AT ANY PARTICULAR PLACE, ETC. Here, then, the method to be employed, and the reasons for adopting it are distinctly stated; on this point, therefore, there is no room for criticism. But it is not insinuated that the method here employed is superior, or even equal to other methods; on the contrary, it is acknowledged at the conclusion of the paper, page 421, that the same thing can be done more expeditiously in numerous other ways, by the aid of subsidiary tables; but these, and all other methods involving radical expressions, it was the writer's deter 235 mined object to avoid, and to confine himself solely to the method of development, and the formulæ arising from it. After describing the process of development, and showing how to reconstruct the trehedral solid, an operation which, from the nature of the subject, is rather lengthy, but not more so than is necessary under the circumstances, the investigation is proceeded with, and the following equation is ultimately arrived at; viz., cos. B*cos. b cosec. a cosec. ccot. b cot. c. This equation is general for determining one of the angles of an oblique-angled spherical triangle, when the three sides are given; and the corresponding symmetrical = equations for the other angles are easily deduced from it, by simply observing the order of arrangement. They are as below: viz., cos. A cos. a cosec. b cosec. ccot. b cot. c. COS. CC cos. c cosec. a cosec. b cot. a cot, b. Now all this, it is manifest, could easily have been done at the time of investigation; and the development might have been so modified as to represent the three angles expanded on a plane, from which expansion the formulæ exhibited above could readily have been deduced: this, however, was foreign to the writer's purpose, and it would also have been inconsistent with the conditions of the problem, which requires nothing beyond the determination of the horary angle, or the angle at the pole, contained between the meridian of the place, and a declination circle passing through the centre of the sun, or a fixed star. Each of the above equations consists of two terms on the right-hand side, connected by the signs minus and plus, the one term containing three factors, or members, and the other containing two. In this form, the theorems are wholly free from surd expressions; but they are not well adapted for logarithmic computation, since it is necessary to compute the terms separately. They are, however, very convenient for assisting the memory, as there are only the three trigonometrical functions, cosine, cosecant, and cotangent, to be remembered, and nothing can be more elegant and concise than the forms which they assume, admitting of a numerical arrangement, exceedingly neat and easy of performance. It is true there are three different tables required in the process when it is performed as far as it can be by logarithms; but since these tables are generally placed in juxta-position, it is not easy to conceive how any difficulty can arise The capital letter H is employed in the original paper, as being the initial letter of the word hour, or horary; but we here employ the capital letter B to denote the angle opposite to the side b, as being more consistent with the notation conventionally used in the science. from this circumstance, more than by referring to different places of the same table. If, however, we choose to dispense with the use of logarithms, and calculate by the rules of common arithmetic, one table will be found sufficient for the purpose; and it will be shown a little farther on, that the extra labour required in this case is not very considerable. With respect to the signs under which the above equations are represented, no ambiguity can possibly arise in the application of them, especially to those who are acquainted with the elementary principles of the science, and it is not expected that those who are not, will attempt a calculation under any circumstances; it may however be useful to attend to the following observations, in order to know when such and such a sign must be employed. Now, it is manifest, that the first term, which consists of three members, will be positive or negative according as that member in it, which is expressed by a cosine is less or greater than 90 degrees, or a right angle; that is, positive when less than 90 degrees, and negative when greater, the other two members being expressed in terms of the cosecant are always positive, and consequently, have no influence in changing the common sign. Again, in the second term of the equations, or that which consists of two members in terms of the cotangents, the sign will obviously be positive when one of the members is less than 90 degrees and the other greater, and it will be negative when both members are less and both greater than the same standard quantity of 90 degrees. These are the conditions to be considered, when the equations are employed in a general way, to determine an angle from the three given sides; but as this was not the object of the original paper, it would have been foreign to the writer's purpose to have noticed them at the time it was drawn up. The design of the paper was, to give a distinct and independent method of calculating the time at any particular place, by means of data derived from observations of the sun or a fixed star, and this method it was intended should be deduced from the expan sion or development of the trehedral solid, a method of construction which, in consequence of its simplicity, cannot be too familiarly known. The formula arising from this development, when adapted to the purpose for which it was originally intended, is as follows, viz.: cos. hor. ang.=sin. alt. sec. lat. sec, dec.=tan. lat. tan. dec. In this expression the minus or plus sign must be employed, according as the latitude of the place and the sun's declination are of the same or of different names; that is, both north or both south, or the one north and the other south; the formula requires no other circumstance to be considered, and this distinction can occasion no difficulty. At page 135, vol. xliii., your critical correspondent, " KINCLAVEN," has given another formula for finding the time, as being better adapted for logarithmic computation; it is as follows, viz.: cos.hor. ang. = √sin. s sin. (s-a) sec. lat. sec. dec. where the symbol s denotes the half sum of the three sides of the triangle, formed by the complements of the data, and a the complement of the altitude, or the side opposite the angle sought. It is here stated with some show of self-gratulation, that this equation is not only well adapted to logarithmic practice, but that it requires only five steps, and moreover, that it supersedes the distinction of cases incident to that which precedes it. Now, if this be so, in what manner is the a+b+c deter quantity c in the expression 2 mined, when the latitude of the place and equally true, that there can be no ambiguity in the result derived from the first theorem, for since the answer comes out in terms of a cosine, the positive or negative affection is perfectly sufficient for determining to which angle that cosine belongs; at all events, the uncertainty to an unscientific calculator, is not greater in the one case than it is in the other; for if the angle, as calculated by the condemned theorem, may be either 77°3′44′′, or 102° 56' 16; why may not that derived from the improved theorem, be either 51° 28' 8", or 128° 3' 44", seeing that the result of calculation is a cosine by both methods; doubling these quantities, we get 102° 56′ 16′′, or 256° 7′ 28′′; so that it appears, an unscientific calculator is more likely to be led into error by the critic's well-known equation, than by the one which it is intended to displace. It now remains to draw a comparison between the antagonist equations in point of facility of application; and for this purpose, it will be convenient to take the data of the example at page, 135 above referred to, in which no seconds occur, and consequently no subsidiary calculation is required on that account. The data are as follows: Latitude 51° 32′ north; altitude of the sun's centre 10° 30', and the sun's declination 23° 20′ north; and the process by the rule deduced from the development of the solid, is thus performed. log. sin. 9.260633 log. sec. 0.037055 0.31905.. log. required angle. Now, if the natural cosines had been given in the table up to log. tan. 0.099914 log. tan. 9.634838 9.503856 -0.54294 log. 9.734752 0.31905 subtract -0.22389 natural cosine of the 180 degrees, this result would have led us to 102° 56′ 16′′ and not to 77° 3′ 44′′, for the AT ANY PARTICULAR PLACE, ETC. cosines of all angles above 90° would in such a case be marked with the negative sign; it is therefore manifest, that the liability to be led into error arises from the limited nature of the tables, and not from any deficiency in the method by which the result is obtained; there can be no ambiguity in cases where the answer is brought out in terms of a cosine. The tabular angle corresponding to the above cosine considered positively, is 77° 3′ 44′′; but since the cosine is actually negative by the nature of the formula, the angle to which it belongs is greater than a right angle, and equal to the supplement of that which the tables supply; the required angle is therefore 18077° 3′ 44′′-102° 56′ 16′′. 237 numbers; there is also a sixth opening required to find the angle corresponding to the calculated cosine, and this is in the table of natural cosines; in all, six openings of the book. Now, it is certainly of very little consequence, whether these six openings are all made in one or in three different tables, provided they are placed together so as to occasion no difficulty or loss of time in referring to them. But besides these six openings of the book, there are two additions and two subtractions, viz., one subtraction in the natural numbers, and one in the angular magnitudes, making in the whole process, not more than ten distinct steps, one of Here then it is evident, that the above which is not required when the angle is a cute. process down to the determination of the In the foregoing process there is no prenatural cosine of the angle, requires five paratory work required, the calculation proopenings of the book, viz., three in the table ceeding immediately with the given quanof logarithmic sines, tangents and secants, tities; not so, however, with the other and two in the table of the logarithms of equation, the operation by it being as below. 10° 30' =79° 30' 51° 32' =38° 28′ 23° 20' =66° 40' Preparatory -Step 1....... а== =90° Step 4...... Step 5... a+b+c=184° 38' .... log. cosec. 0.206168 Step 7. Step 8. Step 13........ hor. ang.=51° 28' 8" log. cos. 9.794446 Step 12. 2 Step 6....... s-a-12° 49' log. sin. 9.346024 ing only four openings of the book. The process is as exhibited at the close of the article. Step 14............ hor. ang. =102° 56′ 16′′ Now, the above are faithful statements and representations of the facts; from which it appears, that instead of the first equation requiring no less than eight steps, and the second only five, as stated by your correspondent; the first requires only ten steps, and the second no less than fourteen, five of which are openings of the book, being one opening less than is required by the first method, and the number of figures is nearly the same in both cases. It was all very well to keep the necessary preparatory work out of view, in order that the remaining part of it might suit his own purpose, by giving an apparent advantage to the method which he employed; but when the whole work is exhibited by both methods, it will be no difficult matter for your readers to decide on which hand the advantage lies. We have now to show, that by dispensing with the use of logarithms, and calculating by the common rules of arithmetic, the result can be obtained from the first equation, by reference to one table only, and requir Now, there is nothing formidable in the process as thus performed, and the extra labour required is not very considerable; but this method is very seldom if ever resorted to in the present state of the science; indeed, it was for the very purpose of avoiding such operations that logarithms were invented at all; the process has only been performed in this place, pursuant to a previous notice, to show that the solution can be made out by means of the equation that has been so much censured, and by reference to one table only. With regard to the second problem, page 4, vol. xliii., it is only necessary to remark that in this the critic has, as in the first case, entirely mistaken the writer's motive; for he says, "After a very long geometrical investigation, he ultimately, at the final equation, cos bcos B, sin a, sin c + cos a, cos c. Now this takes up nearly three columns; but the proof might be made out in two or three lines from his first equation." Granted. The investigation is long, though not more so than necessary to explain all the circumstances, and the proof might be made out in the manner stated; but had "KINCLAVEN" read the preamble to the problem, he would probably have discovered that it was not the intention of the writer to deduce the rule, or the proof of the rule in this way; for it is expressly stated in a passage of the second paragraph, that he had been "induced to propose it in an insulated form, independent of all other problems of a kindred nature," &c. With regard to the number of cases into which the problem divides itself, they cannot be got over by any means, whatever may be the form of the equation by which the problem is resolved, since they refer to the data of the question under different conditions, and not to the results obtained by this or by that equation. Your correspondent will therefore see that on the whole the objections that he has raised are not so valid as he seems to consider them; and if he had rightly consulted and understood the writer's motives, and the circumstances that prompted his choice in the method of solution, he would probably have been induced to reserve his censures for another purpose. The two equations advanced, pages 135 and 150, vol. xliii., on the supposition that they are new, are not so; the demonstration of the first is self-evident, and that of the second is not difficult; they, or properties similar to them, are as old as the time of Hipparchus, but they probably never were represented in the exact form which they now assume, until Leonard Euler established the algorithm which has raised trigonometry to such a distinguished place among the sciences. I shall now take leave of your correspondent by informing him that the object of the second problem is not the determination of the time, but the distance between two places; see page 151, where it is stated that 100° 46′ 18" is the time as before nearly. The inconsistencies that display themselves in the calculation immediately preceding this announcement must be attributed to the printer. Arithmetical Process. nat. tan. 1.25867 Latitude 51° 32' .... nat. sec. 1.60756 .... A. B. Sir, I have explained the general character of one projection of the line of intersection of a cylinder and a cone, at page 143, vol. xliii. Another projection of the same line of intersection, if made on a plane perpendicular to the axis of the cylinder, will be the arc of a circle, of the radius of the cylinder, and which, in that example, can never exceed a semicircle. A third projection of the same line of intersection, if made upon a plane parallel to the axis of the cylinder, but perpendicular to the plane on which the first projection of the line of intersection was made, will give the projection of both branches on one line. That line, thus projected, will resemble one branch of a cuspidated conchoid. The line formed by the first projection, as has been explained, resembles the reciprocal of the conchoid, with both its branches. But, it may be again observed, the third projection gives only one branch of a line resembling a cuspidated conchoid. It may, however, be imagined that the plane of projection may divide, and thus complete the two branches of the resemblance of the cuspidated conchoid. CAPTAIN ERICSSON'S SCREW PROPELLER. This is as easy to conceive as it is to explain. Now, by the simple motion of the common trammel, the form of every possible elliptic section that can be made by planes cutting a cone at every possible angle in every possible direction, can all be produced at once on one plane. Altering the angle of direction, and altering the distance of the tracer, appear to be equivalent to altering the angle and position of the plane elliptic section. I am, Sir, Your obedient servant, J. JOPLING. 29, Wimpole-street, September 29, 1845. CAPTAIN ERICSSON'S SCREW PROPELLer. Emerson v. Hogg and Delamar. Circuit Court of the United States, 1845. This was an action for an alleged infringement of a patent granted to the plaintiff for a submerged wheel, with spiral paddles, intended to propel vessels. The defendants conduct the Phoenix Foundry, in the city of New York, and had fitted various vessels with "Ericsson's propellers." The principal question in the cause was, whether these propellers, claimed to have been patented by Mr. Ericsson, were substantially the same as the wheel patented by the plaintiff. He The plaintiff, Mr. John B. Emerson, produced and read to the jury a copy of his letters patent, dated 8th March, 1834. proved, by Dr. Jones, that, at the time of filing his specification, he deposited in the Patent-office a drawing of the wheel, and also a model. The original specification, drawing, and model, were destroyed by a fire which consumed the Patent-office in December, 1836. The counsel for the plaintiff then produced and offered to read a certified copy of a drawing made by the plaintiff, and filed in the Patent-office on the 28th February, 1844; this was objected to by the defendants' counsel, on the ground that the specification did not refer to any drawing, and that none had been annexed thereto-this objection was overruled, and the drawing was put in evidence. The deposition of Dr. Jones, of Washington, was then read, by which it appeared that the plaintiff came to that city in March, 1844, and had with him the model of his improved wheel; that Dr. Jones was consulted by him, and then advised him that the drawing filed in February was imperfect, and an inaccurate delineation of the wheel, and that thereupon Dr. Jones prepared a new drawing, with references, which was sworn to by Emerson, and filed on the 27th March, 1844. The counsel for the plaintiff 239 then offered to put this corrected drawing in evidence. The counsel for the defendants objected, upon the ground that the Commissioner of Patents had no right to receive and file more than one drawing, and that by the filing of the drawing made by Emerson in February, the power conferred by the Act of 1837 had been exhausted. Court overruled the objection, and the second drawing was put in evidence. The The counsel for the plaintiff then produced the model of a ship, with the propeller wheel, patented by the plaintiff, and then read the deposition of Charles Robinson, who deposed that he had made the said model in the year 1837, in New Orleans, and that it had been publicly exhibited for a year, in the Merchants Exchange of that city, and from thence was taken to the plaintiff's ship-yard. The plaintiff's counsel then called William Serrell, who testified that he was a civil and mechanical engineer. Being shown models of the plaintiff's wheel, and of Ericsson's propeller, he stated that he had examined them, and had been forced into the conclusion that they were essentially the same. This witness was subjected to a very long and minute cross-examination, which strongly exhibited his accurate and scientific acquaintance with the principles of practical mechanics. He stated, in substance, that the two machines were substantially the same in mechanical construction and action; that he could construct the plaintiff's wheel from his specifications. He went into a detailed explanation of the specification, and said, that, taking it as a whole, he considered it sufficiently disclosed that which the inventor intended to construct. The plaintiff's counsel also called James P. Allaire, who testified that he had been engaged for many year in making steam engines and other machinery; that "Ericsson's propeller" was indentical in mechanical construction and effects with the plaintiff's wheel. He examined the specification, and testified that he could from it construct a wheel similar to the models produced in Court. John C. Kiersted testified that he was a practical mechanic, and that, taking the specification, with either of the drawings filed by Emerson, he could construct a wheel similar to the models. He also proved that the defendants had made and applied "Ericsson's propellers" to a large number of vessels. Stephen E. Glover testified that he was acquainted with "Ericsson's propeller;" that he had been interested in his patent, and that the charge for the use of his propeller was three dollars per ton for large |