INSTRUMENTAL ARITHMETIC. of the divisions on duodecimal line for 9 inches (each division between 7 and 8 being 3 inches) on the slide under 1 on the middle of the stock, and under 3 on the decimal line, and three of the divisions on the duodecimal line on the stock for six inches (each division between 3 and 4 in this Example being for two inches), you will have on the slide a little more than 27 on the decimal line on the slide for 27 feet 1 inches, the answer required. Ex. 28.-Multiply 8 feet 5 inches by 4 feet 7 inches. Place 8 on the decimal line, and nearly two divisions on the duodecimal line, (each division between 7 and 8 being three inches,) on the slide under 1 on the stock, and under 4 on the decimal line, and a little more than three divisions on the duodecimal line on the stock, you will have 384 on the decimal line, for 38 feet 7 inches, the answer. Ex. 29.-Multiply 9 feet 8 inches by 7 feet 6 inches. Place 7 feet 8 inches on the slide under 1 on the stock and under 7 feet 6 inches on the stock you will have 77 on the slide for 77 feet 6 inches, the answer required. Ex. 30.—Multiply 8 feet 1 inch by 3 feet 5 inches. Place 8 feet 1 inch on the slide under 1 on the stock, and under 3 feet 5 inches on the stock you will have rather more than 274 on the slide for 27 feet 7 inches, the answer required. Ex. 31.-Multiply 75 feet 7 inches by 9 feet 8 inches. Place 75 on the decimal line under 1 on the stock, and under 9 on the decimal line, and nearly three divisions on the duodecimal line for 8 inches on the stock, you will have 7 and two and half divisions on the decimal line for 730 feet, the answer. The two following Examples will illustrate the method of proceeding when one of the terms is under 1 foot; in which case you must place the number of inches on the line of pence under 12 on the decimal line on the stock, instead of placing it under 1, or unit, as in the foregoing Examples, because 12 is the number of inches in one foot. Ex. 32.-Multiply 9 inches by 14 inches. Place 9 on the line of pence on the slide under 12 on the decimal line on the stock, and count to 14 on the same line, under which point on the slide you will have 10 inches on the line of pence for the answer required. Ex. 33.-Multiply 6 inches by 24 inches. Supposing a board to be 6 inches wide, and 24 inches long, what is the superficial contents? Place 6 on the line of pence on the slide under 12 on the decimal line on the stock, and under 24 on the same line on the stock you will have 12 on the line of pence on the slide for 12 inches, or 1 foot, for the answer required. If this question had been 6 inches wide and 60 inches long, the answer would have been 2 feet 6 inches. Cubical measure requires two operations because the first and second terms are multiplied into each other, and the product multiplied into the third term. Ex. 34.-What is the cubical or solid contents of a stone or box 3 feet 6 inches wide, 4 feet 8 inches long, and 2 feet 3 inches deep? Place 3 feet 6 inches under 1, and under 4 feet 8 inches you will have 16 feet 4 inches on the slide, at which place make a small mark with pencil, and remove this point under 1, and under 2 feet 3 inches you will have nearly 37 for 36 feet 9 inches, the answer. Ex. 35.-What is the cubical contents and amount of freight of a chest of Irish linen 1 foot 8 inches wide, 4 feet 6 inches long, and 1 foot 6 inches deep, at 21. 15s. per ton, from London to Quebec?-Note. Forty cubical feet is 1 ton for measurement goods. Place 1 foot 8 inches under 1, and under 4 feet 6 inches you will have 7 feet 6 inches, which must be removed under 1, and under 1 foot 4 inches you will have 10 feet for the cubical contents. Then place 21. 15s, the rate of freight under 40, the number of feet in one ton, and under 10 feet, the contents of the chest in measurement, you will have 13s. 9d. for the amount of freight. Ex. 36. What is the cubical contents and amount of freight of a bale of woollen cloth 2 feet 11 inches wide, 5 feet 6 inches long, and 2 feet 8 inches deep, at 41. 10s. per ton? Place 2 on the decimal line and eleven divisions on the duodecimal line on the slide under 1 on the stock, for 2 feet 11 inches, and under 5 on the decimal line and two divisions on the duodecimal line on the stock for 5 feet 6 inches, you will have 16 feet on the 250 ANSWERS TO MATHEMATICAL EXERCISES. slide, which you must remove under 1; and under 2 on the decimal line and eight divisions on the duodecimal line on the stock for 2 feet 8 inches you will have 4, and nearly three divisions on the slide for 42 feet 9 inches for cubical contents, the answer. Then place 41. 10s. the rate of freight, under 40, the number of feet in one ton, and under 42 feet 9 inches, the cubical contents of the bale, you will have 41. 16s. 2d for the amount of freight. In the foregoing Examples I have fully illustrated with peculiar minuteness the principles and advantages of the improved sliding-rule, from which it will be readily acknowledged to be an expeditious computator, and a valuable companion either for the pocket or the desk. I would again remind the reader of what has before been'stated; namely, that in using the sliding-rule it is not intended that it should supersede the use of figures in the common mode of making calculations where accuracy is required, but the utility of the rule will be fully admitted as a corrector in all operations in proportion to the size of the instrument: for by adjusting the slide to the respective questions, the solution is instantly obtained, whereby the calculator will be enabled to judge if his computations are correct; if not, the error, if considerable, will be immediately detected. Where great accuracy is not necessary, and only the proximate value or amount of any question required to be instantly ascertained, or in working of low numbers, much gratification will result from the use of this instrument. Improved sliding-rules accurately laid down, either 6, 12, or 24 inches long, may be had on application to me, No. 8, Upper John-street, Commercial-road, THOMAS SADDINGTON. Let ABCD be the wall; E its centre of gravity; EB produced will evidently pass through D: ABD is therefore a right-angled triangle, of which AB 1, AD=2: therefore, by plain trigonometry, 2 rad. 1 cos. of angle A=30°. Whence angle A comp. of 30=60° angle of inclination when the wall is just supported. [Similar solutions have been received from Mr. Hogan (the proposer of the question), Mr. Hughes, A Reader at Liverpool, Cosine, and Adelphi Kappa's answer is wrong by between two and three degrees.] Question Fourth. "To find the centre of gravity of the three squares described on the three sides of a right-angled triangle." By Mr. Thomas Hughes, MansionHouse School, Hammersmith. IS D Q F MECHANICAL AND MATHEMATICAL EXERCISES. 251 EN: NF:: a2: c2, hence EF: NF:: a2 + c2 ; c2 .' . NF = √2 Let ABC be the right-angled triangle, and L, M, and O the intersections of the diameters of the three squares, described upon the three sides of the triangle ABC: then will L, M, and O be the centre of gravia+c, and EN=√× a2÷a2+c2 ties of the three squares. Draw CN perpendicular to AB, join LC and CM; because LCA and MCB are each half a right angle, and ACB is a right angle.. LCM is a straight line. Divide LM in S. To that MS :SL::AN:NB join SO, which bisect in C, then will C be the centre of gravity of the three squares, AD, BF, and A K. in the AEAG, we know AE, AG, and the EAG; hence we can find EG, in a similar manner, in the AGCF, we know GC, CF, and the GCF, hence we can find FG: again, in the AEGF we know all the sides, hence the angle GFN is known; lastly, in ▲ GFN we know GF, FN, and GFN, hence we can find GN, which bisects in H, and it will be the centre of gravity required. [Solutions have been also received of this question from Mr. Gilbert; Herbert D.; P. T., and H. Paynton. Herbert D. remarks truly, that in the same way "the centre of gravity of force, or any greater number of bodies, may be ascertained. The first and second are to be considered as a single body, placed in their common centre of gravity; and the centre of gravity of this body and a third is to be determined. These, in like manner, being considered as united at their common centre, we may proceed to a fourth, and so on.] Solutions of the algebraic exercise in No. 316 in our next. 2 L Let APC be the given right-angled triangle, right-angled at P; describe the squares AI, CK, AL, of which E, F and G are the centres of gravity, join EF, FG, GE; divide EF in N, so that EN: NF:: square CK: square AI; join GN, which bisect in H (because AL AI+CK, (47. E.)) and H is the centre of gravity required. Calculation: put AP=c, PC=a, AC Given the hypothenuse of a rightangled triangle 1000000,000001 feet, and the base 999999,999999 feet. To find the perpendicular? The question is simple, but will be found in its results extremely curious. 252 3. By Senex. CANALS AND RAILWAYS. The inhabitants of a certain street have agreed to have for their exclusive use a fire-pump; but the son of one of them, who is learning at the London University how to set people by the ears, tells his father that, as the houses on one side of the street are only three stories high, and those on the other five, the former ought not to be charged with the expense of the extra forcing power, and extra length of hose, required for the latter. Will some reader of "the Mechanics' Magazine" be pleased to determine how the fact stands, and to say in what different proportions (if they are different) the inhabitants of the two sides of the street should contribute to the expense? ALGEBRAICAL PROBLEM. Proposed by Mr. J. Gilbert, jun. [Answers to be sent in before the 1st of January, 1830.] Let there be given the sum (S) of any given number of terms in geometrical progression, and the sum (S') of the two extremes; it is required to find the terms of that progression when the multiplier, by which they increase, is 2. Lemma, on which the solution depends. Let y be the second term of any geometrical progression, the multiplier being 2. Let x+y++w+l+r=S and r +r=S'; then we shall have 2y+y+*+ w+1=x+r=S'. DECAY OF STONE CORNICES. Sir, It appears to me that the quick decay of the corner parts of mouldings, cornices, &c. in comparison with the upper parts of the same members, (see Inquiry, No. 319, p. 64,) may be attributed to the rain-water; for if we attentively examine any external moulding or cornice after a shower of rain, we shall find that the lower parts remain wet much longer than the upper ones; the lower parts are thereby exposed to the influence of expansion, (during the frequent and sudden changes, from wet to frosty weather, which our climate is liable to during the winter months,) in a much greater degree than the upper parts. The prejudicial effect of rain-water upon projections of this description," does not arise from expansion alone, which would perhaps only prove injurious during the winter months; for even in the mildest weather, the carbonic acid contained in rain-water acts chemically upon the description of stone generally used for building purposes, and to this action the lower parts of cornices, &c. are of course much more exposed than the upper parts. If the above suppositions are wrong, I hope some of your correspondents will contradict them, and state the true cause, for the information of your readers, including your humble servant. PIT. [T. of East-place, Lambeth, has favoured us with a similar explanation.— ED.] CANALS AND RAILWAYS. Sir,-In one of your interesting communications concerning locomotive engines, I think I read something about filling up canals and laying down railroads upon this new surface. Now, why not simply drain the canal, and by a comparatively small use of the spade form a way for the rails on the bottom of it? The expense could not be excessive; and by this method you would obtain room enough for the tallest of the Liverpool engines to pass under the bridges. A gradually inclined plane might be formed at the places where there are locks, or a stationary engine fixed. How soon by these means might we have a greatly-accelerated transit of merchandise every where, and I believe to the great emolument of canal proprietors. Should you think this common-sense idea worth your editorial notice, we may expect to see more of it in your very useful Magazine. Yours, &c. Chelsea, Nov. 19, 1829. WM. D. R. [It was not "the filling up of canals" which we recommended, but the laying down of railways along each canal; on the side, for example, not occupied as the towing-path, so that the advantages of canal and railway conveyance might be combined. We think. this plan preferable to doing away with a canal altogether; for in all cases where a speed of not more than two miles per hour is wanted, twice the weight can be- INQUIRIES. moved on a canal which can be propelled on a railway by the same power. We shall not be at all surprised, however, should the proprietors of some of those canals on which the traffic in heavy goods is unimportant, find it ere long to their advantage, to do even as our correspondent suggests. One or two we do know of, that could do nothing better.EDIT.] RAILWAYS IN LONDON. Sir,-We talk about railways, their utility, the speed with which our gentlemen of the nineteenth century will travel from London to Edinburgh in a twinkling, &c. &c. All this is very well; but how can we, after having been wafted from place to place on the wings of the wind, endure the shaking of London stones. Now we find it a perfect nuisance; then it will be unendurable. Is it impossible to devise a remedy? Suppose we turn the streets of London into rail-roads. It is no Utopian scheme; the plan is a very simple one. narrow streets two, and in broad streets four rows of curb stones, at proportionate distances: bere we have a rail at once, with scarcely any trouble, and at a small expense. How pleasant would it be to ride through the metropolis were this plan adopted! No jolting, no rumbling of wheels; a hundred inconveniencies avoided. I should like to know what obstacle prevents the adoption of this plan forthwith. I am, yours, &c. Nov. 18, 1829. Enquiries. Sink in NEMO. Eolian Harp.-Sir, Your corres. pondent G. J. G., p. 331 of vol. ix. of the "Mechanics' Magazine," in his excellent and familiar illustration of the Theory of Harmonics, has given a slight description of the Æolian harp. I have for a long time had boards cut out, intending to construct one of those simple instruments; but as it is some years since I saw one, and that a very old one, and thinking the description given by G.J.G. not sufficiently explicit for my purpose, I have been deterred from making the attempt. If G. J. G., or some of your correspondents, could give through your useful publication some further information on the subject, I should take it a favour. I particularly wish to know, 1. The thickness of the end boards; and as the boards which form the top, bottom, and sides are so very thin, whether 253 any lining of slips of deal is required to strengthen the joints? 2. Whether the strings are stretched upon the pins and pegs without any support at the ends for the strings to rest upon? 3. What the distance is from the top to the strings? And, 4. whether the pins and pegs are to be of wood, and of what size? When tuned, your correspondent says, that the outside or largest strings should be tuned to E or F below gamut, and the middle or smaller an octave higher. Now supposing the instrument to have eight strings, I wish to know if the six middle strings should be of one sound; if not, what they ought to be? I am, Sir, |