REMARKABLE MATHEMATICAL PROPERTIES OF A CERTAIN PARALLELOGRAM. 397 REMARKABLE MATHEMATICAL PROPERTIES OF A CERTAIN PARALLELOGRAM, WITH A SUGGESTED APPLICATION TO PURPOSES OF GENE RAL UTILITY. Sir,-At one of the meetings of the Royal Society of Edinburgh, in February last, an interesting paper was read by John Scott Russell, M. A., F.R.S.E., containing a notice of the remarkable mathematical properties of a certain parallelogram. The parallelogram which formed the subject of this communication was the rectangle whose sides are to each other in the ratio of the diagonal of a square to its side,-a figure well known to architects, sculptors, and painters, from its beauty, and frequently adopted in the practical arts. The author showed, that if the given rectangle bisected by a line parallel to its shortest side, each segment will be a figure similar in all respects to the original rectangle; and if either of these halves be itself bisected in the same manner, their halves will be rectangles similar to the original rectangle; and so on ad infinitum. The sides of the primary figure and its halves are continual proportionals, represented by the series b. The author endeavours to trace an analogy between the properties of this parallelogram and the logarithmic spiral. A class of figures may be obtained by trisection and by division into four, five, or any number of figures, all of them similar to the primary figure, and capable of division ad infinitum in the same manner.* The following elegant and useful application of the foregoing facts, has been suggested by Sir John Robison, the talented secretary of the Royal Society of Edinburgh, and I think the advantages of the plan proposed, must be evident to all. Many of your readers will be aware, that it has been contemplated to sell stamped envelopes or franks, for the conveyance of letters at a reduced charge. In a letter recently addressed to the Lord Advocate for Scotland, Sir John Robison observes : "If the proposed change in the mode of charging the postage of letters should take adopted for the stamped covers will be a matplace, the forms and sizes which may be ter of some importance, as considerable inconvenience would arise from an injudicious form being selected, or from too great a variety of them being made requisite. The endless variety both of proportion and of dimensions which the makers now give to the writing papers, renders it difficult to determine on a size of envelope which should suit all cases; but if the largest ordinary page of quarto post, be taken as a ground (say 10 by 8 in.) then it may be seen that letters written on such paper, and notes written on the octavo and sexto-decimo forms derived from it, may all be conveniently fitted by two sizes of envelopes. e. g. an envelope 5 by 44 will contain the quarto page folded twice transversely, across the middle; or octavo pages doubled in the same manner; and a smaller one, (4 by 2 will contain octavo notes twice doubled) as above; or sexto-decimo notes one doubled. A full sheet of folio post, if thin, and neatly folded three times instead of twice, may also be contained in the larger sized cover. "A great advantage may, however be gained, by making a small change in the proportions generally given by manufacturers to their papers. The present average proportion of the sides of an expanded folio sheet of writing paper is in the ratio of 20 to 16. If this were to be slightly modified, and if the longer side was to bear to the shorter one the proportion of the diagonal to the side of a square, this convenient consequence would ensue, that, however often such a parallelogram should be doubled by folding across its middle, its subsidiary divisions would always have the same proportions, and be symmetrical; and two sizes of envelopes of symmetrical proportions would serve to contain all sizes of letters." The differences between the present and the altered forms are as follows: Quarto page. Octavo page. 8 by 5 .. 16mo p. * From proceedings of the Royal Society of Edinburgh. NEW AMERICAN ACT FOR THE REGULATION OF STEAM VESSELS. The numerous accidents and the immense loss of life in the United States from the bursting of the boilers of steam vessels, have produced in the last Session of Congress some attempt at a legislative remedy. As all hints of the kind must be valuable, when we ourselves are considering a proper remedy for the same evil, we subjoin the enactment on the subject which was passed by the American Congress on the 7th July last : CIRCULAR. Custom-house, Collectors's-office, New York, The undersigned has received from the Treasury Department of the United States as a certified copy of a law approved the 7th day of July last, entitled "An act to provide for the better security of the lives of passengers on board of vessels propelled in whole or in part by steam," (a copy of which is annexed,) with instructions to take immediate and suitable measures within the limits of this district, to make the owners and masters of vessels propelled as above stated, ac quainted with the nature of the provisions of said law, and of the absolute necessity of their being complied with on or before the first day of October next, and also of the settled determination of the department rigidly to adhere to all of these provisions sofar as depends upon the power conferred by law upon the officers of the Government. In furtherance of the desire of the head of the Treasury Department to carry out the will of the legislature as expressed in said law, the undersigned hereby gives notice of its provisions, and of the instructions he has received in relation thereto, the execution of which, so far as devolves upon him, will be promptly enforced. J. Horт, Collector. An act to provide for the letter Security of the lives of Passengers on board of Vessels propelled in whole or in part by Steam. Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That it shall be the duty of all owners of steam boats or vessels propelled in whole or in part by steam, on or before the first day of October 1838, to make a new enrolment of the same under the existing laws of the United States, and to take out from the collector or surveyor of the port, as the case may be, where such vessel is enrolled, a new license, under such conditions as are now imposed by law, and as shall be imposed by this act. Sec. 2. And be it further enacted, that it shall not be lawful for the owner, master, or captain of any steam-boat or vessel propelled in whole or in part by steam, to transport any goods, wares and merchandise, or passengers, in or upon the bays, lakes, rivers or other navigable waters of the United States, from and after the 1st day of October, 1838, without having first obtained, from the proper officer, a license under the existing laws, and without having complied with the conditions imposed by this act; and for each and every violation of this section, the owner or owners of said vessel shall forfeit and pay to the United States the sum of 500 dollars, one half for the use of the informer; and for which sum or sums the steam-boat or vessel so engaged shall be liable, and may be seized and proceeded against summarily, by way of libel, in any district court of the United States having jurisdiction of the offence. Sec. 3. And be it further enacted, that it shall be the duty of the district judge of the United States within whose district any ports of entry or delivery may be, on the navigable waters, bays, lakes, and rivers of the United States, upon the application of the master or owner of any steam-boat or vessel propelled in whole or in part by steam, to appoint, from time to time, one or more persons skilled and competent to make inspections of such boats and vessels, and of the boilers and machinery employed in the same, who shall not be interested in the manufacture of steam-engines,, steam-boat boilers, or other machinery belonging to steam-vessels, whose duty it shall be to make such inspection when called upon for that purpose, and to give to the owner or master of such boat or vessel duplicate certificates of such inspection; such persons before entering upon the duties enjoined by this act, shall make and subscribe an oath of affirmation before said district judge, or other officer duly authorized to administer oaths, well, faithfully, and impartially to execute and perform the services herein required of them. Sec. 4. And be it further enacted, that the person or persons who shall be called upon to inspect the hull of any steam-boat or .vessel, under the provisions of this act, shall, after a thorough examination of the same, give to the owner or master, as the case may be, a certificate, in which shall be stated the age of the said boat or vessel, when and where originally built, and the length of time she has been running; and he or they shall also state whether, in his or their opinion, the said boat or vessel is sound, and in all respects seaworthy, and fit to be used for the transportation of freight or passengers; for which service so performed upon each and every boat or vessel, the inspectors shall each be paid and allowed by said master or owner applying for such inspection, the sum of five dollars. Sec. 5. And be it further enacted, that the person or persons who shall be called upon to inspect the boilers and machinery of any steam-boat or vessel, under the provisions of this act, shall, after a thorough examination of the same, make a certificate, in which he or they shall state his or their opinion whether said boilers are sound and fit for use, together with the age of the boilers; and duplicates thereof shall be delivered to the owner or master of such vessel, one of which it shall be the duty of the said master and owner to deliver to the collector or surveyor of the port whenever he shall apply for a license, or for a renewal of a license; the other he shall cause to be posted up, and kept in some conspicuous part of the said boat, for the information of the public; and for each and every inspection so made, each of the said inspectors shall be paid by the said master or owner applying, the sum of five dollars. Sec. 6. And be it further enacted, that it shall be the duty of the owners and masters of steam-boats to cause the inspection provided under the 4th section of this act to be made at least once in every 12 months; and the examination required by the 5th section at least once in every six months; and deliver to the collector or surveyor of the port where his boat or vessel has been enrolled or licensed, the certificate of such inspection; and, on a failure thereof, he or they shall forfeit the license granted to such boat or vessel, and be subject to the same penalty as though he had run said boat or vessel without having obtained such license, to be recovered in like manner. And it shall be the duty of the owners and masters of the steam-boats licensed in pursuance of the provisions of this act to employ on board of their respective boats a competent number of experienced and skilful engineers, and in case of neglect to do so, the said owners and masters shall be held responsible for all damages to the property, or any passenger on board of any boat occasioned by an explosion of the boiler, or any derangement of the engine or machinery of any boat. 400 MR. J. J. O. TAYLOR'S APPARATUS FOR PROPELLING STEAM-SHIPS. Sec. 8. And be it further enacted, that it shall be the duty of the owner and master of every steam vessel engaged in the transportation of freight or passengers at sea, or on the lakes of Champlain, Ontario, Erie, Huron, Superior, and Michigan, the tonnage of which vessel shall not exceed 200 tons, to provide and to carry with the said boat or vessel, upon each and every voyage, two long-boats or yawls, each of which shall be competent to carry at least 20 persons; and where the tonnage of said vessel shall exceed 200 tons, it shall be the duty of the owner and master to provide and carry as aforesaid not less than three long-boats or yawls of the same or larger dimensions: and for every failure in these particulars the said master and owner shall forfeit and pay 300 dollars. Sec. 9. And be it further enacted, that it shall be the duty of the master and owner of every steam vessel employed on either of the lakes mentioned in the last section, or on the sea, to provide, as a part of the necessary furniture, a suction hose, and fire engine and hose suitable to be worked on said boat in case of fire, and carry the same upon each and every voyage in good order; and that iron rods or chains shall be employed and used in the navigation of all steam boats, instead of wheel or tiller ropes; and for a failure to do which, they, and each of them, shall forfeit and pay the sum of 300 dollars. Sec. 10. And be it further enacted, that it shall be the duty of the master and owner of every steam boat, running between sunset and sunrise, to carry one or more signal lights, that may be seen by other boats navigating the same waters, under the penalty of 200 dollars. Sec. 11. And be it further enacted, that the penalties imposed by this act may be sued for and recovered in the name of the United States, in the district or circuit court of such district or circuit where the offence shall have been committed, or forfeiture incurred, or in which the owner or master of said vessel may reside, one-half to the use of the informer, and the other to the use of the United States; or the said penality may be prosecuted for by indictment in either of the said courts. Sec. 12. And be it further enacted, that every captain, engineer, pilot, or other person, employed on board of any steam-boat, or vessel, propelled in whole or in part by steam, by whose misconduct, negligence, or inattention to his or their respective duties, the life or lives of any person or persons on board said vessel may be destroyed, shall be deemed guilty of manslaughter, and, upon conviction thereof before any circuit court in the United States, shall be sentenced to confinement at hard labour for a period not more than ten years. Sec. 13. And be it further enacted, that in all suits and actions against proprietors of steam-boats for injuries arising to persons or property from the bursting of the boiler of any steam-boat, or the collapse of a flue, or other injurious escapes of steam, the fact of such bursting, collapse, or injurious escape of steam, shall be taken as full prima facie evidence, sufficient to charge the defendant, or those in his empioyment, with negligence, until he shall show that no negligence has been committed by him or those in his employ. Approved July 7, 1838. A true copy compared with the Roll in this office. A. VAIL, Chief Clerk. Department of State, July 9, 1838. MR. J. J. O. TAYLOR'S APPARATUS FOR Some very interesting models of vessels, propelled by an apparatus which is meant to supersede the use of paddle-wheels, and the inconvenient and unsightly appearance of paddle-boxes in steam-ships, have just been exhibited by the inventor, Mr. J. Jephson O. Taylor, at No. 51, Gracechurch-street, at which place the models are now open to the inspection of captains in the navy, engineers, and all persons connected with the shipping interest, or generally concerned in the improvement of science. The new apparatus, when adapted to actual steam-ships, is to be worked by the power of steam, in the same manner as the paddle-wheels attached to such vessels are now worked; in other words, steam is to be the power employed, but its operations are to be directed to different machinery. The difference is this:-The power of the steam engine will be brought to bear upon a horizontal iron shaft, which will pass from the engine or closely in position with it, beneath the deck of the main cabin, through the stern-post of the vessel; at the extremnity beyond the stern-post two blades, in shape like the blade of an oar, will be fixed, not perpendicularly, but at an angle of 22 degrees to the perpendicular stern-post, and beyond these blades, which occupy but little space, will be affixed a false stern-post, secured to the real stern-post at top and bottom by transverse timbers and iron knees, &c., for the sake of strength, and to prevent accident in case of grounding. The rudder will of course be attached to the false sternpost. The space between the real stern-post and the false one will be very trifling; in the models it is not sufficient to destroy the symmetry of the vessel, though sufficiently large for the purposes required. The iron The shaft being put in motion by the power of the steam-engine, revolves with great rapidity, and at each stroke drives the blades through the water. The vessel is thus propelled forward in precisely the same manner as a wherry is seen to be frequently propelled in the river by a man at the stern using one oar or scull to force it forward. experiments are made on the models in a large tin trough of water, of about 30 feet long. The power used in them is that of the common clock spring wound up, steam being of course out of the question in models of a foot or two feet long. An experiment was first tried on a model with paddle-wheels to prove her speed; by using the same power, she moved from one end of the trough to the other in 115 seconds. An experiment was then tried on her, using the same power applied to the new apparatus of stern blades, and she performed the distance in 18 seconds. The patentee of this invention insists that the following advantages belong to it, and if in a large vessel it should prove as satisfactory on trial as it as on the experiments with the models what he contends for must be admitted. He says there will be a great saving, not only of expense, but in the construction of the machinery, the waste of coals and the employment of engineers, because an engine of 60 horsepower will be able to do as much under the new system as one of 80 under the old. There will be a great deal of room saved by removing the paddles and paddle boxes, and a straight uninterrupted gangway for guns in vessels of war. There will be no swell by the use of the blades, so that accidents to boats or small craft in rivers will be avoided, and the banks of canals uninjured in canal navigation by steamers. There will be space for masts and rigging to carry any quantity of canvass, and as the blades at the stern are beneath the surface of the water, there will be less chance, or rather no danger at all, of their being shot away in action, or in an attack by hostile vessels or batteries. In the same room a naval officer has a model of a brig of war, to which he has attached blades on each quarter; they are made to lie flat to the quarter when not required, and can be carried out in a few minutes by means of a very simple but efficacious apparatus, when required; they are to be worked by means of manual labour at a windlass. Nothing can be more "snug," to use a nautical term, than the whole apparatus; it is cheap in construction, simple in operation, and takes up scarcely any room whatever. Nevertheless, in keeping a vessel from drifting under a battery, or on a rocky shore, getting her head round, and enabling her to manœuvre in a small space, it is of important use.-Times. SAFE-PLUG FOR STEAM-BOILERS. The Society of Arts lately voted a sil er medal to Mr. Isaac Dodds, of Masborough, near Rotheram, for his safe-plug for steamboilers; the following description of which we quote from the last part of their Transactions. In some steam-boilers a hole is made at the bottom, and is afterwards filled up with a plug of lead or fusible metal, in order to serve the purpose of a safe-valve; for, by duly proportioning the ingredients of which the plug is composed, its melting point may be lowered from that of 612° Fah., the temperature at which pure lead becomes fluid, to 212° Fah., the ordinary boiling point of water, which is more than sufficient to melt common fusible metal, composed of eight parts bismuth, five lead, and three tin. Mr. Dodds objects to the usual position of such safe-plug, because, being inserted in the lowest part of the boiler, it will remain covered with water, even when so little is left in the boiler as to allow part of the bottom and sides to be uncovered, and therefore liable to become red hot. Now, since the plug in this position cannot acquire a higher temperature than that of the water by which it is covered, it may continue unmelted while the steam and other gaseous contents of the boiler are accumulating, till an explosion takes place. This hazard, in the opinion of Mr. Dodds, will be prevented by placing over the hole in the bottom of the boiler a cap of iron, of such height that its top shall be left uncovered, while the whole bottom of the boiler, and those other parts that are exposed to the direct action of the fire, shall still be under water. The top of the cap is to be perforated by a hole, which is to be filled up by a plug of fusible metal. When, therefore, from defect in the feeding apparatus, or from neglect on the part of the engine-man to work the supply-pump, the water in the boiler is so far evaporated as to be a little below the top of the cap, the heat, not |