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unquestionably, exists in the solution, in passing to the state of deutosulphate, deposits its base, which gives up its oxygen and acid, to form the new salt. It is evident that the revival of the copper may be effected, in this manner, without the assistance of any iron; and, in fact, there are no traces of that metal in the interior of the tub. It is not, however, this part of the phenomenon that appears to me most remarkable, but the cohesion, acquired by the copper, so precipitated, from the midst of a solution: a cohesion, which is so great as to allow the metal to be hammered in the cold, and reduced to thin leaves; and whose specific gravity is equal to that of fused copper, viz. 8.78. I have, moreover, filed a morsel of this copper, and have produced a surface as brilliant, and free from pores, as could have been obtained, by similar means, with an ingot of common copper. M. CLEMENT.

PROVINCIAL COPPER TOKENS.

Sir, I have often regretted that the issuing of provincial copper tokens was discontinued. I know not whether they were suppressed by law, as was the case with the provincial silver, a few years ago. Their continuance would have given employment to many persons, as die-sinkers and others connected with the subject, besides giving an opportunity for im provement in the art of design and beanty of execution. It has often occurred to me, that tradesmen would do well to have tokens struck, if not issued as money, to be given in lieu of cards of address. Such a token might contain, on one side, a view of some picturesque object (and there are very few towns that do not afford some beautiful buildings) or other curious subject of local interest; and on the other, the person's address. The very implements of manufacture or articles of trade, would themselves, in many cases, form very pretty objects. Medals of this description would be considered as curiosities by many, and would be preserved with care, when a common card is usually destroyed almost immediately. I think in one way they might be made a source of

profit to some trades; such as grocers, bakers, &c. We all know how anxious youth of both sexes are in the pursuit of novelties. I would say, 'If you purchase goods at my shop to the amount of a shilling, or any sum that will be a sufficient remuneration for the token above the usual profit, you shall be entitled to one.' I have no doubt but this would produce plenty of young customers, whose parents would be importuned to allow them to deal with the person where they could procure one of those pretty pieces.

I

These hints suggested themselves to me from looking over a great number of tokens, which I have preserved for years, some of which bear devices which are really very beautiful. am not concerned in trade myself; otherwise I would soon carry my wishes into effect. If you should think the above worth insertion in your valuable publication, you will much oblige your constant reader, and obedient servant, G. A. E. Chichester, Sept. 21, 1829.

REMARKS ON HENRY D'S SELF-
ACTING RAILWAY CARRIAGE.

Sir, It is a fact well known, that no body by descending an inclined plane can gain velocity sufficient to carry it to the summit of another plane of equal altitude. Nor can a pendulum acquire a velocity by falling through one half the arc it de scribes equal to the whole resistance it has to overcome in passing through the other half. These cases are quite analogous; and the same causes that prevent the perpetual oscillation of the pendulum, namely, friction and atmospheric resistance, also obstruct the ascension of a body up a plane by the mere power it has acquired by descending one of equal height. Hence no angle of inclination-no form or disposition of planes can produce the progressive motion contemplated by Henry D; and therefore, until the laws of motion are changed, his self-moving railway carriage will not "assuredly continue to roll along in one undeviating course till time shall be no more."

Of all planes, a level one in a right line will be traversed with the least possible expense of power: for a direct line being the shortest connecting any two points, the least distance is travelled over in communicating from point to point reciprocally, with friction and resistance proportionate; whereas the proposed series of planes would materially add to the distance, and, consequently, to the friction, &c. without giving a carriage or other body one atom more power to continue its motion than it possesses on a level plane.

I have been induced to make these remarks from the analogy you have supposed (at first sight I make no doubt) to exist between Henry D's and Mr. Sylvester's speculations. Mr. Sylvester supposes a plane destitute of friction and other resistance; and that

245

if a carriage or other body be put in motion thereon with any given velocity, it will continue to move uniformly for ever. This is a correct hypothesis, and accordant with the laws of motion; but here is no analogy, for Henry D has both friction and resistance. Next, Mr. S. supposes his plane to have an uniform friction, and that if a carriage or other body be put in motion thereon by some power tra velling with it, sufficient to overcome this friction, the carriage or body will continue to revolve round the earth for ever. This is also correct; but again here is no analogy, for, though Henry D's plan is subject both to friction and resistance, it has no power to overcome it.

I am, Sir,
Your humble servant,

WM. GILMAN, 3

Sir, It really does seem to me quite impossible that Mr. Henry D's "selfimpelling railway carriage" should move of its own accord otherwise than as most other bodies will, down hill; and that being the case, it must, I presume, sooner or later, come to a stand still. It is very well known, the double cone cannot ascend (as it is called) the inclined plane, unless its centre of gravity descends, which may be effected by the two bars or rails composing the inclined plane, making an angle more or less acute; and it can easily be shown, that for any given cone there is a perpendicular height of the plane which cannot be exceeded: the limit of which height is the radius of the greater circle of the cone, as an inspection of the preceding figure will show. Such being the fact, suppose the cone, of which ob is the greater radius, has rolled up the plane or rail, its centre of gravity will have traced out the line oa, which necessarily is a descending line, and it

will also be supported by its smallest diameter, because the rails are then at their greatest width asunder; from a to X they are parallel, and the cone will, of course, roll down them to X, at which point it will be a whole radius below its original level: it will not therefore be in statû quo, and the next pair of rails must commence so much below the horizontal line Xb, which, to my mind, amounts to much the same thing as if it had descended a single inclined plane from b to p. The state of the case will be something better, if the parallel descending rails are done away with, and a second pair, similar to ab, commence at S; even then the cone must necessarily begin something below Xb: the fact is, that the centre of gravity de scends, and there exists the same difficulty in raising it to its former height, as in all other " perpetual motions." I am, Sir, Yours, &c.

London, Nov. 10, 1829.

R. C. JON.

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INSTRUMENTAL ARITHMETIC. DESCRIPTION AND USE OF AN IMPROVED SLIDING-RULE, ADAPTED TO ALL PURPOSES OF COMMON ARITHMETIC. BY MR. THOMAS SADDINGTON. (Concluded from page 155.)

Compound Multiplication.
Examples.

Ex. 1.-Eighteen yards of cloth at 9s. 6d. per yard.

Place 9 for nine shillings, and one and a half division for sixpence, on the duodecimal line on the slide under 1 on the middle of the stroke, which must here be called 10, and count eight of the long strokes towards 2, which will be 18 for the number of yards, under which you will have 8, and nearly three divisions (which are in this case four shillings for each stroke or division), for Sl. lis. the answer required.

Note. Keep in mind the rule laid down for estimating the value of any question or answer, viz. that where I is increased tenfold in the question, and called 10, so must the answer be increased tenfold; otherwise the point under 18 would be only seventeen shillings and one penny on the duodecimal line

Ex. 2.Ninety-seven hundred weight of cheese, at 11. 5s. 3d. per

ewt.

Draw the slide out towards the right hand, and place I for one pound, and two of the long strokes and the next short stroke for five shillings; then esti mate a quarter of the space towards the next division for three-pence under 1 on the middle of the stock, which must be called 10, and count on to 2 for 20, &c., until you come to 9 for ninety, and three and a half divisions for ninetyseven on the stock, under which you will have on the slide 1, and two long strokes and nearly half a subdivision or short stroke. By calling 1 for 100, and each of the long strokes for 10, will give a little more than 122 for 1221. 9s. 3d. as near as can be estimated. Here you observe, that 1 is increased to 10, and also 1 on the slide is increased to 10, which now becomes increased another tenfold, and is called 100, and each of the long strokes will be 10, and the short strokes 5.

Ex. 3.-Forty-three dozen pounds of candles, at 6s. 4d. per dozen.

Place 6 on the line of shillings, and one division on the duodecimal line for

six shillings and four-pence under I on the middle of the stock, and under 43 you will have 13, and rather more than half towards the next division, for 131. 12s. 4d. the answer required.

Ex. 4.-Four hundred and seven yards of cloth, at 3s. 94d. per yard.

Estimate 3s. 94d. on the slide by the rules laid down, and place it under 1, and look for 407, under which you will have a little more than 7, and three and half strokes for 771. 3s. 24d. the an

Ex. 5.-Five thousand seven hundred and four pounds at d.?

Placed. on the line of farthings and pence on the slide under 1 on the stock, and call 5 for 5000, and three and a half divisions for 704, under which you will have 5 on the decimal line for five pounds, and four and a half divisions (each being valued at four shillings), for 57. 18s. 10d.

Note. In this example you ascend in a tenfold degree three times. Suppose the question was 5 at d. you would have the answer on the same line, viz. 1d. but increase the question ten times, and call 5 for 50, then the answer would be on the line of pence, 12 d. Increase it a second time ten times, and call 5 for 500, then the answer would be on the line of shillings, 10s. 5d. Increase it a third time ten times, and call 5 for 5000, then the answer would be 5 on the decimal line, and a little more than one division, for 5l. 4s. 2d. the answer.

Ex. 6.-Five thousand four hundred and seventy yards at ad.?

Placed, under 1, and under 5470 you will have 11 for pounds, and nearly a short division, for 11. 7s. 11d. the amount required.

Ex. 7.-Two thousand seven hundred and ten yards at 24d.?

Place 24d. which is on the line of pence (over one farthing on the line of farthings and pence being a tenfold increase of one farthing) under 1, and under 2710 you have a little more than 28 on the decimal line, for 281. 4s. 7d. the answer.

Ex. 8.-Seven thousand five hundred and twenty-four pounds at 1s. 6d.

Here you must take 18 on the line of pence for 1s. 6d. and place it under 1, and under 7524 you will estimate rather more than 5, and three of the divisions, for 5647. 6s. the answer.

The answer to this question exemplifies the remark made in a former part of this subject, on the impossibility of estimating the exact sum in large amounts, unless the rule were made of a portionate length of radius.

Ex. 9.-Three thousand one hundred and fifty yards at 3s. 4d.?

Place 3s. 4d. under 1, and under 3150 you will find 5, and one division and a quarter, which gives 5257. for the answer required.

Compound Division.

Ex. 10. The clothing of 35 charity-boys comes to 571. 3s. 7d., what is the expense of each?

Estimate 571. 3s. 7d. on the slide, and place it under 35 on the stock, and under 1 on the stock you will have 17. 12s. Sd. on the slide for the answer.

Ex. 11. If 20 cwt. of tobacco comes to 271. 5s., at what rate is it per cwt. ?

Place 271. 5s. on the slide under 20 on the stock, and under 1 on the stock you have 1, and a little more than three and a half strokes, for ll. 78. 3d, for the

As this is a very extensive and useful rule in arithmetic, I shall introduce here a few examples generally classed under other denominations, such as interest, commission, brokerage, profit and loss, &c., which are only modifications of the rule of proportion.

Ex. 13. If I gave 4l. 18s. for 1 cwt. of sugar, at what rate did I buy it at per lb.?

Here you must note that 112lbs. is equal to 1 cwt.

Place 41. 188. under 112, and under 1 you will have 10d. for the answer.

Ex. 14.-If 1 cwt. of cheese cost 1l. 14s. 8d., what must I give for 34lbs.? Place 17. 14s. 8d. under 112, and un

247

der 34 you will have 13d. for 1s. Id. the

answer.

Ex. 15.-What is the interest of 3751. for one year at 5 per cent. ?

Call 1 in the middle for 100, and place 5 on the slide under it for the rate per cent., and under 375 for lbs. you will have 18 for 187. 15s. for the

answer.

Ex. 16.-My correspondent writes me word that he has bought goods to the amount of 754/. 16s on my account, what does his commission come to at 24 per cent.?

Place 27. 10s. for 24, the rate per cent. under 1 for 1007. and under 7547. 16s. you will have nearly 19, viz. for 18. 178. 4 d. for the answer.

Ex. 17. -If I employ a broker to sell goods to the value of 2575l. 17s. 6d., what is the brokerage at 4s. per cent.?

Place 4s. the rate per cent. under 1 for 1007., and under 25751. 17s. 6d. you will have 57. 3s. for the answer.

Ex. 18.—If a broker is employed to buy a quantity of goods to the value of 975l. 6s. 4d., what is the brokerage at 6s. 6d. per cent.?

Place 6s. 6d. the rate per cent., under 100, and under 9751. 6s. 4d. you will have three and nearly two divisions for 31. 3s. 4d. the answer.

Ex. 19.-What is the interest of 2571. 5s. 1d. at 4 per cent. for one year and three quarters?

Note. The solution to this question requires two operations, because the question is for a year and part of a year.

Place 4 for 41. the rate per cent. under 100, and under 2571. 5s. 1d. make a small mark with a soft black lead pencil (which may be easily rubbed out again) on the slide, then remove this mark under one and three quarters, that is to say, at 17, between the figures 1 and 2, under which you will then have 18 for 187. Os. 1d. the answer.

Ex. 20.-What is the interest of 2591. 13s. 5d. for twenty weeks at 5 per cent.?

Place 5, the rate per cent., under 100, and under 2591. 13s. 5d. you have the amount of interest for one year; at that point make a small mark, as before directed, on the slide, and then remove it under 52 for the number of weeks in one year, and under 20 you will have nearly 5, viz. 47. 19s. 10‡d. for the

answer.

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Ex. 21.-If one yard of cloth is bought for 11s. and sold for 12s. 6d., what is the gain per cent.?

Note. This, and all similar questions where shillings and pence or fect and inches are the first term, will illustrate the advantage derived from the upper part of the rule, being divided and subdivided into shillings and pence in the same manner as the slide.

Place 12s. 6d. on the slide under 11s. on the stock and under 1, which is now 100, the rate per cent., you will have 113 on the slide, being 13 above 100, which is the rate per cent. profit, the true answer is 137. 12s. Sd. per cent.

Questions of this nature may be worked another way. The cloth is bought for 11s. per yard, and sold for 12s. 6d., which leaves a profit of 1s. 6d. per yard. Then say, if 11s. gives 1s. 6d., what will 1004. give? Place 18d. for 1s. 6d. on the slide under 11s. on the stock, and under 100 on the stock you will have a little more than 13 on the slide, the rate per cent., for the answer required.

Ex. 22.-If 1lb. of tobacco cost 16d. and is sold for 20d., what is the gain per cent.?

Place 20d. on the slide under 16d. on the stock, and under 100 on the stock you have 125, which is 25 above 100, for the answer.

This question may be worked in the same manner as the last, and say, if 16d. give 4d. profit, what will 100%. give?

Place 4d. on the slide under 16d. on the stock, and under 100 you will have 25 the rate per cent., for the answer.

Ex. 23.-Bought 124 yards of linen for 321., how should the same be retailed per yard to gain 15 per cent. profit?

Place 32 for pounds on the slide under 124 for yards on the stock, and under 1 on the stock you have 5s. 2d. on the slide for the cost price of one yard, and under 115, being 15 above 100, you have 5s. 114d., the answer for the value o fone yard when sold to gain 15 per cent. profit.

This question may be worked another way, and say, if 100 gives 5s. 2d. the cost price of one yard, what will 15, the rate per cent,, give?

Place 58. 2d. under 100, and under 15 you will have 91d. for the profit to be added to the cost price to gain 15 per cent.

Ex. 24. If 1 yard of cloth cost

13s. 4d., what must I sell it for to 'gain 20 per cent.?

Place 13s. 4d. on the slide under 100 on the stock, and under 120 on the stock you have 16s. on the slide for the price of 1 yard when sold to gain 20 per cent. profit.

This question, as the last, may be worked another way, and say, if 100 gives 13s. 4d. the cost price, what will 20, the rate per cent. profit, give?

Place 13s. 4d. under 100, and under 20 you will have 2s. 8d. to be added to the cost price to gain 20 per cent.

Ex. 25.-What is the interest of 240/. for 120 days, at 4 per cent. per annum ?

Place 4 for the rate per cent. under 100, and under 240 on the stock make a small mark on the slide (as before directed in Examples 19 and 20), then remove this point under 365 for the number of days in one year, and under 120, the number of days, you will have 34. 3s. 1d. for the answer required.

Ex. 26.-What is the interest of 3647. 18s. for 154 days, at 5 per cent. per annum?

Place 5, the rate per cent. under 100, and under 3611. 18s. make a small mark on the slide, then place this mark under 365, the number of days in one year (which you will perceive is already done iu this Example as near as can be estimated), and under 154 you will have 74. 14s. for the answer required.

Cross Multiplication.

Having explained the advantages to be derived by the use of the improved sliding-rule in transactions of common arithmetic, I shall now proceed with a few examples in duodecimal arithmetic, or what is generally called cross multiplication and squaring of dimensions by artificers and workmen in superficial and cubical

measure.

In finding the superficial or cubical contents, you must make use of the decimal line for inches; when the terms are between one foot and ten feet, you must work with the decimal line in the same manner as if you were working with integral numbers. Again, if either of the terms are under one foot, you must use the line of pence for the parts of a foot required.

Ex. 27.-Multiply 7 feet 7 inches by 3 feet 6 inches.

Place 7 on the decimal line and three