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405

Journal of a Voyage to the Hebrides, &c.

406

security in the moment of alarm, is Monday, 4th.–At eight in the mornuncertain; their forms are generally | ing we set sail, with a very faint breeze, similar, inclining to the oval, and but the tide of flood carried us fortheir situations best adapted for de- ward at a very considerable rate. Passfensive operations. We afterwards ing the end of Scarba, we obtained a visited the remains of a church, but it distant view of Colonsa and Oranhad nothing remarkable about it: one sa, and the back of Jura, with one of grave-stone has a large claymore (the the Paps, rising to a great degree of emblem of a chieftain) engraven upon elevation. it, and is said to be the burial-place of To the north-east, were seen the coone of the M‘Leans, of the neighbour-nical hills about Bun Awe; and faring island of Shuna. The others were ther to the northward, the towering merely plain flags, which served to height of Ben Murs, and the vast point out where

mountains about Fort William and the “Each in his narrow cell, for ever laid,

Linnhe Loch. Close on our right, was The rude forefathers o'the hamlet sleep.".

the small island of Eysdale, famous This island, like most of the Hebrides, for its fine slate quarries; and to the is destitute of wood, though traces of left, the rugged cliffs of Mull. its having borne oak-trees of consider- At noon, we entered the sound with able magnitude were frequently dis- a favourable breeze, which followed cernible; and, consequently, its pre- us as we passed round the point. On sent open state does not arise from any the left soon appeared Castle Dowart, local incapacity, but merely from the situated upon an abrupt cliff. Its inattention of the inhabitants to the mouldering walls and buttresses fringprofitable, as well as necessary, busi-ed with ivy, formed a pleasing object, ness of planting, and from their too finely backed by the gloomy hills in the great aptitude to be discouraged by interior of the island. This castle was imperfect attempts.

formerly the seat of M‘Lean, but is in the formation of a large planta- now garrisoned as a fortress. On the tion, great care is necessary to guard right, are the bold craggy shores of the young trees as much as possible Morvern, the country of the celebrated from the bleak sea-breezes, which, in Fingal. these unsheltered islands, blow with After passing the small remains of great severity. To effect this, it is the castle of Ardlornish, (formerly a requisite to circumscribe its limits, by seat of the Earls of Ross) which stands a broad margin of such trees as are on a small headland, jutting out from rapid in their growth; and it would be the coast of Morvern, the sides of the still better to do this a year or more sound become tame and uninteresting, previous to the intended plantation. rising with a gradual and uniform

Thus the inner plants, from being slope on either hand. At times, insheltered while young and tender, will, deed, the smoke of the kelp-fires as they increase, mutually protect each yielded a pleasing variety, breaking other, and at length attain a degree of the formality of the nearer outlines, luxuriance equal to those of a milder and lightly diffusing itself over the disclimate. This is evident in several tant parts of the strait. parts of the Highlands of Scotland; As we proceed, however, it imthe ancient forests yet remaining, and proves on the side of Mull, the emi. the modern ones, formed by the atten- nences swell into greater grandeur, tion of the dukes of Argyle and other and the contour becomes more broken noblemen, are proofs sufficient. Nor and diversified: but still we vainly can it be urged as an excuse for not look for the beauties of vegetation; planting, that it occupies too much no woods either wave on the summit of land; since some trees, and particu- the bold headland, or adorn the side of larly the Scotch fir, will grow in places the sweeping vale: heath, small patches that are totally useless for every other of pasture, and naked rock, form the kind of vegetation: and the rocks only varieties of the surface, and about Dunkeld, are at once a monu- spread their monotonous tints on every ment of the effects of persevering in- side. dustry, and a general reproach to the This sound has doubtless heretofore proprietors of the present barren been a place of great importance, since islands, and uncultivated mountains of it formed a grand inlet to the Danes the Highlands.

and Norwegians to the interior coas

66

MAGAZINE.

of the Highlands; and, most probably, master's attention; and that it had so has frequently witnessed the generous operated upon his curiosity, as to instruggles of freedom with rapine and duce him to open it, and read the ambition. In the short run of 18 love-told story of the amorous footman. miles, we counted five castles or forti- It was in vain that James pleaded to fied houses, though these we may per- have it restored. “ No, no," said Mr. haps consider of more modern con- Wortley, you shall be a great man, struction than the times to wbich we James, and this letter shall be publishhave above alluded.

ed in the Spectator.” It was accordIn the afternoon we arrived at Tober ingly shortly afterwards communicated Morey, in Mull, the best harbour in to Sir Richard Steele, and printed as the sound. The present town was it had been actually written. But alerected by the society instituted for though James was unfortunate in his the improvement of the Highlands; it letter, he was not finally unsuccessful is not yet completed, but some public in his attention to the person to whom buildings and several houses are erect- it was addressed. He soon found ed, on a plan, plain, neat, and uni- means to remove that cruelty of which form ; and, along the front, an exten- he complains in this epistle; but besive quay or wharf has been formed, at fore their wishes were completed, the a considerable expense. The ground death of Betty put a period to their rises extremely from the front range mutual prospects of felicity. of houses, and the remainder of the town is situated on a considerable eminence, which overlooks the harbour

TO THE EDITOR OF THE IMPERIAL and sound.

At this place, another ship belonging to the Spanish armada was burnt,

DEAR SIR, but whether by accident or design is When the Cube Root of a number is uncertain.

to be extracted to many places of (To be continued.)

figures, the common rule is always found very prolix and tiresome; on

this account, several approximating ANECDOTE.

rules have been devised by eminent In vol. 1st of the Spectator, No. 11, mathematicians, to abridge the labour there is a letter beginning with Dear attendant on the common process: the Betty," which found its way into that best of these is the one recommended publication, under the following cir- by Dr. Hutton, and used in his “ Course cumstances :

of Mathematics,” and other works. The real author of this letter was About ten years ago, I discovered a James Hirst, who, in 1711, lived as a very useful improvement of the comservant with the Hon. Edward Wort- mon rule, which was published in the ley. The epistle was intended for a Imperial Encyclopædia, and which person with whom he was passionately several persons, who had been accusin love; but, unfortunately, instead of tomed to the old rule, have employed falling into the hands of his mistress, with much satisfaction, finding that the it reached the Spectator in the follow- operations were performed by it with ing manner.

much greater facility than by the usual It happened one day, while James method. On considering the subject was delivering a parcel of letters to his lately, it appeared to me that the promaster, that, through mistake, he gave cess might be still further simplified ; his own, which was intended for “ dear and this probably will be manifest to Betty;" and, without adverting to the others in the application of the followfact, kept back one of his master's in ing rule, which is doubtless by far the its stead. On making a discovery of easiest yet published, being much betthis mortifying blunder, he hurried ter adapted to practice, even than that back to his master, delivered up the already mentioned by approximation. letter he had detained, and solicited If you judge that this lucubration the return of that which was designed will be acceptable to your numerous for the inspection of other eyes. James, readers, I shall be pleased to see it however, had the misfortune to learn, inserted in the pages of the Imperial that this ill-fated epistle had been Magazine. among the first which had arrested his

Thos. ExLEY.

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409 A new Rule for extracting the Cube Root. 410 A new, easy, and very concise Rule, for number, with its annexed figure, add extracting the Cube Root of Numbers. double that annexed figure for a new

1. Make a point over the place of reserved number, to which annex the units, and over every third figure from new quotient figure; multiply and set it, to divide the number into periods ; down as before, and add the product find the nearest less cube to the first to the two lines over it, together with period, which subtract from it, plac- the above-mentioned square, for the ing the root in the quotient; and bring next true divisor :--repeat the operadown the next period for a dividend. tions as far as necessary according to

2. Put down three times the quotient the foregoing directions, bringing down figure for a reserved number, which periods of ciphers for decimals where multiply by the quotient figure, for a requisite. trial divisor, and find how often it is Note.- When the divisor is not concontained in the dividend ; neglecting tained once in the dividend, a cipher units and tens, for the next quotient is annexed to the reserved number, figure, which annex to the reserved two to the divisor, and a period of number, and multiply the result by it, three figures or ciphers to the remainplacing the product under the trial der or dividend; also, in trying for the divisor, two figures farther to the right, quotient figure, some allowance is to and add for the true divisor, which mul- be made for the increase of the trial tiply by the quotient figure, and sub-divisor, to form that of the true one, tract, as in division; bring down the as in the common rule. next period, and try as before, by the cially to be noticed in trying for the last divisor, for the next figure of the second and third figures, since in these root.

the proportionate increase is greatest ; 3. Over the product of the reserved thus, the third divisor is to be connumber, place the square of the an- sidered as something more than the nexed figure; also to the reserved second, with the line above it added. EXAMPLE 1. Let it be required to extract the Cube Root of 122615327232.

122615327232(4968 Reserved No. 4881

64

This is espe

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The different parts of the operation | it, from the place under which it in this example will be easily under- stands, for the divisor; and from the stood, by comparing them with the remainder cut off one figure less than rule. The superposited square num- those omitted in the old divisor, and bers have a mark over them, to shew work by the method of contracted they do not belong to the divisors after division. which they are placed.

The figures employed in the followWhen the cube root of a number is ing example, are about 480, and are required to many places of decimals, all that are necessary to be put down having formed half of them, the rest in the whole work, which gives the may be obtained by the rule of con- answer true to the last figure. The tracted division of decimals. Thus, rule by approximation, already noin example 2d, having found the first ticed, is the best of those which have ten figures of the root, try for the next come into general use; but even this figure, which here is 8; put down in rule, in order to obtain the answer its proper place the last figure of the true, to the same extent as in our exproduct of this quotient figure, and ample, will require nearly 2000 figures, the reserved number, (which here is or at least four times as many as are 3,) and add it and the two lines aboye | here introduced.

Ex. 2. Let it be required to extract the Cube Root of 2, true to 20 places of

figures.

2.000(1.2599210498948731647 304

1

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3

42615678063, 61919351 4762203152

4518052846
232070006
41581880
3484255
150713

7847
3085
228
37

4 If one figure more than the half of those required in the root be found by the rule, the increase or addition to the last divisor will be unnecessary, as in the following example.Ex. 3. Let it be required to extract the Cube Root of 22, true to 10 places

of figures.

22.000(2.802039330 1264

8

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21977, 6074573
778
71

0 These examples will be sufficient to elucidate the Rule, and to shew its

advantages.

TO THE EDITOR OF THE IMPERIAL

413
Cube Root.Buffer's Singular Wager.

414 The truth of the Rule may be easily

Buffier's Singular Wager. shewn by the resolution of a binomial cube, from which it will appear, that as the work proceeds, the greatest even

MAGAZINE, cube is always taken from the periods Sir, of the given number as far as the work That there is a something in us which has been extended.

we call Liberty, or Free-will, is as Thus, if two figures of the root be certain, as that we possess an immatefound, let the value of the first be de- rial principle which we call our Soul ; noted by 10 a, and that of the second and though there are persons who by b, then it will appear that a’ is taken espouse a contrary opinion, according from the first period, according to the to whom, man is nothing but a curious first article of the rule; or, which machine, and of course no more acamounts to the same, 103 a’ is taken countable for his actions than a windfrom the two first periods. Again, the mill; yet the conduct of these men trial divisor (units and tens being sup- is ever found to be at variance with plied) is 3.102 a”, and the product the principles they profess: they are arising from the reserved number is pleased or offended with the actions of (3.10 a + b) Xb=3.10 ab + b2 ; which others, and praise or blame them, just being added to the trial divisor, as the as those do who believe themselves rule directs, becomes 3.102 a2 + 3.10 and the rest of their species to be free ab + b2 ; and this, multiplied by b, agents. From this circumstance it is gives 3.102 a2 b+3.10 ab? + 63, which, justly inferred, that a consciousness of according to the rule, is taken from our freedom is as deeply rooted in our the remaining part of the two first pe- nature, as the love of being itself; and riods; hence, in the whole has been if it be possible entirely to erase it, it subtracted, 103 a3 + 3.102 a2 b + 3.10 can only be done by unnatural vioab2 + b3, or the cube of 10 a+b, that lence. The arguments against the freeis, the cube of the two first figures of dom of human actions, are of the same the root : and this is the greatest cube complexion as the arguments of cercontained in the two periods b, being tain philosophers, against the existtaken the greatest number the opera

ence of matter and motion, and should tion will allow. Now, suppose another be treated in the same way; that is, figure found; then to the former re- opposed by an appeal to matter of fact. served number, with its annexed If we refer to our inward consciousfigure, viz. 3.10 a +b, 2 b is added, ness, we shall find, that we are but making 3.10 a + b for a new reserved little less certain that we are free, than number, which is three times the two that we think; nor do I doubt, that if first figures of the root; also having we could but fathom the bottom of added according to the rule, as part their minds, who advocate the docof the next divisor, the former divisor, trine of“ fixed fate,” we should find a 3.10^a+3.10 ab + bo, the former pro- latent, though to themselves unnoticed, duct by the reserved number, 3.10 ab + consciousness that man is free; and be, and also b2, we have 3.102q2 +6.10 which, in certain circumstances, would ab + 3 b?, or 3.10 a + b x 10 a +b, actual expedient, something like the

discover itself even to themselves. An which is the new reserved number, multiplied by the two first figures of the foliowing, would probably verify the root. Hence the new reserved num

correctness of my views.

“ You say that I am not free,” says ber, and that part of the new divisor which corresponds with the former trial Buffier, addressing himself to a Necesdivisor, are found from the two first sitarian, “ and that it does not depend figures, as those were from the first

on the mere determination of my will figure of the root; and the subsequent and choice whether I shall move my

hand or not. If that be the case, it procedure is as in the former case. Hence, in the result, the greatest even

must necessarily be decreed, that cube in the first three periods is taken within a quarter of an hour hence, I from them: in advancing, there is only either shall or shall not raise my hand a repetition of the work, and the proof thrice successively: I cannot, thereis the same; and therefore the truth of fore, alter this necessary determinathe Rule is demonstrated.

tion. This being supposed, in case I

lay a wager on one side rather than Bristol, June 8th, 1819.

the other, I can be a winner only

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