That the telegraphic signals now employed in the navy originated in this way, may be inferred from this circumstance, that Sir Home Popham, to whom the service is directly indebted for them, was a midshipman under Capt. Thompson, when the latter acted as commodore on the coast of Guinea station; as was also the late Captain Eaton, who preserved a copy of the above literal signals until his death. Sir Roger Curtis, who has with much ingenuity contrived a plan of nautical correspondence, similar to that introduced by Sir Home Popham, but who has not been equally successful in its adoption, likewise served under Captain Thompson. Thus did the literal signals, which, among other uses, had the singular application described above, apparently lead to the telegraphic signals, the utility of which is now so generally acknowledged. The latter were, at the glorious battle of Trafalgar, the medium by which the memorable sentence, "England expects every man to do his duty," the conception of the greatest hero our naval annals record, was re-echoed throughout the fleet already prepared "to conquer or to die." S. To the Editor of the Monthly Magazine. SIR, Astenem,ind of desc pubg the Can S the method of describing the Ca Arches and Abutment-piers, may excite some discussion, during the erection of the bridges proposed to be built over the Thames, from the Strand and Vauxhall; and as the properties of this line are admitted, by all writers on arches, to be of the utmost importance in determining the relations of an arch, the following comparison of the line investigated by Dr. David Gregory, in his Paper on the Catenaria, in the Phil. Trans. Aug. 1697, with the line shewn in the publication alluded to, may not be uninteresting to many of the readers of your Magazine. Previous to the erection of Blackfriar's bridge, there arose much controversy, through the periodical publications, relative to the principles of equilibration, and the proper form of the arches of that bridge; and a gentleman then stated, that the catenary was the best form; the absurdity of this position was soon detected, and, by some unfortunate circumstance, the following passage has crept into Dr. Hutton's Mathematical Dictionary, under the article Catenary. "In 1697, Dr. David Gregory published an investigation of the properties before discovered by Bernouilli and Leibnitz, in which he pretends, that an inverted catenary is the best figure for an arch of a bridge."-This mistatement should be corrected. The following is an extract from Dr. Gregory's paper:-" It appears from mechanics, that three powers are in equilibrio, when they have the same ratio as three intersecting right lines, which are parallel to their directions, or which are inclined to them in a given angle, being terminated by their mutual concourse. Therefore, if D d denote the absolute gravity of the particle D d as it must be in a chain of uniform thickness, then'd will represent that part of the gravity which acts perpendicularly on Dd by which it happens (because of the flexibility of a chain moving about d) that d D Jendeavours to reduce itself to a vertical direction; therefore if dd or the fluxion of the ordinate BD, be supposed constant, the action of the gravity exerted perpendicularly on the corresponding parts of the chain d D will also be constant or every where the same.” No one will accuse Dr. Gregory of having pretended, that an inverted catenary is the best figure for an arch of a bridge, who has read merely this quotation from his invaluable paper. It is almost needless to say, that Dr. Gregory has never advanced such a position, nor can any work of his lead to the supposi tion, that he would be so loose in his conclusions as to say, that the catenary, or any other arch, is the best figure for a bridge, knowing, as he must have done, how variable are the forms of the extradosses of bridges. This is not the only aspersion which Dr. Gregory's paper has met with. The author of the Treatise of Arches and Abutment-Piers, in his introductory definitions and remarks, accuses him of having said, that "the invered curve of a catenaria, composed of equal rigid po lished spheres, in a plane perpendicular to the horizon, would keep its figure in the one situation as in the other."-It is true, he says in a note," although Dr. Gregory does not say equal, he evidently means it; and it has been so understood by others, and refers to Dr. Hutton's Recreations in Mathematics,"-If such a statement had been made, it must be a misprint, as Dr. Hutton states, that a catenarian arch may have an horizontal extrados, and be an arch of equilibration, which is irreconcileable with an arch equally Let it be admitted, that the mode, by which the author of a Treatise of Arches and Abutment-Piers describes the catenaria, is correct, and that X A Y is the line so described, ci being the diame ter of the generating circle and e A the constant right line. By Prop. 7. Corol. 3. of Dr. Gregory's paper-If A R be taken equal to the chain A D; and the right line BR be drawn and bisected, and from the Font of bisection a right line at right See page 37, Dr. Hutton's Principles of angles to BR be drawn, intersecting Bridges. BA protracted in C: C will be the centre centre of the conterminate equilateral hyperbola C A its semi-axis, and CR will be equal to CB. By Corol. 4. If the angle BDT be made equal to ACR, the right line DT will touch the catenaria in D. By Prop. 2. Corol. 1. If A H. be the conterminate equilateral hyperbola, and AP a parabola, whose parameter is equal to four times the axis of the hyperbola; B F, the ordinate of the catenaria, will be equal to the parabolic curve A P. less BH, the ordinate of the hyperbola. By Corol. 2. The curve of the cate naria AD is equal to BH, the corresponding ordinate of the conterminate equilateral hyperbola. By Prop. 6. and Corol. 1, 2, 3, 4, 5, and 6. If VA be the evolute, VO will be the osculatory radius; and OZ a tangent to the catenaria at the point O. AC CN NE: OM, and the right line NC will be equal to M V. The evolving right line VA will be a third proportional to the lines AC and CN. The radius K A of a circle equi-curved with the chain, will be equal to the semiaxis AC of the conterminate hyperbola, and the chain AD, and the hyperbola AH will have the same degree of cur. vature at the vertex A. The curve VA less AK will be a third proportional to the right line AC, and the curve AL or the right line NE. The right line KQ will be double A N. catenaria in D or F, and let CR be made equal to ID or W F, that is, to CB. Then will AR be the semi-difference of the lines required IU W G; as ID or CR, is their semi-sum, or CR+AR, and CR—A R, are the members W G or IU. ID the semi-sum of the ordinates IU WG of the logarithmic curve, applied perpendicularly to IW at I gives the ordinate of this catenaria; so the semi-difference AR applied perpendicularly to CA in B is the ordinate of the equilateral hyperbola BH described within the centre C and vertex A, and is equal to the catenaria A D. Now it appears, that this new mode of describing the catenaria does produce the same curve, called the catenaria by Bernouilli, Leibnitz, and Gregory; and that any geometrician, whether acquainted or not with algebra and fluxions, may verify the fact. What important results, in the other branches of mixed mathematics, may be deduced from the simplicity of this mode of construction, a little time may probably shew. I need not make any apology for submitting the following observations on the theory (which Dr. Hutton has called an attempt towards perfection, and which he has acknowledged to be hastily composed, but which one solitary individual has rather inconsiderately called "the true theory,") by the celebrated author of the Prop. 7. Corol. 1, 2, 3. If U AG articles, River, Roof, and Arch, in the be a logarithmic curve, whose subtan- Encyclopædia Britannica." But we gent WS is equal to A C; and if a point beg leave to say, with great deference A be taken, whose distance AC from to the eminent persons who have prosethe assymptote I W be equal to the sub-cuted this theory, that their speculations tangent; and from the points I W any how taken in the assymtote, equally dis#ant from the point C, and if ordinates WG and IU be erected to the logarith mic curve, to half the sum of which ID or W F be made equal; the points D and F will be in the catenaria, corresponding to the right line A C. If A C be unity, whose logarithm is equal to O. To find the logarithm of CA, or of the ratio between CA and CA. To the right lines CA and CA let the third proportional be Ca; and let half the sun of CA and Ca be CB. The ordinate to the catenaria from B (that is BD) is the logarithm required. On the contrary, if from the logarithm given, Clor CW, the correspondent mumher IU or W G be required, or the ratio WG to CA, or IU to CA from W or I, let fail a perpendicular necting the have been of little service, and are little attended to by the practitioner. We venture to allirm, that a very great majority of the facts, which occur in the failure of old arches, are irreconcileable to the theory. The way in which circular arches commonly fail, is by the sinking of the crown, and the rising of the flanks. It will be found, by calculation, that in most cases it ought to have been just the contrary. But the clearest proof is, that arches very rarely fail, where their load differs most from that which this theory allows. We hope to be excused, therefore, by the mathematicians, for doubting the justness of this theory." Of the theory of abutment-piers, perhaps the gentleman, who, intuitively we presume, knows it to be "the true theory," through your Magazine will ex 1809.] On the Pronunciation of the Londoners and Provincialist. 173 plain why Professor Robison has neg lected to notice it? Why Sir Christopher Wren's testimony, respecting the failure of the pillars, and especially the angular pillars, of the crosses in the Gothic cathedrals, and the futility of the immense weight of the towers themselves, as substitutes for abutment, is of so little worth? and why, from high authority, it has been lately objected to, and recommended to mathematicians for reconsideration. Permit me to recommend those who are desirous of obtaining a just knowledge of the principles of equilibration, to refer to the paper of Dr. David Gregory, as a fountain-head, and not to suffer themselves to adopt a theory which depends on what has been called "certain and peculiar modifications," by which it is to be understood, as a land-surveyor would say, "that it is true by coaxing;" and permit me also to call the attention of those, who are desirous of determining a really true theory for the construction of abutment-piers, to refer, particularly, to the 5th Corollary of the 2d Proposition of the same paper. LAPICIDA. To the Editor of the Monthly Magazine. I SIR, SHOULD have no objections to make to your correspondent's animadversions upon the affected pronun ciation of the Londoners, if he did not scem to recommend in the place of it the dialect of the North. "The inhabitants of the more northern counties," he says, pronounce the words abovementioned properly." Some of those words are butter, come, duck; which are pronounced in the North, booter, coome, doock, only giving the oo rather a shorter sound than usual. When I say they are so pronounced, I mean by the generality of people: the lowest vulgar are by me, and I conceive by your corre spondent R. J. excluded from consideration. I am afraid that the observation, that men of a liberal education have no dialect, is not so generally true as might be wished: it can be said of those only, who, before it was too late to direct the organs of enunciation, have taken pains with themselves in this respect, and avoided the disgusting parts of the dialects of the different provinces. In the great schools, this matter is too much neglected for every young man to come out of them with a pure pronunciation: and, in addition to this, the masters themMONTHLY MAG, No. 189, selves are often men whose dialect is "Dum vitant vitia in contraria currunt." There is a strange perverseness in the northern and north-western pronunciation. Though they call pie, poy, mind, moind, &c. they say aysters, or cysters, for oysters. Rejice, in Cheshire, and with a most curious twang, for rejoice. Good, they pronounce gudd, foot, futt; and, on the contrary, but, boot, much, mooch, judge, joodge, there, theere, and the verb tear, teer, &c. ad infinitum. These faults of the pronunciation of both centre of the conterminate equilateral hyperbola CA its semi-axis, and CR will be equal to C B. By Corol. 4. If the angle BDT be made equal to ACR, the right line DT will touch the catenaria in D. By Prop. 2. Corol. 1. If A H. be the conterminate equilateral hyperbola, and AP a parabola, whose parameter is equal to four times the axis of the hyperbola; B F, the ordinate of the catenaria, will be equal to the parabolic curve A P. less B H, the ordinate of the hyperbola. By Corol. 2. The curve of the catenaria AD is equal to BH, the corresponding ordinate of the conterminate equilateral hyperbola. The By Prop. 6. and Corol. 1, 2, 3, 4, 5, and 6. If VA be the evolute, VO will be the osculatory radius; and OZ a tangent to the catenaria at the point O. AC CNNE: OM, and the right line NC will be equal to M V. evolving right line VA will be a third proportional to the lines AC and CN. The radius K A of a circle equi-curved with the chain, will be equal to the semiaxis AC of the conterminate hyperbola, and the chain AD, and the hyperbola AII will have the same degree of cur. vature at the vertex A. The curve VA less AK will be a third proportional to the right line AC, and the curve AL or the right line NE. The right line KQ will be double A N. catenaria in D Now it appears, Prop. 7. Corol. 1, 2, 3. If UAG be a logarithmic curve, whose subtangent WS is equal to AC; and if a point A be taken, whose distance AC from the assymptote I W be equal to the sub-cuted this theory, that their tangent; and from the points IW any have been of little service, how taken in the assymtote, equally dis- attended to by the practit tant from the point C, and if ordinates venture to allirm, that a ver WG and IU be erected to the logarith- jority of the facts, which of mic curve, to half the sum of which ID failure of old arches, are irre or W F be made equal; the points D to the theory. The way in w and F will be in the catenaria, corre- lar arches commonly fail, is by sponding to the right line A C. ing of the crown, and the ri-. flanks. It will be found, by ca that in most cases it ought to just the contrary. But the proof is, that arches very rar where their load differs most fre which this theory allows. We h be excused, therefore, by the mat ticians, for doubting the justness theory." If A C be unity, whose logarithm is equal to O. To find the logarithm of CA, or of the ratio between CA and CA. To the right lines CA and CA let the third proportional be Ca; and let half the sun of CA and C a bc CB. The ordinate to the catenaria from B (that is BD) is the logarithm required. On the contrary, if from the logarithm given, Clor CW, the correspondent munber I U or W G be required, or the ratio WG to CA, or IU to CA from W or I, let fail a perpendicular mecting the Of the theory of abutment-piers, | haps the gentleman, who, intuitively presume, knows it to be "the t theory," through your Magazine will e pla. - ག ་ ༈ |