mory; and such evidence weakens by the lapse of time, even while the original record remains; and it weakens on a twofold account, if the record must from time to time be replaced by copies. Nor is this destruction of evidence arising from the uncertainty of the copy's being accurately taken, any where greater than in the case of copied numbers.

It is useful then to contrive new and easy methods for computing not only new tables, but even those we already have. It is useful to contrive methods by which any part of a table may be verified independently of the rest; for by examining parts taken at random, wę may in some cases satisfy ourselves of its accuracy, as well as by examining the whole.

Among the various methods of computing logarithms, none, that I know of possesses this advantage of forming them with tolerable ease independently of each other by means of a few easy bases. This desideratum, I trust, the following method will supply, while at the same time it is peculiarly easy of application, requiring no division, multiplication, or extraction of roots, and has its relative advantages highly increased by increasing the number of decimal places to which the computation is carried.

The chief part of the working consists in merely setting down a number under itself removed one or more places to the right, and subtracting, and repeating this operation; and consequently is very little liable to mistake. Moreover, from the commodious manner in which the work stands, it may be revised with extreme rapidity. It may be performed after a few minutes instruction by any one who is competent to subtract. It is as easy for large numbers as for small; and on an average about 27 subtractions will furnish a logarithm ac curately to 10 places of decimals. In general 9x

will be accurate to 2n places of decimals.’

2

subtractions

The method employed may be thus stated: suppose x to be any number, then

In the author's plan, a is 10, B either 100 or 1000, &c. and r", supposing it to be the last remainder, is to be equal to 1, followed by as many cyphers as the number of decimal places to which it is intended to work. Thus, suppose r = 1.0000 3141, then h. .=.0000 3141 if we work only to 8 figures.

bl

a and being taken powers of 10, it is plain that b. l,

B

В

as x

&c. are readily calculated; and the operations, such

x

a

or x- are performed by the aid of the de

cimal notation with the greatest ease.

In our instance, we have only employed a, ß, and made six subtractions with a and two with : but it is obvious that we might employ a, ß, y, &c. and make, generally, m subtractions with a, n with ß, s with y, &c.

17

If the quantities be greater than two, they must be reduced by division; thus, if the b. l. 17 were required, 17 = 2+. 16 = 2* (1.0625.) .. b. l. 17 = 4 b. l. 2 + b. l. (1.0625) and any umber between 1 and 2 is easily found by the method above escribed.

The principle of this method of computing logarithms is ry simple, and the practice is both safe and easy. If the nstruction of new logarithmic tables were required, it would valuable; and to us it is estimable for the skill and ingeity with which it has been invented and constructed.

Observations on the Permanency of the Variation of the ComPat Jamaica. By Mr. James Robertson. In Halley's ti, the variation of the compass at Jamaica was 61 degrees E it is such at the present moment, and has remained the sal ever since the grants of land in 1660. This circum sta is clearly and satisfactorily established by the following sin and brief account :

M

I resided,

I resided (says Mr. R.) at Jamaica, as a King's Surveyor of Land, upwards of 20 years there decided by ejectments, in the Supreme Court of Judicature, Disputes at law about boundaries of lands are by the evidence and diagrams of King's surveyors of land. This is different from the practice in England, because the mauner in which grants of land from the Crown are made, in the two countries, is different. In Jamaica, to every grant of land a diagram thereof is annexed to the patent. This diagram is delineated from an actual survey of the land to be granted, having a meridional line, according to the magnetical needle, by which the survey was made, laid down in it. No notice is taken of the true meridian. The boundary lines of the land granted are marked on earth, (as it is denominated,) by cutting notches on the trees between which the line is run through the woods. These trees being mostly of hard timber, the notches will be discernible for 30 years, or more. surveys these lines are kept up and, when the cultivation, on both By repeated resides, renders it necessary to fell the marked trees, (which can only be done by mutual consent, it being otherwise death by the law,) logwood fences are planted in the lines dividing the properties thus cultivated and many of these fences have been regularly repaired, and kept up, to the present time. Lands were granted from the Crown soon after the Restoration, in 1650; and every succeeding year, the number of patents increased. The old estates have been often re-surveyed, and plans of them made, and usually annexed to deeds of conveyance, or mortgage, which must be enrolled, within a limited time, in the office of the Secretary of the Island; where, also, all the patents, and diagrams annexed to them, are recorded. In all disputes at law about boundary lines, where the keeping up of the old marked lines on earth has been neglected, surveyors are appointed to make actual re-surveys of all the old marked lines on earth, (preserved in the manner before mentioned,) and to extract from the Secretary of the Island's office, correct copies of all such dia grams annexed to patents, and to deeds of conveyance, or of lands in the neighbourhood where the disputed boundary is, a mortgage they may think necessary for the investigation thereof. They the compare the lines, and meridians, of these original diagrams wit those in their diagrams delineated from their own re-surveys recent! made; when it is always expected that the lines, and meridians, of th former will coincide with those of the latter coincidence could not happen if any variation of the magnetial It is evident that the needle had taken place in the intermediate time elapsed between te making of the first, and of the last, survey. My business being wry extensive, I was frequently applied to in disputes at law about boundry lines, and I had, besides, abundance of opportunities, on other surveys, to ascertain this fact satisfactorily. From all which I have discovered that the courses of the lines, and meridians, delineated on the original diagrams annexed to patents, from 1660, downwards to the present time, and of the re-survey diagrams thereof, annexed to deeds, coincide with, and are parallel to, the lines and meridians delineated on the new diagrams from recent surveys made by the magnetical needle, of the same original marked lines on earth, preserved as be

fore

fore described; so that whatever course is laid down for the line on the diagram annexed to the patent, (and let it be supposed, for example, to be north and south, or east and west,) upon setting the compass in the old marked line on earth, and directing the sights north and south, or east and west, according to the magnetical needle, the said marked 'line on earth, originally run by the magnetical needle 130 or 140 years ago, has been found by me to be exactly in the line, or direction with that of the compass; consequently no alteration of the variation could have taken place during the whole, or any part, of that period of time in Jamaica.'

Observations on the Variation and on the Dip of the Magnetic Needle, made at the Apartments of the Royal Society between the Years 1786 and 1805 inclusive. By Mr. George Gilpin.This memoir contains, perhaps, the most exact table of the variation of the compass that has been yet submitted to the public. During sixteen months, every day successively, and at several periods of time in each day, Mr. G. observed the compass. The hours were 7 A.M., 8 A.M., 10 A.M., 12 A.M., 1P.M., 2 P.M., 4 P.M., 6 P.M., 8 P.M., 10 P.M., and II P.M.-At 7 A.M. and 8 A.M., the daily variation was at its minimum: at 1 P.M. and 2 P.M. at its maximum; and at & P.M. at its maximum again. The daily observations are comprized in tables occupying sixteen pages.

A table is also given for the mean monthly variation:

Table IV. (says Mr. Gilpin) contains the differences for 12 years, viz. from 1793 to 1805, between the observations of the variation made in the months of March, June, September, and December, or at the times of the vernal and autumnal equinoxes, and summer and winter solstices; by a mean of these 12 years, the variation appears to increase or go westward, from the winter solstice to the vernal equinox o.80; diminishes or goes eastward from the vernal equinox, to the summer solstice 1.43; increases again from the summer solstice to the autumnal equinox 2'.43; and continues nearly the same, only decreasing o'.14, from the said equinox to the winter solstice.

'These differences at the times of the equinoxes and solstices have been noticed by M CASSINI, in his observations made at the Royal Observatory at Paris, between the years 1783 and 178, but the effect was considerably greater in his observations, than in those mentioned above; his results however were, in my opinion, drawn from too few observations, being from only 8 days observations about the times of the equinoxes and solstices, which differ considerably among themselves; and experience teaches us, that magnetical observations made for a period so limited are not sufficient for minute purposes: I have therefore, in the results here given, taken the mean of the observations made during the whole month in which the equinoxes and solstices fall, which appear to me likely to furnish results more satisfactory; and all the foregoing observations are to be considered

as the results or mean of a great many, by way of arriving at greater accuracy than could be obtained without; this, however, was found to be more necessary at some times than at others; sometimes, the needle would be extremely consistent with itself, so as to return exactly to the same point, however often it might have been drawn aside; at other times it varied 2 or 3', sometimes S, to', or even more; this uncertainty in the needle arises principally, I believe, from changes in the atmosphere, for a change of wind, from any quarter to another, almost always produced a change in the needle from steady to unsteady, and vice versa, but it was generally more unsteady with an easterly wind, than when it blew from any other quarter, and most steady whes the wind was south or south westerly. An Aurora Borealis always produced considerable agitation of the needle.'

The variation of the compass at present, according to Mr. Gilpin, is 24° 8' West, and its annual increase is exceedingly small; so that we may conjecture the needle to have reached its greatest westerly declination.

Mr. Gilpin is eminently intitled to the gratitude of the philosophic world, for the care and assiduity bestowed on this subject, and (as it should seem) for the great nicety with which bis observations were made..

On the Declinations of some of the principal fixed Stars; with a Description of an Astronomical Circle, and some Remarks on the Construction of circular Instruments. By John Pond, Esq.With a circle constructed by Mr. Troughton, Mr. Pond made observations on the declinations of those stars, the right ascensions of which are observed at Greenwich; and his labours on this subject may be useful to Astronomers.-He suggests an ingenious and a very simple mode of examining the latitudes of places. If the declinations of the same Stars examined at different places do not agree, then the latitudes of the places of observations must have been inaccurately determined; and a correction therefore will be requisite. After having stated his method, and the corrections proposed to be made to the colatitudes of Greenwich, Armagh, Palermo, and Westbury, he makes the following observation, which well merits attention;

I consider this comparison as interesting likewise on another account; it is an object deserving of curiosity to examine the present state of our best astronomical instruments, and to ascertain what may reasonably be expected from them. The superiority of circular instruments is, I believe, too universally admitted, to render it probable that quadrants will ever again be substituted in their place. But the Greenwich quadrant is so intimately connected with the history of astronomy, the observations that have been made with it, and the deductions from those observations, are of such infinite importance to the science, that every circumstance relating to it cannot fail of being interesting,