance of a mistake. Hence the surveyor would be greatly benefited by the immediate publication of a five-place trigonometrical canon, arranged in the decimal way.
JOC/EFR August 2006
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