12. Our last article regards a repetition of some of our hints: first, as to the importance of a wise plan for the construction of our buildings, so as to prevent the commencement of a conflagration in them, and then the extension of that mischief. In a country where the facility of obtaining lumber is great, we often see brick walls for buildings, the roofs of which, are covered with wood; and various other hazards of the kind are wantonly incurred in the U. States. When we add to this, the neglect of guarding staircases and passages. from fire, in the French manner, so sagaciously pointed out by Dr. Franklin, we must perceive, that we have yet some important lessons to learn as to conflagrations in buildings.-Omitting to speak of frequent fires among the frail buildings of the Asiatics, and of the burning of Rome under Nero, and of Moscow, in our day, where design had the chief share in the catastrophe; we must admit, that the greatest conflagration known in history of a casual description, was that of London, the metropolis of our ancestors, in 1666. The ground floors of the houses then burned, were indeed in many instances covered with rushes ;* but a considerable fire occurred in London in the last century, the London tavern being built on a part of the ruins; and numerous fires still occur in that city, although many useful regulations to prevent it are by law, constantly imposed on builders. But in England, they have not yet applied to use, Dr. Franklin's discovery above mentioned; the principles of which, are perhaps, not universally understood in France itself; and these principles, probably, are as little known in England, as in the U. States. But it is time that they should be known in both countries; and particularly in the U. States, where the increase of population will make the houses in large towns every day more and more to approach each other, so as to favor the spreading of fire in them.† "Erasmus (in * Dr. Jortin in his life of Erasmus, Vol. i, p. 77, has this passage. letter to a friend,) ascribes the plague, from which England was hardly ever free, and the sweating sickness, partly to the incommodious form and bad exposition of the houses, to the filthiness of the streets, and to the sluttishness within doors. The floors, (says he,) are commonly of clay, strewed with rushes, under which lies unmolested, an ancient collection of grease, fragments, bones, &c."-Dr. Jortin in his second vol. (pp. 341, 342,) has given the original letter by Erasmus, which is still more pointed than the above summary.-Neither plague nor sweating sickness, has occurred in London, since 1666. ↑ Even "Rome itself, (according to the proverb,) was not built in a day." Martial, in his time saw great improvements made in it, even as to the streets; for which But I leave the subject of conflagration by land, to say secondly, a parting word as to the dreadful destruction of life and property, consequent upon the use of badly managed Steam boats; in which species of calamity, we seem to surpass all nations in the world. Mr. Webster's motion on this subject, it is hoped, will produce some effect before the present session of Congress shall expire; and that something may be done to save the public from these disasters, but yet not all (I am sure,) that Mr. Webster would wish. As regards myself, I may be allowed here to take my leave of these topics. 1 By way of return, however, to those who have followed the wanderings of the writer of these pages, his piece will conclude with an interesting and authentic anecdote, respecting a conflagration, which was the origin of the present Cathedral of St. Paul's in London; which is certainly the most splendid building, not in a Gothic style, known in the christian world among the seceders from the church of Rome. The anecdote which I have to furnish, is extracted from Mr. Evelyn's dedication to Sir Christopher Wren, of his account of architects and architecture; and is as follows; being prefaced by a well known and not unmerited compliment. "If the whole art of building (in the Greek and Roman style,) were lost, (says Mr. Evelyn,) it might be recovered and found again in St. Paul's, the Historical Pillar,* and other monuments of your happy talent and extraordinary genius. I have named St. Paul's, and truly not without admiration, as oft I recal to mind the sad and deplorable situation it was in, when (after it had been made a stable for horses * * * consult his amusing epigram; (Epig. 60, book 7,) the concluding line of his last couplet being Nunc Roma est: nuper magna taberna fuit. Even Nero, however, is said to have issued proper orders for the construction of the new buildings, expected to appear in the restoration of Rome. Evelyn, in his dedication of his translation of Freart's parallel of ancient and modern architecture, to Sir John Denham, (the poet and a knight of the Bath,) when Denhain was superintendent and surveyor of buildings and works, to Charles II, relates, that the first orders for paving the streets of London, were issued by Sir John Denham, only two years before the great fire of London in 1666. Boston is an American city, in which the most resolute and fundamental changes have lately been made, on principles introduced by its late spirited Mayor, Mr. Quincy, now President of Harvard College; but still the great rule of Dr Franklin, bor. rowed from what the Dr. observed in France, is probably to this moment, unknown even in Boston. * The "Monument," so called, on Fish street Hill, London. and a den for thieves,) you, with other gentlemen and myself, were by the late king Charles, named commissioners to survey the dilapidations; and to make report to his Majesty, in order to a speedy reformation. You will not, I am sure, forget the struggle we had with some, who were for patching it up any how, (so the steeple might stand,) instead of new building which it altogether needed;—when, to put an end to the contest, five days after, that dreadful conflagration happened [namely, the great fire of London,] out of whose ashes this Phoenix is risen. The circumstance is so remarkable, that I could not pass it over without notice." N. B.-On revising the above Supplement, a seeming neglect will be observed. It is said, that steam boats have not been liable to suffer by lightning, which is correct; and is owing to the influence of the edges and points of the iron tubes of steam boats, and to the vapor thrown forth by them, &c. and perhaps to masts. But, when the steam engine is withdrawn, it will be found, that the principal vessel, and the little vessel in the wake of this principal vessel, will then be left unprotected as to lightning. Masts and rigging for these two vessels, were not hinted at, although they might have been; nor will they now be insisted upon; but if provided, it is plain in what manner these vessels may be guarded from lightning. March, 1834. ART. V. On the Parallelogram of Forces; WHATEVER moves a body, or tends to move it, or alters its motion in any manner, is called force. The direction of the force, is that in which it tends to affect the motion of the body. Two forces are equal, when being applied to a material point in opposite directions, they destroy each other's effects. If any number of equal forces, each represented by unity, are applied at once to a material point in the same direction; then if a denotes their number, the point is said to be acted on in that direction by the force xX1; or simply, by the force. When any number of forces acting in any directions, are simultaneously applied to a material point, if they do not balance each other, the point will evidently move, or tend to move, in a certain direction, (by their action,) in the same manner that it would do by the action of some single force in that direction; the single force is called the resultant of the applied forces, and its direction is that of the resultant; also each of the applied forces is said to be a component of the resultant. It is evident that if the resultant is applied to the point in the opposite direction, it will balance the components; and that it will produce the same effect on the point in any direction, as its components; therefore the resultant may be substituted for the components, and reciprocally, the components for the resultant in any calculation. If two forces are applied to the point in the same direction, their resultant evidently equals their sum, but if in opposite directions it equals their difference; if the directions of the forces form an angle, the resultant will manifestly be in the same plane with the components, and its direction will be intermediate between their directions; if the components are equal, the direction of the resultant will obviously bisect the angle formed by their directions. We will now proceed to determine the direction and quantity of the resultant. Suppose then, that two forces x and y, whose directions form a right angle, are at once applied to a material point M ; to determine the direction and quantity of their resultant. Put P-3.14159 etc. the semi-circumference of a circle whose radius=1; let z denote the resultant, the angle which its direc tion makes with that of x, then makes with that of y. P 2 — the angle which its direction If x and y are changed to nx and ny, it is evident z will become nz, and that will not be changed, or if is invariable, will be in variable; but if varies, must vary; reciprocally, if varies, Ꮖ must vary hence the relation between and 6, may be expressed by 2=q(^), (1), also by changing ◊ into P 2=9(2−1), (2); where ç(1) denotes a function of e, whose form θ will manifestly remain the same, however may vary. Again, we may suppose z to be the resultant of two equal forces R and S; R acting in the direction of x, and S in the plane of x and y, its direction being on the same side of x with that of z, and making an angle with the direction of x, which equals 28; then z evidently bisects the angle 24, formed by the directions of R and S, it also equals the sum of their components which act in its direction; hence and by (1), we have ;=¢(8), .:. x=(R+S). (4)2=zp(4)= the resultant re R+S solved in the direction of x; by resolving S in the direction of x, and adding R which acts in that direction, we have R+Sp(28)= the components resolved in the direction of x, which must equal the resultant resolved in the same direction; hence we shall have (R+S).¢(@)2=R+Sp(28), or since RS we shall have 24(4)2 = 1+(28), (3); which must evidently be an identical equation. a It is manifest by (1), that if 8=0, (8) and (28) will each =1; hence supposing (4) to be converted into a series, arranged according to the ascending powers of e, its first term must = 1, and the powers of must be positive, for should any of them be negative, () would be infinite when =0, instead of being =1, as we have proved it must be ; .. (8) must be of the form, (4)=1+A8a+ B9b+Cdc +Dj+, &c. (4), and by changing into 28, we shall have the expression for ¢(24); by substituting the values of ¢(4), ¢(24) in (3), we have 2(1+A4+, &c.)2=2+2a. A8+, &c., or 4A8 +2(A*A* @ + 2B))+4(AB}+b) +Cốc)+2(B*82*+2D® + 2AC(a+c))+, &c.=2a. Aa +2°. B11 +3°. CA© +2d. Dad +, &c. (5). Since (5) is to be identical, (so that may be indeterminate,) it is evident that the coefficients of 8", must be equal; .. 4=2o, which gives a=2, but A remains undetermined; by substituting the value of a, and comparing the next higher powers of 8, we have 2(A381+2B)')=26 B♪", which requires that b=4, .'. A2+2B =8B, or B= ; in the same way we find c=6, d=8, &c. C= a A3 2.3 erally be less than 1, .'. put A= and we have B C= ,י 2 2.3.4 uation is manifest; hence by substituting the values of a, b, &c., A, B, &c., in (4), we have (8)=1– + k282 K484 Кодо +, &c. which is the well known expression for cos. k; hence (4)=cos. kê, which substituted in (1) gives, xz cos. ke. To determine k, let |