Sir,-The accompanying sketches are perspective representations of two pieces of my astronomical apparatus which I term Geo-chorions, or Season Illustrators, which were both invented by me several years ago. They illustrate, in a very pleasing and novel manner, the rotation of the earth on its axis. The motion of fig. 1 is accomplished through the agency of magnetism; and that of fig. 2, by a circular kind of friction motion, on an inclined surface of glass. The instruments are of very simple construction, and will be readily understood from the following description.

Fig. 1.-G represents a circular base, which supports a mahogany pillar, A, the top of which is bevelled off at such an angle, as to allow the ball of the earth to perform its circuit round the sun in an oblique path of 234 degrees with the ecliptic: a bent wire is made fast in the top of the pillar, on which is placed loosely a large gilded ball, S, representing the sun.

The bent wire just mentioned is at right angles to the oblique plane of the pillar; on this, as a centre, the horizontal bar, B, has its motion. On the extreme end of this bar a perpendicular pillar of brass, C, is made fast, which carries a horizontal slip or stage of brass, L, having a circular perforation large enough to admit a box, M, containing a strong bar magnet, the lower end of which protrudes at K, on which is suspended, by a steel axis, a small terrestrial globe of very light material, such as card, pith, or cork, with a map of the world delineated on it. The globe which I use is made of extremely thin indiarubber; in the centre of it is a circle of wood, to which is made fast the steel axis of the earth. The circle or disc of wood is firmly cemented to the interior equator of the rubber ball. After this is done, the substance is blown into a globular form, and then a map of the countries of the world laid down on it. Such a ball is scarcely of any weight, and consequently well adapted for this apparatus. When the ball E is finished, allow its steel north polar axis to come into contact with the point of the bar magnet inclosed in the box, M, and the magnet will support its weight. Take hold of the south pole at O, and give the ball E a twirl in the proper direction, and it will rotate for some time, and all

the while keep itself suspended to the magnet. Were the ball E spun on its axis with a cord, and then lifted up to the magnet, it would continue to move round on its axis for several minutes, and display a very pleasing effect. Dur ing the rotation of the ball, take hold of the handle, H, and give motion to the horizontal bar, B, and thereby cause the earth to circulate round the sun, S. By the obliquity of the top of the pillar, P, the earth, E, through the medium of the bar, B, will move in a corresponding oblique path round the sun. In the engraving, the vertical solar ray, P, of the sun, S, is directly over the tropic of Capricorn, and consequently our winter: remove the earth 90 degrees from this position, and the vertical solar ray will point over the equator. When at a point diametrically opposite, the solar ray will be over the circle 1, or the tropic of Cancer, &c. In the bar, B, a wire, F, is fixed, and ascends a little way into a perforation in the sun; this causes the ball, S, to follow the motion of the bar, and thereby the solar ray, P, is always kept pointing to the surface of the earth, E, thus showing where the sun is vertical at different positions of the earth in its orbit. The apparatus, therefore, shows the diurnal rotation of the earth on its axis, (as if suspended on nothing); its annual and oblique motion round its orbit; and, consequently, the cause of the different lengths of days and nights throughout the year, and the beautiful vicissitudes of the seasons.

An observation of the following phenomenon first suggested the idea of causing a light ball to rotate on its axis. Provide an ivory, bone, or wood “teetotum;" drill a hole in the top of the handle, to the depth of about a quarter of an inch, into which insert firmly a steel pin, with a tapering point; this done, make the tee-totum to spin on its axis, and, whilst it is in motion, apply the end of a bar magnet to the steel pin you inserted in the drilled hole, and you will find that the magnet, if strong, will lift up the tee-totum, which, notwithstanding, will continue in motion for a considerable time. When the tee-totum is made light, and caused to spin after the manner of French humming-tops, the tee-totum will rotate for a long time, suspended in space by the magnet.

Fig. 2.-As far back as the year 1823, I used to observe the rotation of a watchglass on the surface of a looking-glass, &c. A few years ago, whilst engaged in completing the apparatus just described, I applied a globe of india-rubber to the watch-glass, by which means the ball was charged with motion. A few remarks will make evident the operation of this second apparatus. Let L L represent a looking-glass: describe a kind of an elliptical path on the surface of the glass, as o o, and delineate a sun in the centre, or what is better, (if you wish to have a glass entirely for the purpose,) erect an axle, and place on its point a large ball, say of 4 inches in diameter, to represent the sun. Next procure a common watch-glass, (the more convex its surface the better); put a drop of water or oil on the surface of the lookingglass, and place the convex point of the watch-glass on it; this done, gradually elevate the looking-glass, and the watchglass will, at a certain point of elevation, become endowed with a rapid rotary motion; when the watch-glass is in motion, care must be taken to incline the lookingglass, so as to let the watch-glass rotate on a descending plane, which will easily be accomplished after a few trials. I have a thin india-rubber ball cemented with gum to the watch-glass, and on its surface a map of the world is delineated, as at E, fig. 2. This affords a pleasing and very simple illustration of the motion of the earth on its own axis, and of its circuit round the sun.

I am, Sir, your obedient servant,
E. HENDERSON.

London, December 20, 1842, 10, Powell-st., West. [The astronomical apparatus, the two pieces of

complete:-"Divide as in whole numbers, and make as many decimal places in the quotient as the number of places in the dividend exceeds that in the divisor." Let us apply this to divide 1 by ·001. Divide, as in whole numbers, 1+1=1. Now, there are in the dividend O decimal places, and 3 in the divisor; therefore, by the rule, we are to give 0-3 decimal places to the quotient. Algebraical extension is necessary to carry this rule into practice, while poor arithmetic looks on, and wonders what they are doing with her. The rule can only be arithmetically applied by annexing ciphers to the divisor at pleasure; thus, 100000. Now we have 100000 ÷1=100000, and 5-3, or 2, is the number of decimal places, giving 1000.00 or 1000, the correct answer.

A rule ought to be given in which the calculator can readily tell beforehand where his decimal point will fall, in order that he may prepare accordingly for the requisite number of decimal places. I subjoin two rules which have this advantage; the first is the best, but the second is not without its interest.

Rule 1.-Move the decimal point to the right of the divisor, and carry the decimal point of the dividend as many places to the right as you did that of the divisor. Annex to the dividend as many ciphers as will make its decimal places the number you wish to have in the quotient; divide as in whole numbers, and cut off the number of decimal places prepared for.

Example 1-To divide 179062 by 00012 to 4 places of decimals. Change the divisor into 12, and carry the decimal point 5 places to the right in the dividend,

tion, will be familiar to many of our country readers, as that used by Dr. Henderson to illustrate his popular lectures, at different institutions, on astronomy, horology, and other branches of science. Altogether, it is the most original, ingenious, and complete collection of the sort which has ever come under our notice.-Ed. M. M.]

DIVISION OF DECIMAL FRACTIONS.

Sir,-If the advancement of correct computation be one of your objects, the following remarks may not be without their use. I have always found that, if there be one rule in which more errors are made by beginners than in any other, it is that for the division of one decimal fraction by another.

The rule, as commonly given, is in

mal places is 1790620-0000, which divided by 12 gives 1492183333, whence 149218-3333 is the answer.

Example 2. To divide 174 by 64217.99 to eight places of decimals. Here the dividend, properly prepared, is 17:40000000, and the divisor is 6421799. The quotient is 271, whence the answer is 00000271.

Rule 2-Let like numbers be those which are both greater, or both less, than unity; and let the number of places from the units place, exclusive, to the first significant figure be called the index of a number. ADD or subtract the indexes of the dividend and divisor, according as

they are UNLIKE or like, and make the result the index of the quotient; but remember always to put the decimal point one place more to the left than in the rule, whenever the first significant figures of the divisor (without reference to their local value) are higher than those in the

numerator.

In the first example preceding, the indexes are 1 and 4, and the numbers are unlike; and 1+4=5, the index of the quotient, or there are five places preceding the units place. No rule is necessary to point out whether the index is that of more or less than unity, which is shown immediately by the values of the divisor and dividend. In this case the supplementary process is not wanted.

In the second example, the indexes are 1 and 4, and the numbers unlike; accordingly, 5 as the provisional index, or the first significant figure, would be in the fifth decimal place; but, because the divisor is greater in its commencing significant figures than the dividend, the first figure is in the tenth place.

Example 3.-In 1674 113÷127.29 the indexes are 3 and 2, and the numbers like, 1 is the index of the quotient, the supplementary process is not needed, and 1, the first figure of the quotient, means

But no rule is so good as simple perception, to those who can perceive. All who practise this, or any other rule, should first endeavour to get such an idea of the answer as unassisted common sense will give; and the result will be, that rules will often be superfluous. All who have used computation often must remember instances in which rules, used carelessly or in absence of mind, led them to most absurd results, which were only detected by their own impossibility.

I am, Sir, yours obediently,

A TEACHER OF MATHEMATICS. London, December 19, 1842.

ON MR. C. W. WILLIAMS'S PATENT FOR THE COMBUSTION OF GAS.

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Sir,-As it may be desirable to have the means of distinguishing between my mode of consuming gas in furnaces, and those of others for " consuming smoke," I propose here to describe my own principle and practice, by reference to the only document which can be taken as legal evidence, namely, the enrolled specification. I have already stated, in a former communication, that in my tempt to imitate in the furnace what was done in the Argand gas-burner, I obtained success by merely reversing the process; that is, instead of bringing the furnace gases in jets to the air, I brought the air in jets to the gas." Here not only is the object, sought to be accomplished, but the mode of effecting it clearly indicated; and this it is which has been made the subject of the patent.

The preamble of the enrolled specification states, distinctly, the object of the patent to be" to cause a thorough admixture of the atmospheric air with the gas;" and as it is inflammable gas, and not smoke, that is evolved from the furnace, our attention should be mainly directed to the producing this "thorough admixture," such being the sine qua non in effecting combustion, and preventing all or any of those gases from passing away either wholly unconsumed or partially so in the form of true smoke. It is manifest, indeed, that no atoms of air and gas can be chemically united in combustion, until they have first been mechanically brought into contact, or at least within the range or sphere of their respective attractions. The specification also states that this operation is intended "for the same purpose as air is admitted into the centre of the body of gas issuing from an Argand gas-burner;" and hence a furnace so supplied with air has been called an "Argand furnace.”

The specification goes on to show how this thorough admixture" may be practically effected, namely, "by introducing the air by means of small jets issuing from numerous apertures." The value of this mechanical contrivance is further illustrated by stating that, "where the required quantity of air is introduced in a body, through large pipes or orifices, the gas and the air (requiring time for mixing or effecting the necessary contiguity of their parts) have been found

not to be adequately incorporated until they had passed beyond the igniting temperature of the furnace." And again, "before the mixture had passed into the flues beyond the influence of the high temperature essential to ignition."

With respect to the quantity of air required, the specification furnishes the following particulars:

"The carburetted hydrogen gases require, as the condition of their complete combustion, 1st, that they be intimately blended with an appropriate volume of atmospheric air, which volume varies with the nature of the combustible gas; and 2nd, that after having been so mixed with the air, the mixture be then heated to its temperature of accession, or be brought in contact with flame.

"Again, as one cubic foot of carburetted hydrogen gas requires two cubic feet of oxygen, or about ten cubic feet of atmospheric air, while one cubic foot of bicarburetted hydrogen (or olefiant gas) requires three cubic feet of oxygen or fifteen of air, to effect complete combustion, these proportional volumes of air must be supplied, and so intimately blended, as to bring the combustible gases and atmospheric air within the sphere of their reciprocal chemical attraction."

Here the specification states, not only the quantity of air required, but the object of its introduction in a divided form, in terms which admit of no doubt, namely, the effecting such a degree of juxta-position among the atoms of the gas and air (that is, the being "effectually blended,") that chemical action may become operative, and chemical combination, which is combustion, be effected.

With respect to controlling the quantity of air so admitted, nothing is required practically beyond what the specification describes in the following words:

"The admission of air may be regulated by slides, as no more air should be admitted than will be found sufficient to consume the combustible gaseous and fuliginous matters, and prevent the deposition of the carbon and formation of smoke in the flues."

The specification, in conclusion, sums up in what is called "the claim," and which is generally referred to as the most explicit description of a patentee's rights, by stating that,

"I specially and exclusively claim as my invention, first, the use, construction, and application of the perforated air distributors, by which the atmospheric air is more imme

diately and intimately blended with the combustible gases generated in the furnace."

As to the situation, number, shape, size, or other modification of the apparatus for dividing the body of admitted air, and the use of pipes, plates, tiles, or other materials and contrivances for effecting its required division, these are matter of detail, and almost of indifference. To prevent cavil, however, in such details, the specification contains the usual saving clause, as follows, viz.:

"As I do not confine myself to the particular number, dimensions, or situation of the several parts here described, they may be varied to suit the constructions of furnaces and boilers, and the circumstances under which they may be placed, and which may be effected by any competent persons; -these, my inventions, being applicable and intended to apply to all descriptions of furnaces, stoves, or boilers, where coal is consumed."

In the specification, I have, as is customary, described "one mode of applying my invention to land boilers," and selected one which was best adapted for displaying the peculiar effect produced when air is made to enter in thin films, or in a divided state. In this example the air distributors are placed in a conspicuous situation, and one most favourable for ocular inspection. The effect of dividing the air on its approaching the gas is there so palpable as to carry conviction to all who see it. In fact, without this visible mode of exhibiting the effect, I foresaw I should be discredited, the chemical action denied, and the whole classed among the numerous "smokeburning" expedients, whose patentees allege that "smoke is burned," and other effects produced, but which they can neither prove, visibly or chemically, and where in truth the gas is often not consumed, but merely rendered invisible.

The specification embraces other useful objects connected with furnaces. These, however, need not here be referred to. The main features, then, may thus be recapitulated:-1. The introduction of the necessary quantity of air. 2. The incorporating or mixing this air with the gases before it be too late, that is, before they are cooled down below the required temperature. 3. The effecting this incorporation by any of the ordinary contrivances which will cause the air to enter in a divided form, the object being to