Introduction.-It will be observed I have used the common method of counting the intervals of the diatonic scale, viz. by using Roman figures; and the chromatic scale is marked with small common figures, counting the interval from the first note named (but not included) to the next note required.

Of Keys.-The rule when not in use, i. e. the slide even with the ends of the stock (see fig. 1,) represents the only natural keys, viz. the upper part of the stock and slide show C not to require any sharps, because I. on the slide is opposite C on the stock; II. is opposite D; III. opposite E; IV. opposite F; V. opposite G; VI. opposite A; VII. opposite B; and VIII. opposite C.

At the same time, I. on the (lower part of the) slide is opposite A on the lower part of the stock; II. opposite B; III. opposite C, and so on to VIII. opposite A; likewise A to be the relative minor key to C major. To find how many sharps or flats are required, and on which notes in any proposed key, you have only to set I. on the slide to the proposed key

note.

See fig. 2, where E major key is proposed, you will find I. on the slide under E; II. under F sharp; III. under G sharp; IV. under A; V. under B; VI. under C sharp; VII. under D sharp; and VIII. under E, the octave. Showing E major key requires four sharps; viz. F sharp, G sharp, C sharp, and D sharp; likewise the chords to the key note are seen at sight; suppose then opposite III. is G sharp, and opposite V. is B ;-if, then opposite IV. is A, and opposite VI. is C sharp.

Without removing the slide, it is seen that D flat is the relative minor, and requires five flats, viz. D flat, E flat, G flat, A flat, and Bflat, when ascending; be. cause, when a minor ascends to the octave, the semi-tone shown between the V. and VI. is removed to between the VII. and VIII; but in descending the semitones fall in their natural places.

Intervals, either diatonic or chromatic, are read off similar to the chords described above. In fig. 1, from C to A is a VI., or nine semi-tones.

Inversion is easily explained, as, for instance, see fig. 1: suppose a flat be found under E in the base-query, the thorough bass and its chords? It will be

seen under III. on the slide is a flat, which being under E on the stock, shows C to be the thorough bass note, E the III., and G. the V., &c. &c.

The chords of the seventh and its inventions are as easily applied.

Thorough bass and fundamental bass are explained by a similar process.

Chords by supposition, as the ninth, eleventh, and thirteenth are as easily managed, since the general use of the rule are to show the chords at sight, by having a figured base, or having a bass and an accompaniment to find the proper figures.

On the stock of the rule under the slide are figures (not shown in the drawings), showing the proper accompaniment to each individual note in the octave, either major or minor, ascending or descending.

An example or two by way of applica tion:

In fig. 1, all this is explained without altering the slide.

The curve lines from I. to III., and III. to V., denote that the chord is from the tonic. On my rule I have three different coloured curves-the first denotes, as in fig. 1, the chord of the tonic; the second (not on the drawing) denotes the chord is from the subdominant; the third, the chord from the dominant.

W. ANDREWS. Near Ivinghoe, Bucks, Oct. 1, 1836.

MR. HOWARD AND MR, SYMINGTON.

Sir, I observe that Mr. Symington has not stated as I requested, "without reserve, where and in what manner , my method of condensation had been in practice forty years. Why has he omitted to answer the latter part of the question? He taxes me with being out of humour, and with using towards him an authoritative tone. Suppose it to be so (which I do not however admit), does he not give me cause for it?

To bring the correspondence in your Journal on this subject nearer to an end, allow me to direct the attention of your readers, and more particularly of Mr. Symington himself, to the following extract from his original communication in your 677th Number, for I have not yet had leisure to inspect his specification :

By cooling down the water in the hot well to the temperature of the external water, by means of a pipe so placed outside the vessel, as to receive the direct action of the water, in order that condensation might be effected by injecting again and again a portion of the same water." Mr. Symington further refers in that communication to the superiority of this plan of condensation by injection, as a new method over that by external cold applied to the steam itself. Now this is precisely, and, I aver, beyond all cavil, the same process of condensation as that previously invented and carried into effect by myself, and of which Mr. Symington ought, at least, not to have been ignorant. There is this difference only in the detail, that he proposes to lead the cooling pipe outside the ship; whereas, I lead it through the cold water within the ship. He may deem this an improvement in the arrangement-I beg to differ with him; but an improvement in the effect, I know by practice it can hardly be, for my plan accomplishes all that can be desired. However, he is welcome to it, apart from the appropriation of my invention, if in itself it be found new or advantageous.

Does not Mr. Symington perceive that his repeated reference to the drawings accompanying our specifications is admitting the weakness of his cause? My drawing is that of an arrangement by which the invention is rendered practically efficient; and the pumps there shown, by which the refrigerating pipe is supplied and the injection effected,

present an advantage over the (I admit rather more simple,) plan of making use of the air-pump and hot cistern, as usual, in so far as the power required to work this air-pump (say 1lb. per square inch on the steam piston,) is nearly all saved. But it happens that the process under this more simple form has been practised, and is at work in the Vesta steam-packet, now running daily to or from London, and Southend, and Sheerness. Perhaps in the next vessel, or may be on land, I may think proper again to alter the detail. But the principle or method remaining the same, and having shown in my specification, and since publicly carried into effect, a perfectly efficient arrangement, which I thought the best at the time of enrolment (and I still think it so on the whole), I appeal to you, Mr. Editor, from your knowledge of patent business, if under such circumstances (putting it merely as a general question,) I am not entitled to vary the details, and, at the same time, to prevent others from appropriating the invention itself by doing so; and further, if this be not the true reading of the patent law, whether, in your opinion, any patent whatever for a new invention can be worth one farthing ?*

I may just add (Mr. Symington having alluded to the subject in his communication before referred to), that by means of a simple contrivance (in the Vesta) I have obviated the liability to accident arising from the condenser or the airpump, in the common mode of withdrawing the injection water by it, becoming choked with water under the unequal motion of the engines in a heavy

sea.

MANUMOTIVE-CARRIAGES.

Sir,-By the extract in your last Number from a Dublin paper, we earn thai a whitesmith of Enniscorthy is employed, as several other persons at this time are, in constructing a manumotive-cariage. In the present instance the vehicle is

To the question as put by Mr. Howard, in the general," we have no hesitation in giving our answer in the affirmative. He gives a very correct view of the law on the subject. How far it deter mines the case in hand we presume, not to say,— ED. M.M.

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propelled by an iron handle, which the guide moves to and fro with the right hand."

One would think it was almost impos. sible to hit upon any scheme for this purpose possessing much novelty, so many and so various are the plans that have been tried. When velocipedes were so much the rage in London a few years since, much ingenuity was exercised to produce manumotive-carriages in which the softer sex might ride; but without success, and since that time this has been the favourite hobby of many individuals. It is unfortunately a fact, that too many persons are apt to imagine, that the success of their machine depends upon the quantity and complexity of the mechanism employed, and this mistake generally proves fatal to their success.

When discussing this matter in a previous Number (635), I stated that the simplest, and therefore of necessity the best way of constructing manumotivecarriages, was to "fix a spur-wheel on the axle of the propelling-wheels, and drive them by a pinion duly proportioned to the inverse quantities of time and power. The man's labour applied to the pinion by means of a winch-handle, would produce all the effect to be derived from such a source, and more than could possibly be obtained by any more complicated train of meclianism."

3

By applying the manual power to two cranks placed on the axle of the pinion, but opposite to each other, working the one with the right hand and the other with the left, the greatest possible effect would be produced. The guiding could easily be managed with the feet. By a machine so formed, favoured with all the advantages of the best workmanship, a person might propel himself at a tolerable good speed; but for any great distance, I apprehend it might be walked over in about the same time with less bodily exertion, and therefore with greater ease.

There are cases, however, where persons have not the free use of their legs, while their arms retain all their wonted vigour; to such parties a machine of this kind would be of infinite service. It is desirable, therefore, to put those persons in the right road who are wishing to construct such a machine.

In the case of Mr. Nicholson, he appears to be adopting the plan, familiar to the inhabitants of this metropolis, from its having been frequently seen in

our streets; a carriage was constructed, and propelled at the rate of five or six miles an hour by six men, who each pulled a lever" to and fro" with a mation very similar to rowing, which acting upon cranks placed on the axle of the driving-wheels, turned them round and thereby urged the carriage forward. The conversion of rectilinear into circular motion, in this case, is attended with a great waste of power, and the plan I have pointed out would in practice be found more convenient as well as much more efficient.

There are cases, as I have already stated, in which manumotive vehicles would prove eminently useful; in general, however, to persons in full possession of all their natural powers, the marrowbone stage will be the best conveyance. Yours respectfully,

London, Oct. 4, 1836.

WM. BADDELEY.

CIRCULATING DECIMALS.

Sir, I observe that the 25th volume of the Mechanics' Magazine is now completed, without your correspondent, "A Country Teacher," having named the "two different vulgar fractions" he had in his mind's eye" at p. 175, the first seven decimal places of which are 4256781. Mr. Peacock, p. 197, gives two fractions, which, he says, "if not identical, very closely resemble those referred to." However, p. 254, “A Country Teacher" denies them to be the two he had fixed upon, though without giving himself any farther trouble about them afterwards. Surely, after Mr. Peacock had answered his challenge, though erroneously, it would only have been courteous to have pointed out where the error lay. I must confess that I have fallen into the same error myself, though perhaps from a very different mode of calculation; the two fractions to which Mr. Peacock alludes being respectively the seventh and ninth of the series of continued fractions produced by the decomposition of the given number. It was my intention to have written a communication to that effect, but Mr. Peacock having forestalled me, rendered it unnecessary.

Though "A Country Teacher" has given us what he terms a hint, p. 254, yet his wording of it necessarily involves a contradiction. He says, p. 175, that the one is a pure, the other a mixed circulate; but, p. 254, he says neither con

Sir, I wish that some of your correspondents would turn their attention to.. wards the invention of a method by which several copies of the same letter could be taken off on letter paper. I had imagined, from seeing so many advertisements of copying-machines, that I should find no difficulty in obtaining something that would answer the purpose I have described; but on searching the London shops I find, to my surprise, that nothing of the sort is known. There are many modes of taking single copies of letters that a writer sends to his correspondent, and also of taking several copies on tissuepaper of the same letter; but not one by which a writer is enabled to take eight or ten copies of the same letter on letterpaper, to be sent to as many correspondents.

My business requires me to send every now and then eight or ten circulars, all precisely the same. It would be a great expense to have types set up for so small a quantity; and, besides, it is frequently requisite that I should observe secrecy; I consequently cannot trust the printer with the letter. It would employ eight or ten clerks to get all these circulars written out in sufficient time-and here, again, the secret must be confided to every one of them. I remember some years ago at Dublin, in a public office, a serious inconvenience arose from the want of some such invention as I have described. Twenty copies of a paper, which it was of the utmost importance to keep secret, were required, when no printer could be found whom it was thought safe to trust. 1 believe the Ministry have a lithographicpress in the Home-office; here, then,

there is obvious risk of Cabinet secrets creeping out through the means employed to obtain the few copies wanted for the use of Ministers, which I believe never exceed twenty. You may remember an instance, some years ago, of an important paper getting to the public, I presume, through this channel.

It seems to me that the fairest prospect of attaining the desired end is presented by the art of lithography. Would it not be possible to invent some small portable machine, by which a letter, after being written in some sort of composition by the hand, might be transferred by the lithographic process to as many sheets of paper as might be required? Perhaps some of your ingenious correspondents will turn their attention to this point; and I will promise them that success will be attended by both fame and fortune. I am, Sir, yours, &c. LETERA.

Brighton, Oct. 10, 1836.

MR. MACKINTOSH'S FOURTH LECTURE ON THE ELECTRICAL THEORY OF THE UNI VERSE.

Mr. Mackintosh remarked, that at first view it appeared parodoxical that two effects so diametrically opposed to each other, as the formation and vapourisation of so hard a body as a diamond, by which he had illustrated the contraction and expansion of matter, should be produced by one and the same agency. But galvanism being a branch of electricity by the help of the electrical machine, the phenomena might be magnified so as to enable us to obtain a more distinct view of the mode of operation. If a body be expanded so as to occupy more space, this expansion must result from repulsion amongst the atoms constituting that body. If a body be contracted, the contraction must result from attraction amongst the atoms. This being admitted, it is evident that the expansion and contraction of matter must be effected by an agency containing within itself the principles of repulsion and attraction, and by no other.

Now let us turn to our magnifying glass, that is, the electrical machine. Here we find that two bodies positively charged, repel each other; that two bodies negatively charged repel each other, and that two bodies, the one charged positively and the other negatively, will attract each other; if instead of two bodies positively or negatively charged, we were to take a million, the same effect would follow; and if we sugpose those bodies a million of atoms, the repulsive force exerted amongst them will furnish us with a