fifth step with the previously computed test values. In this way immunity from error was obtained, excepting in the last places, where small errors are inevitable.

In performing the multiplication sinna.2 vera stopping at the thirty-third place, the last figure of each partial product may err in excess or in defect by; now it is possible, though not likely, that all of these errors may be on one side, and therefore there is a possible error in the last place of the total product, of half as many units as there are lines in the multiplication. By using the table of multiples up to one thousand, we reduce the number of lines to one third, and therefore the possible amount of residual error in the same ratio; so that the auxiliary table both saves labour and augments the exactitude of the result.

These final-place errors were corrected at each fifth step by altering the last figures of the second differences, and thus the accumulation of those errors was prevented. In the whole calculation of the sines of the quarter degrees, it was not found necessary to alter any one second difference to so much as the limit of possible error, and therefore we may hold that the manuscript table of the sines of these arcs is absolutely correct to within the prescribed degree of precision, namely, unit in the thirtieth decimal place.

The next quinquisection, conducted in the same way, gave the sine and cosine of 5'; the functions of whose multiples were obtained as before, and compared, at each fifth step, with the previous work. The table of sines to every fifth minute is already well advanced.

The third quinquisection gave the sine and cosine of the single minute of the decimal division. A table of the multiples of 2 ver 1' has been constructed up to 1000, and has been used in forming a good beginning of the canon of sines to each single minute.

For the purpose of preventing all error in the record of these calculations, the second differences only were copied from the duplicate scroll calculations, and the successive first differences and sines were thence recomputed on the record sheet. Since any error in copying, or even in the original computation, was necessarily continued and extended into the after part of the record, its detection was rendered certain, so that the recorded results may be implicitly relied on. To make the record more secure, each page

was copied on thin transfer paper, on which no alteration can be made without being obvious.

The method of computing might have been extended to differences of the fourth order, in which case the common multiplier, 4 ver a2, would have been much smaller, and the terms of the products fewer. But, on the other hand, the entries in the record would have been more, and the effects of the residual errors would have been much greater and more troublesome in correction. For the interpolation of each quarter degree, the saving obtained by the use of fourth differences would have been unappreciable; for those of each fifth and of each single minute, it would have been hardly such as to compensate for the inconveniences just mentioned; but for the future interpolations of each tenth second and of each single second, the fourth differences may be advantageously used.

When the canon of sines shall have been completed, the computation of the working table, that of logarithmic sines, will be easy, particularly by help of my fifteen-place table of logarithms.

Beginning, as John Nepair did, at the sine of the whole quadrant and proceeding downwards, the computed logarithmic sines may be verified by their differences, which are small. When the work has been brought down to the log sin of the half-quadrant, the farther progress is easy and rapid, for the formula

2 sin a. cos a = sin 2 a

gives log sin a = loglog sin 2 a log cos a, so that the differences of any order may be got at once from the previously tabulated differences of that order, and, what is most worthy of remark, may be used without the fear of an accumulation of residual error.

The table of logarithmic tangents follows as a matter of course.

3. On the Precautions to be taken in recording and using the Records of Original Computations. Edward Sang, Esq. The real utility of tables of numerical results is only secured by making them accessible to those computers who may require them;

and the essence of their utility lies in this, that the labour of a single computer saves that of many others.

It is indispensable that those who use the tables be able to rely implicitly on the accuracy of the tabulated numbers, and that they have a ready means of detecting any error should the existence of one be suspected.

I do not mean, at present, to say a word on the mechanical arrangements of setting up the types, of stereotyping, revising the proofs, and printing; these have already often been discussed, but I shall take the matter up at this critical point:-The investigator has in his hands a set of printed or of manuscript tables, which he means to use in his researches, and he wishes to know whether the individual book be or be not to be trusted. His confidence must necessarily be influenced by the history of the book and by its corelatives. Thus, if it be a stereotype copy of a work in extensive circulation, he may accept the general opinion as to its accuracy; but if the table be one seldom used, such as those which serve as the foundation of working tables, this source of confidence fails him. The nature of the case may be most clearly seen from an example :

We propose to extend the logarithmic canon beyond the limits to which it has been already printed; this extension must be founded on the logarithms of the prime numbers; now Abraham Sharp computed, to 61 places, the logarithm of every prime number up to 1097; these were printed in Sherwin's collection, and thence reprinted by Callet in his Tables Portatives; shall we then build our more extensive tables on the computation by Sharp? Sharp was known as a most zealous and careful computer; both Sherwin and Callet would take care that the numbers be correctly copied; yet for all that, we cannot venture to found on Sharp's work because there is an essential omission.

If we were to proceed to compute, by help of these, the logarithms of larger primes, and if, after a lengthened series of operations, we were to find a disagreement, we should be left in doubt as to which of the many logarithms that had been used may be in fault; we should have to recompute such of Sharp's logarithms as might be implicated, while the labour and irksomeness of the search would become intolerable.

In all such calculations we seek to arrive at the result by two independent processes. All the use, therefore, that can be made. of Sharp's tables, is to hold his work as constituting one of these processes; a great use certainly, yet, at best, only half of what it might have been.

Now, in the computation, to twenty-eight places, of the logarithms of the prime numbers, no error whatever was discovered among those given by Callet; so here we have an instance of records, in themselves quite exact, and yet insufficient to obviate subsequent recomputation.

The means of readily verifying the record are awanting; these means must necessarily vary with the nature of the tabulated functions.

In the volumes containing the computation of the logarithms to twenty-eight places of all primes below ten thousand, which was laid before the Society, the articulate steps of every calculation are recorded and indexed, so that if an error be suspected in any one logarithm, we have the means of instantly verifying the table, or of detecting the source of the error. Had such a record accompanied Sharp's admirable table, the need for subsequent re-calculation would have been entirely obviated.

The vast majority of tables have their arguments arranged with equal differences, consequently the functions progress gradually; and, for the most part, these tables have been constructed by help of differences. It is then sufficient to record the differences along with the values of the functions. For the canon of sines I have found it convenient to place the first difference, with the sign +, and then the second difference, with the sign, below the preceding sine, as shown below:

:

This arrangement enables the computer to examine any sin which he wishes to extract, so as to guard against any typographical error; and, if the table of the multiples of 2 verl' were appended, to check readily the computation itself. When, however, the differences have only a few figures, the ordinary method, of placing them in separate columns, is to be preferred; it saves room, while the practised calculator has no difficulty in performing the requisite additions or subtractions. The utility of this arrangement has been long recognised.

In the case of tables for common use, which are, in general, abbreviations of original calculations to a greater number of places, it is enough to give the first differences when these vary much.

When such means of verification have been provided, the user of the table can make sure that the number which he extracts contains no error; and if all users were to habitually make this examination, printed tables, and above all those printed from stereotype, would be gradually freed from errors of press.

4. On an Unnamed Palæozoic Annelid. By Professor Duns. (Plate IV.).

CYMADERMA* (nov. gen.).—Generis Characteres:-Corpus cylindricum, elongatum, nudum, striatum; striæ tenues, in ordinem undi latæ, et ubique corpus cingentes, ita ut cutis subrugosa videatur; linea dor sualis continua, alternasque cavaturas ovatus et vertices ostendens; nulla verorum articulamentorum indicia; vestigium tortuosum, bisulcum.

CYMADERMA (new genus).—Generic Characters:-Body cylindrical, elongated, naked, striated; striæ minute, waved symmetrically, encompassing the form in all parts, and giving to the skin a subrugose appearance; a continuous dorsal line, showing alternate oval depressions and slight ridges; no traces of true articulations; track, tortuous bisulcate.

This annuloid fossil was obtained from upper Carboniferous strata, in a cutting on the Midland Railway, valley of the Ribble, near Settle, Yorkshire. Peculiarities characteristic of the striation and the median dorsal line led me to conclude that the form was probably rare, or one that had not been described. It was thus of importance to get the other part of the slab on which the impres

Etymol. kuua, unda, dépμa, cutis.