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If m, mi, be integer numbers, then it is easily proved that, (1 + v) i tmut V? + &c.

2

m'm' - I (1 tv)m' = i + mn'v +

w? t &c.

2

If these series be multiplied together, there results a series

It Av + Av? + A"' q + &c. in which

A=mt mt'
m2 + m'?—m + m + 2mm (m+m') (

mm)

[blocks in formation]

1.2.

(m+m') (m+m-1) (m+m->)

&c.

3 Now although the formulas for (1 + v)m (1 + over have been deduced on the supposition of m, m' being whole numbers, yet if such series as it mut &c. and it m'u &c. be mul. tiplied together, values of A, A', A”, will be precisely the same with regard to form, whatever 12, ni are, and consequently will be the same if », »' be fractions. · The actual multiplications,' says Mr. R. will end in the same powers of m and m', the same combinations of them, and the same numerals, whether we consider in and mi' as whole numbers or s fractions.'-By virtue of this property, if : +1v1.1.1-10+ &c. be assumed the 4th root of 1 tv; then, if we multiply this series into itself, the result will be 1 + 2+1 (-1) 1.2 + &c.; and if this series be multiplied into itself, the result will be

itivt!vit &c.

Itu

Eitv and consequently, the 4th root of 1 to 2 was rightly assumed.

A similar method may be used for the 1" root. Now, if this demonstration of the binomial has any claim to originality, it must be in the remark relative to the property of the coefficients A, A', A", &c. remaining the same in regard to form, whatever be the indices.- In the 19th volume of the Novi Commentarii Petropol. we have a demonstration of the binomial by Euler; and in the summary which always precedes the memoirs themselves, the reporter thus speaks of Euler's solution :

"itnx+&c. de quâ quidem serie constat, eam casu n numeri integri, equalem esse (1 + x), generatim, vero hujus serici valorem indicandum, indicat Illust. Auctor signo [n]. Quodsi nunc duo talia signa [m] [n] in se invicem ducantur, et productum exprimi supponatur per seriem

indicat mony: 1)S

1+ 4x + Bx? + &c. coefficientes A, B, C, per litteras met n determinari evidens est, patetque modum quo hæc determinatio sit, ab indole met n non pendere, ideoq. eundem esse sive m, n, supponantur integri, sin fracti. At si m, n, numeri integri, est omnino

[m] [n] = (1+x) min = [m+n] inde generatim quoque

[m] [n] = [m+n]” &c. &c. If we consult the memoir itself, we shall find that the me. thod of demonstration essentially rests on this principle: “ hic autem imprimis observari convenit, hanc compositiones rationem non ab indole literarum met n pendere, sed perinde se esse habituram, sive literæ met n denotent numeros integros, sive alios numeros quoscunque. Hoc ratiocinium non vulgare probe notetur, quonium ei tota vis nostræ demonstrationis innititur._We think that these extracts prove, beyond the possibility of a doubt, that the principle of Mr. Robertson's demonstration is precisely the same with that which was adopted by Euler thirty years ago. It is not our wish, on the sameness of the principle and the simplicity of the two demonstrations, to found a charge of plagiarism against Mr. R.: but he must abandon all claim to originality or priority of invention; which must be awarded to Euler.

The honor of the invention, however, does not deserve to be violently contested. Euler's demonstration, when reduced to plainness from his uncouth symbols, is not a direct demonstration : he assumes a series for the case in which the index is a fraction, and then proves that this series was rightly assumed: this is very different from assuming

(1+x)= itax+b*** &c. and then deducing b = a. " IC=b.

= &c.

.3 New Methed of computing Logarithms. By Thomas Manning, Esq.

* If there already existed (says the author) as full and extensive logarithmic tables as will ever be wanted, and of whose accuracy we were absolutely certain, and if the evidence for that accuracy could remain unimpaired throughout all ages, then any new method of computing logarithms would be totally superfluous so far as concerns the formation of tables, and could only be valuable indirectly, inasmuch as it might shew some curious and new views of mathematical truth. But this kind of evidence is not in the nature of human affairs. Whatever is recorded is no otherwise believed than on the evidence of testi

M 3

mocy; 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. we may in some cases satisfy ourselves of its accuracy, as well as by examining the whole.

! Among the various methods of computing logarithms, nane, 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 "+" subtractions will be accurate to an places of decimals.'

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

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a

a

a

again ;

' &c. and so on, suppose, to go"r"". Again; r"

B
B-

=gri &c. and so on.
B

3 Hence if we stop at all

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a

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qP &c.

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B2 q' &c. =

-1)

(B-1)3 Hence he log. *=5 bolaning tab. i to kolena {á + }

+ + &c.} +2{ $ +

gør + Rc.}

=5

1

1

+

2

+ h.pro

as *

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In the author's plan, a is 10, ß either 100 or 1000, &c. and r", supposing it to be the last remainder, is to be equal to i, followed by as many cyphers as the number of deci. mal places to which it is intended to work. Thus, suppose m = 1.0000 3141, then hol. = .0000 3141 if we work only to 8 figures. a and ß being taken powers of 10, it is plain that helemaan

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

, , -, 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 B: but it is obvious that we might employ a, 3, 7, &c. and make, generally, m subtractions with a, n with ß, s with y, &c.

If the quantities be greater than two, they must be reduced by division ; thus, if the bel. 17 were required, 17 = 2 = 2 (1.0625.) .. h.b. 17=4 h... 2 + b.l.(1.0625) and any lumber 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 Comp. at Jamaica. By Mr. James Robertson. - In Halley's ti, the variation of the compass at Jamaica was 64 degrees E, it is such at the present moment, and has remained the sal ever since the grants of land in 1660. This circuma stą is clearly and satisfactorily established by the following sin and brief account:

• I resided,

17

M4

• I resided (says Mr. R.) at Jamaica, as a King's Surveyor of Land, upwards of 20 years Disputes at law about boundaries of lands are there decided by ejectments, in the Supreme Court of Judicature, by the evidence and diagrams of King's surveyors of land. This is different from the practice in England, because the manner in which grants of land from the Crown are made, in the two countries, is different. In Jamaica, tu 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. By repeated resurveys these lines are kept up: and, when the cultivation, on both sides, renders it necessary to tell 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 increascd. 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 mortgage of lands in the neighbourhood where the disputed boundary is, a they muy 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. It is evident that tłs coincidence could not happen if any variation of the magnetial needle had taken place in the intermediate time elapsed between die making of the first, and of the last, survey. My business being ery 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 dis. covered 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 withı, and are parallel to, the lines and meridians delincaled on the new diagrams from recent surveys made by the magnetica! needle, of the same original marked lines on tarth, preserved as be

fore

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