proper investigation,) but to answer the challenge it contains, and very briefly remark upon a few of the statements. This gentleman has been very fortunate in discovering his verbal mistake of February 7, which occurs in page 53, is positively affirmed in page 54, and of which expression he is reminded in a most pointed manner, page 58, col. 2. It would, however, have been quite as well had Mr. Ibbetson noticed this "verbal mistake" in his subsequent letter, or in any number, before its incorrectness was proved by another person four months after its publication; and, had not this been done, I am really afraid this gentleman would never have corrected this mistake, which is of so much importance.

I am happy to find that Mr. I. allows I have shewn how to produce cycloidal figures on the lathe, and, consequently, that I must at least know something about cyclcids. In one part of this communication it very unfortunately happens, that by an assertion (see page 364, col. 2) Mr. I. (perhaps by mistake in important names) has given me an opportunity of proving that my apparatus is on the same principle as the geometric chuck; and of shewing him once more that there are persons who know something, even more than himself, about mechanical combination. I beg here to remark, that more than 40 years ago I turned the cardioid upon the very chuck which I have before stated to have been taken to America. Mr. I. says, No. 291, page 54, "Describe an ellipsis, in the usual way, on any surface of wood affixed to the chuck; then make a small hole any where on the line of the ellipsis, and fix a bit of slate pencil in it; hold a slate steadily in front of the chuck, so as just to touch the point of the pencil; then move the lathe round, and the point will of course draw a line on the slate." And in No. 310, page 364, "And this is the cardioidal curve which invariably accompanies elliptical motion. I cannot obtain this curve through the medium of the geometric chuck, nor by means of any other chuck working in the lathe." Accepting this declaration as a challenge, I will attempt to shew plainly

how this may be done by an apparatus applied to the lathe; in speaking of which, Mr. I. says, "Mr. Child knows nothing in the world about it." Let the wheel that is fitted on to the back of the chuck, and is concentric with the mandril, (and in which also the back part of the chuck itself turns round,) drive a second wheel, equal to itself both in size and number of teeth; the axis of this second wheel comes through the plate of the chuck, &c., and on its projecting end there must be fixed a third wheel, which drives the fourth; this fourth wheel has only half the number of teeth that the third has, and may be set either eccentric to, or concentric with the mandril; and must, for the present purpose, carry the work on its axis properly affixed to it: face the work in the common way and set it to what eccentricity you please; push in the slide, which is on the face of the fore puppet, to stop the first wheel, by means of the pin or pins which are in it; then put the lathe in motion, and, by holding a point against the face of the work, the geometric oval will be produced; and, if the point be fixed against the proper part of the work, the figure will assume the cardioidal form. This figure may also be produced by a different arrangement of the same apparatus. In Mr. I's list of impossibilities the spiral is mentioned; this, for the present, I leave, that he may exercise his ingenuity and skill in mechanism in attempts to produce it by means of a chuck working in the lathe. I shall merely observe, that in a late Number of the "Mechanics' Magazine" I gave a method of turning both the cardioid and spiral, though not in terms quite so plain as the above; yet in such as any clever mechanic might understand. In the beginning of the year 1825, and in December 1826, I transmitted to the Editor of the "Me

* The nature and properties of these curves have lately been partially investigated by the scientific pen of Mr. Jopling and others, in the pages of the Magazine. I beg to thank these gentlemen for their labours, and to assure them that their investigations are interesting to myself as well as to others, and advantageous to science.

chanics' Magazine" a great number of specimens drawn by the apparatus, and accompanied them with a very brief account of the method in which they were drawn. About two years afterwards the figures turned by Mr. I. were given to the public: and after this, I applied to the editor as a private gentleman to return the several packets I had sent: the reply to this application may be seen in the Magazine (vol. viii. p. 344). In the account accompanying the specimens, I stated that my first attempts to describe curves were made by a combination of pullies and strings. This first instrument is the very model which Mr. I. endeavours to make his friend say was a chuck intended for a lathe. Before the publication of his specimens I had never heard of Mr. Ibbetson, and had stated that I was self-taught; that the workmanship of my models was rude, as might be expected, and consequently the execution of the figures drawn by them imperfect. I had more pleasure in studying the principles than in improving the execution of my work. In conclusion, Sir, I beg to say, that I have not submitted to you any statement, in reference to these things, which, when I think it necessary, I shall not be able to prove; and am too indifferent about them, as my communications have been made only for the amusement or improvement of others, to condescend so far as to notice personal abuse.

I am, Sir,

Your most obedient servant,
K. CHILD.

Shaw-Lane, July 25, 1829.

I beg to recommend to the notice of your scientific readers the last two paragraphs of Mr. Ibbetson's letter, that they may observe the singular results of this “sifting to the very bran.”

BRITISH ALMANAC.

Sir, The excellence of the British Almanac is deservedly allowed; but it is not without its faults, and has consequently been severely criticised. One error, however, appears to have escaped notice. The total amount of evaporation for the year (having added the proportions for each month) appears to be 23'871 inches, whilst the

409

rain supplied is only 22∙199 inches: the consequence of this would be, that the surface of the earth is continually becoming drier.

Trusting the above will lead to a
correction of the error in the next
Almanac of the Society,
I am, Sir,

Your obedient servant,
GEO. MANNING.

Olney, June 30, 1829.

SUBSTITUTE FOR OIL FOR LUBRICATING THE MACHINERY OF CLOCKS AND WATCHES.

It is well known that the amount of friction, even in the best clocks and watches, depends in a great measure on the purity of the oil with which the different parts are lubricated; and that it has been hitherto a matter of great difficulty to obtain oil sufficiently pure for the purpose. A very celebrated

maker informed Mr. Reid, of Edinburgh, that he had a regulator" which had been found to go to a greater degree of accuracy (though not to a second in two months, as has been said of others,) than even that of Verona, as observed by the Astronomer Cagnoli, or that at Manheim, as observed by Major, but which was found to perform very indifferently after being cleaned, and at the end of three or four months stopped altogether, owing entirely to the application of bad oil." Several years ago, it was suggested that plumbago, being of an unctuous nature, might, in all probability, be advantageously substituted for oil; and a trial of it having been made with an astronomical clock made by Herbert, that clock was lately taken to pieces and examined by a committee of the Society of Arts, whose report on the subject is so satisfactory as to leave little doubt that plumbago will now be the material universally used. All the rubbing surfaces which had been covered over with the plumbago were found to have acquired an extraordinarily high polish; no part of them was in the least clogged; and they did not appear, on examination by magnifiers of very high power, to have undergone the slightest wear. The plumbago is recommended to be prepared by repeatedly grinding and washing it, till all the gritty particles, which occur even in the best black lead, are removed. It is then to be applied with a camel-hair pencil, either in the state of powder or mixed up with a drop, or two of pure spirit of wine. M. S. A.

side of the lower one; F the steampipe from the boiler; G the eduction pipe; I the rod which moves the slide; K the spanner by which the rod I is attached to the upper piston. In the drawing the piston is represented at the bottom of the cylinder, at the bottom of which the steam is entering from the boiler by the pipe M, to drive the piston to the top of the cylinder, while the steam which remained at the top from the last stroke is escaping down the pipe EE to the condenser, by the eduction pipe G. Now, suppose the piston arrived at the top of the cylinder, and the position of the slide changed (by the usual means of eccentric motion, &c.), so that the under surfaces of the pistons are above the apertures to the pipes LM, the steam would then enter the top of the working cylinder by the pipe L and drive the piston downwards, while the steam remaining at the bottom would escape by the pipe M and the eduction pipe G to the condenser (which is not represented in the drawing, to prevent occupying too much space).

NEW RAILWAY CARRIAGE.

A new locomotive carriage, invented by Mr. Stephenson, has recently made its appearance on the Liverpool and Manchester railway, which is said not only to burn its own smoke completely, but to be constructed on such a selfregulating safety principle that explosion is almost impossible.

DEVELOPMENT OF CERTAIN NEW PROPERTIES OF NUMBERS.

BY THOMAS TAYLOR, ESQ. Sir,-You will oblige me by inserting in your valuable Magazine the following properties of certain numbers, which Ï believe are no less novel than extraordinary.

I. It appears to be universally true, that, in every circulating decimal, the sum of the digits of the circulator will always be equal to 9. Thus, resolved into a decimal will be 0142857, 0142, &c. And the sum of the digits of •142857 is 27, and 2 + 7 = 9. If 9, also, is multiplied by 7, the denominator of the fraction, the product will be 63, and 6+3=9. And an infinite series of 0999, &c. may be considered as equal to 1. Thus, too, is resolved into a decimal will be

411

In like manner, resolved into a decimal will be '0588235294117647, 058823, &c. And the sum of the digits of the circulator is 72, and 7 + 2 = 9. Also, the sum of the digits of 17, the denomi nator of the fraction, is 8, and 9 × 8=72, and 7+29.

II. If 9, or any multiple of 9, divides any number which, after the division, leaves a remainder, the sum of the digits of the remainder will be equal to the sum of the digits of the divided number; as is evident from the following examples. 9)1568978

174330, and the remainder is 8. And the sum of the digits of the dividend is 44, and 4 + 4 = 8. 18)432789537(24043863

36

72

72

78 72

69 54

155 144

113 108

57

54

၂ က

143427 139968

34599 17496

171035 157464

135718

122472

132466

122472

99946 87480

124662 122472

21901 17496

4405=13=1+3=4

And the sum of the digits in the dividend is 67. But 6+7=13, and 1+3=4.

III. If one number be subtracted from another, and the number subtracted consists of the same terms as those of

(1)

6967214

1246976

5720238

Here the sum of the digits of the remainder is 27, and 2+7=9.

(2)

6271496 4127696

2143800

The sum of the digits in this remainder is 36, and 3+6 = 9.

Thus, also, if from 43521, 12534 be subtracted, the remainder will be 30987, the sum of the digits of which is 27, and 2+7=9. And the same thing will be found to take place when the order of the terms is irregularly changed in both the subtrahend and the number subtracted.

Thus, likewise, if any number be multiplied into itself, and the terms of this number in an inverted order be also multiplied into themselves, and the less of the two products be subtracted from the greater, the sum of the digits of the remainder will always be 9. Thus, 61 × 61 = 3721, and 16 × 16 = 256; but 3721 256 3465, the sum of the digits of which is 18, and 1 + 8 9. Thus, too, 56 × 56 = 3136, and 65 × 65 4225; but 4225-3136 = 1089, and the sum of the digits of this remainder is 18, and 1-8-9. Again, 241 × 241 58081, and 142 × 142⇒ 20164; but 58081 2016437917, and the sum of the digits in 37917 is 27 = 2+7 = 9.