ON THE USE OF THE SLIDING RULE-BROWN'S GAS ENGINE. We 244364 tons 10., 18..0..204. have made no allowance for friction, which, in this construction, must be very little, and it may be remembered that we applied a power of only 28 lbs. This may, however, be increased by lowering the adjusting screw, and a much greater effect produced. Taking advantage of the suggestions on the envelope of the Supplement to the third volume of the Mechanics' Magazine, I have to state, that the above presses are to be had only at the manufactory, Nopu40, Macclesfield-street, Canalbridge, City-road. Super Royal, plattin 20 by 261, 557. 50 Cash......... 50%. Warranted two years. Other sizes in proportion. .1 RULE. 163 Set, on B to the number on A, then against the number on B is the square on A. EXAMPLE I. What is the square of 76 ?-Set 1 on B to 76 on A, then against 76 on B is 5776 on A, the square required. EXAMPLE II. Required the square of 144?-Set I on B to 144 on A, then against 144 on B is 20736 on A, the square required. PROBLEM IV. To extract or find the square root of any number. RULE. Set 1 or 100 on C to the 10 on D, then against any number on C stands its root on D. NOTE. If we account the numbers on C as tens, those on D will represent units; if on C hundreds, those on D tens, and so on. EXAMPLE I. What is the square root of 144 ?— Set 1 on C to 10 on D, then against 144 on C is 12 on D. EXAMPLE II. What is the square root of 20736? -Set 100 on C to 10 on D, then against 20736 on C is 144 on D. We here notice that we must, may in this case, reckon the 10 on D as 100, and the 20 as 200, &c.; because we have reckoned on C as thousands, as in the First Example we reckoned on Cas hundreds, so on D we reckon as tens, and if we had reckoned the 90 Divide 31360 by 64.-Set 64, the divisor on B, to the dividend 3136 on A, then against 1 on B is 49, the number on C as tens, we must call quotient on A. ( 164 ) PROPOSITION I.-PROBLEM. INVESTIGATION AND SUMMATION OF A NEW SERIES, EXPRESSING THE LENGTH OF A CIRCULAR ARC. (Continued from page 153 of our last Number.) In the mean time we shall show how the aproximate sum may be determined by means of an infinite series, in terms of the tangent of the given arc, in the following manner : when the law of the coefficients, or numerators, of the serieses in the vertical column is very obvious; the first column being the sum of a series of units, the second the sum of the triangular numbers, the third the sum of their squares, the fourth the sum of their cubes, the fifth the sum of their biquadrates, &c. &c. Whence, by taking the successive differences of each series, we find that the 1st, 3rd, 5th, 7th, 9th, &c. orders of differences become respectively equal to nothing, and, consequently, we can assign the sum of n terms of each series by means of the well-known differential series: Let, then, a, b, c, d, &c. denote the sums of the 1st, 2nd, 3rd, 4th, &c. serieses respectively, and we shall have a = -, b = n.n-1.n-2+ A e=+ n.n-1.n-2.n-3.n-4.n-5.n-6+C 7 ni 17 n.n-1.n-2. n−3. n--4. n−5. n-6. n −7. n−8 + D 9 no to &c.; whence the capital letters, A, B, C, D, &c. represent the sum of all the terms, minus the last, which arise from substituting the successive differences in the aforesaid differential series. Now, from the nature of this series, it is evident that the highest power of n in each of the functions, A, B, C, D, &c. will be one dimension less than the highest power of n in the corresponding denominator. It follows, therefore, that when n is indefinitely great, the successive quotients arising from dividing the said functions by the respective denominations being equal to a series of finite, divided by a series of infinite quantities, will be equal to nothing. In this case, also, n = n - 1 = n 2, &c.; because an infinite quantity is not decreased by subtracting a finite one from it. Hence equal to any given multiple arc whose tangent can be found in terms of the radius, the series will become known, which, being repeated as often as BN is contained in the whole circumference, will give the length of the circumference in terms of the diameter. This is the very identical series first investigated, for this purpose, from the fluxional calculus, by Mr. James Gregory, the ingenious inventor of the reflecting telescope, which was sent to Mr. Collins in a letter of February 15th, 1671, and inserted in the Commerc. Epistol. (To be continued.) བའི་ཉལ།།* ON THE LINE OF DRAUGHT IN CARRIAGES. SIR,-The following observations may, perhaps, exhibit nothing whatever but the ignorance of the writer. Dubious as he is, however, of his abilities and knowledge, in respect at least of mechanism, the possibility of his observations furnishing a hint which may be practically developed by others into an experiment, is enough to make him hazard the exposure. Travelling, the other day, on the top of a stage-coach, and observing how the leaders drew, it struck me that the principle on which they drew might be exchanged for a better. Give me leave to state my opinion. Suppose, in the above figure, a k to be the pole, be the swinging-bar, hi the splinter-bar, and ed, fg, the trace-bars of the leaders. Now, be being moveable at the point a, it is evident that when the leader at A pulls, he pulls the end b forward, ergo, the end e backward, and, ergo, the leader at B backward; hence the leader B must exert himself to overcome the drawback of the leader A, and therefore this part of leader B's force is lost to the vehicle at C. In the same way some part of leader A's force is lost to the vehicle. It is not so with the wheel-horses at D and E, for the splinter-bar is fixed (at least nearly so), « and therefore the whole force of each 166 NEW MODES OF SCREW-CUTTING. wheel-horse is devoted to the vehicle. When the swinging-bar is exactly at right-angles to the pole, that is, when the leaders are both up to their traces," then, indeed, be may be considered as fixed, and then, perhaps, there is no force lost; but how seldom is this the case? how often do we see the coachman obliged to whip up one end of the swinging-bar? and until it is up, one leader is, in fact, not so much pulling the vehicle as pulling back the other leader. I submit, Mr. Editor, that if there be any truth in this assertion, it would be well to remedy the imperfection I allude to. It is with the utmost diffidence I propose any innovation, but why may not the leaders draw like the wheelhorses? why may not the traces, fl, gm, &c. be continued on, in the dotted lines, to the splinter-bar? and the superfluous machinery of swinging-bars, &c. be thus done away with? The leaders would then exert their whole 'force on the vehicle; nor does my ignorance allow me to perceive any inconvenience which would result from this reform in stagecoaches, bnt, on the contrary, a considerable benefit, not only to the horses themselves, but to their owners. It may be said that it is the swingingbar which directs the coachman which leader to whip up. I am no Jehu, but I believe it is by the collars that a good coachman judges whether a horse be performing his duty. What directs him with the wheel-horses? will be very uncertain. By the tra versing mandrel, only a certain number of threads can be cut, which depends on the number and size that is on the end of the mandrel. The traversing chuck is a new invention, but is very complicated, and cannot be used in some cases. These objections have induced me to adopt two other contrivances for screw-cutting in the lathe, and they surpass all others for simplicity and perfection. The most inexperienced artist may cut a screw to the greatest nicety, and to any length or size he may think proper, from the 100th part of an inch to an inch and half thread. It has also the advantage of cutting lefthanded screws, and may be performed by any well-made lathe in the following manner A wheel must be made of brass, with any convenient number of teeth, say 36; in the centre of this wheel is a hole for the screw on the chuck of the lathe to pass through into the mandrel, and which may be screwed on with any of the chucks most suitable to hold the work; on the puppet or collar of the lathe is screwed another wheel, of the same size and number of teeth as the former; in the centre of this wheel is fixed a tube, about an inch and a half in length, to receive the head of the screw, which is turned in a globular shape, similar to the pin of a vice; it has a notch sawed rather beyond the centre of the head, and brought up to an angle; a hole must be drilled through the head, crossing the saw-cut a little from the angle, to allow room for a pin to pass through between the angle and the hole you have just drilled; a rivet must be put in the hole, and filed off level to the head of the pin ; the globular head of the pin being fastened in the tube by means of a small pin passing through its diameter, has liberty to act as a universal joint; the screwed part of the pin acting on the screw-tool, or parallel rest, draws it steadily along the work, and produces a regular thread; the two wheels act together, both turning inwards, consequently, if a righthand thread be fixed in the tube, the produce will be left-handed, or vice It is my intention to give, in versa, SIR, Observing in your very useful publication of last Saturday, a communication respecting a chanical mode of obtaining scientific results" in calculating Interest, at five per cent., I will, with your permmission, make a few observations on it, because I conceive it to be any thing but useful, besides being tedious in its application. 41 1 "His rule is," multiply the sum by the days, and divide the product by 365, the quotient will be the answer in shillings. Now, if your Corre 167 terest of which is required for one day, be divided by 12,80 and 17,500, and the quotients added to the sum, the answer will be given in decimals, inserting the decimal point four figures from the right hand.-Required the reason. Example-Suppose the interest of 10,000 for 100 days is required= 1,000,000. 1,000,000 83,333 = 12,500 57 1-12th 109,5890-£109 11 9 I do not give the above as a short, though a correct method, but merely, as it is original, to amuse the curious. It is much easier to reduce the interest from five per cent. than to work as above. The five per cent. method shall be communicated in my next, as it is the shortest and most correct spondent intended to give the world method that can be used. a short and expeditious mode of solving questions in interest, he has erred very widely. I would ask, whether it be easier to divide by 365, to get the answer in shillings, &c. and then divide by 20, to bring that result into pounds, shillings, or pence, or to divide by 365 × 20=7300, at once? Every schoolboy knows as well that 11. is the interest of 73001. for one day, as they know that a shilling is the interest of 3651. for the same pe riod. Your learned mathematician would make us believe that there is something of importance in his communication, when it is told in his pompous and pedantic style. I never knew that it was easier to divide by 365 than by 73. He will tell us next, I expect, to divide by 30,416, and the quotient will be the answer in pence. For the present, allow me to ask the following:-If any sum, the in I am, Sir, June 6th, 1825. G. U. A. BROWN'S GAS ENGINE. SIR, A Correspondent of yours, who signs himself "A Burnt Retort," informed me, in Number 67 of_your Magazine, that Mr. George Frasi was making one of Mr. Brown's Gas Engines on the piston principle, which was to be finished in about fourteen days; but it is now six months since, and I have not heard what progress has been made with it. I should be much obliged if your Correspondent would inform me if it is completed, and working any kind of machinery; and if so, where it might be seen. I remain, Sir, yours, &c. Commercial-road. |